To the theory of semi-linear equations in the plane

In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z),whereas its reaction term f(u) is a continuous non-linear function. Assu...

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Datum:	2019
Hauptverfasser:	Gutlyanskii, V.Ya., Nesmelova, O.V., Ryazanov, V.I.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2019
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spelling	irk-123456789-1694342020-06-15T01:26:40Z To the theory of semi-linear equations in the plane Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z),whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak C(Ď )∩ W¹,² loc (D) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semi-linear equations of mathematical physics, arising from modelling processes in anisotropic and inhomogeneous media. With a view to further development of the theory of boundary value problems for the semi-linear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. 2019 Article To the theory of semi-linear equations in the plane / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 105-140. — Бібліогр.: 74 назв. — англ. 1810-3200 2010 MSC. Primary 30C62, 31A05, 31A20, 31A25, 31B25, 35J61 Secondary 30E25, 31C05, 34M50, 35Q15 http://dspace.nbuv.gov.ua/handle/123456789/169434 en Український математичний вісник Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z),whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak C(Ď )∩ W¹,² loc (D) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semi-linear equations of mathematical physics, arising from modelling processes in anisotropic and inhomogeneous media. With a view to further development of the theory of boundary value problems for the semi-linear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity.
format	Article
author	Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I.
spellingShingle	Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. To the theory of semi-linear equations in the plane Український математичний вісник
author_facet	Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I.
author_sort	Gutlyanskii, V.Ya.
title	To the theory of semi-linear equations in the plane
title_short	To the theory of semi-linear equations in the plane
title_full	To the theory of semi-linear equations in the plane
title_fullStr	To the theory of semi-linear equations in the plane
title_full_unstemmed	To the theory of semi-linear equations in the plane
title_sort	to the theory of semi-linear equations in the plane
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2019
url	http://dspace.nbuv.gov.ua/handle/123456789/169434
citation_txt	To the theory of semi-linear equations in the plane / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 105-140. — Бібліогр.: 74 назв. — англ.
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fulltext	Український математичний вiсник Том 16 (2019), № 1, 105 – 140 To the theory of semi-linear equations in the plane Vladimir Gutlyanskĭı, Olga Nesmelova, Vladimir Ryazanov Dedicated to the memory of Professor Bogdan Bojarski Abstract. In two dimensions, we present a new approach to the study of the semi-linear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z), whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak C(D) ∩W 1,2 loc (D) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semi-linear equations of mathematical physics, arising from modelling processes in anisotropic and inhomogeneous media. With a view to further deve- lopment of the theory of boundary value problems for the semi-linear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. 2010 MSC. Primary 30C62, 31A05, 31A20, 31A25, 31B25, 35J61 Sec- ondary 30E25, 31C05, 34M50, 35Q15. Key words and phrases. Semi-linear elliptic equations, quasilinear Poisson equations, anisotropic and inhomogeneous media, conformal and quasiconformal mappings. 1. Introduction The study of linear and non-linear elliptic partial differential equations in two dimensions by the methods of complex analysis and quasiconfor- mal mappings with concrete applications in nonlinear elasticity, gas flow, hydrodynamics and other sections of natural science has been initiated by M. A. Lavrentiev [54], C. B. Morrey [60], L. Bers [8], L. Bers and L. Nirenberg [9], I. N. Vekua [72], B. Bojarski [11], J. Serrin [68] and others, see the references therein. The history of such equations actually Received 29.03.2019 ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України 106 To the theory of semi-linear equations in the plane goes back as far as the celebrated works by d’Alembert on the Cauchy- Riemann systems in hydrodynamics, by Gauss on geometry of surfaces, by Lobaczewski on non-Euclidean geometry, and by the pioneer paper of E. Beltrami [7]. A rather comprehensive treatment of the present state of the theory is given in the excellent book of Astala, Iwaniec and Martin [4]. This book relates the most modern aspects and most recent developments in the theory of planar quasiconformal mappings and their application in conformal geometry, partial differential equations and nonlinear analysis. The book contains also an exhaustive bibliography on the topic. Among the variety of deep results on this topic we single out the fundamental Harmonic factorization theorem, see [4], Theorem 16.2.1, for the solutions to the non-linear uniformly elliptic divergence equations divA(z,∇u) = 0, z ∈ D ⊂ C, (1.1) and the corresponding regularity results. In particular, the factorization theorem claims that every solution u ∈W 1,2 loc (D) of the equation (1.1) can be expressed as u(z) = h(f(z)), where f : D → G is K-quasiconformal and h is harmonic on G. In series of our recent papers [33–40], we have proposed another ap- plication of the quasiconformal mappings theory to the the study of semi- linear partial differential equations of the form div [A(z)∇u(z)] = f(u), z ∈ D, D ⊆ C, (1.2) the diffusion term of which is the divergence form elliptic operator L(u), whereas its reaction term f(u) is a non-linear function. Here the sym- metric matrix function A(z) = {aij(z)}, detA(z) = 1, with measurable entries satisfies the uniform ellipticity condition 1 K \|ξ\|2 ≤ 〈A(z)ξ, ξ〉 ≤ K\|ξ\|2 a.e. in D, 1 ≤ K <∞, (1.3) for every ξ ∈ R2. The set of all such matrix functions we denoted by M2×2 K (D). In the cited papers we have studied the composition properties of L(u) first for sufficiently smooth functions and then in the Sobolev spaces making use of the fundamental compositional theorems established in [28,70]. It was shown that by the chain rule for the function u = U ◦ω the following basic formula holds div [A(z)∇(U(ω(z)))] = Jω(z)△U(ω(z)), (1.4) where Jω(z) stands for the Jacobian of the quasiconformal mapping ω, agreed with the matrix function A(z). This formula, which is understood V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 107 in the sense of distributions, takes place, in particular, if U ∈ W 1,2 loc (G), A ∈ M2×2 K (D) and ω : D → G is a quasiconformal homeomorphism satisfying the Beltrami equation ωz̄(z) = µ(z)ωz(z) a.e. in D. (1.5) Here the complex dilatation µ(z) = 1 det (I +A) (a22 − a11 − 2ia12), (1.6) satisfies the uniform ellipticity condition \|µ(z)\| ≤ 1 +K 1−K . (1.7) Vice versa, given measurable complex valued function µ, satisfying (1.7), one can invert the algebraic system (1.6) to obtain A(z) = ( \|1−µ\|2 1−\|µ\|2 −2Im µ 1−\|µ\|2 −2Imµ 1−\|µ\|2 \|1+µ\|2 1−\|µ\|2 ) . (1.8) The compositional property (1.4) for the operator L(u) can be applied to the study of a wide range of problems arising in the contemporary analysis in the plane. For instance, (1.4) is useful for the study of such semi-linear partial differential equations in anisotropic case as the heat equation ut − div [A(z)∇u(z)] = f(z, u), (1.9) (the same equation describes the brownian motion, diffusion models of population dynamics, and many other phenomena), the wave equation utt − div [A(z)∇u(z)] = f(z, u), (1.10) and the Schrödinger equation iut + div [A(z)∇u(z)] = k\|u\|pu, (1.11) as well as their stationary counterparts. Note also a very interesting recent preprint [27], where the authors have developed a method for the study of spectral properties of the operators L(u) with the Neu- mann boundary condition in (non)convex domains in the complex plane. The suggested method is based on the composition operators on Sobolev spaces with applications to the Poincare inequalities. 108 To the theory of semi-linear equations in the plane The composition property (1.4) for the operator L(u) implies that the study of the semi-linear equations of the form (1.2) is decomposed into the research of a proper quasi-linear Poisson’s equation △U(w) = J(w)f(U), w ∈ G = ω(D), (1.12) where the weight J(w) stands for the Jacobian of the inverse quasicon- formal mapping ω−1 : G → D, and the study of the mapping ω(z) agreed with the matrix function A(z). In other words, every weak solu- tion u(z) to the equation (1.2) in a domain D is represented in the form u(z) = U(ω(z)) where ω : D → G stands for a quasiconformal mapping generated by the matrix function A(z) and U(w) is a weak solution to the quasilinear Poisson equation (1.12) in the domain G = ω(D), see [36], Theorem 4.1. On the one hand, this opens up new possibilities for the study of (1.12), because we can apply a wide range of effective methods both of the potential theory and genuinely nonlinear methods which did not belong to the world of classical harmonic analysis, see e.g. [16, 26, 30, 51, 52, 64, 68], and the exhaustive bibliography therein. On the other hand, a comprehensively developed theory of quasiconformal mappings in the plane, see e.g. [1, 4, 12, 13, 42, 55], and also [41, 43], allows us to study in detail both the regularity properties for solutions to the equations (1.2), (1.12) and the proper representation of such solutions. A rather comprehensive treatment of the present state of the theory concerning the semi-linear equations of the form (1.12) is given in the excellent books of M. Marcus and L. Véron [58] and L. Véron [73]. Note an important family of quasi-linear Poisson’s equations that in- volves an absorption term such that uf(u) ≥ 0. Such equations are of particular interest because, in particular in them we have two compet- ing effects, observed in a number of applications: the diffusion expressed by the linear differential part and the absorption produced by the non- linearity of the right hand side. Among the verity of model semi-linear equations in the plane recall also the Liouville–Bieberbach equation △u = eu, (1.13) investigated by Bieberbach in his celebrated work [10] related to the study of automorphic functions in the plane. The Liouville–Bieberbach semi-linear equation is one of the principle model equations in the theory of non-linear partial differential equations and their applications, see e.g. [58] and the references therein. Note that the equation appears also as a model one in problems of differen- tial geometry in relation with existence of surfaces of negative Gaussian curvature [72] and in studying the equilibrium of a charged gas. V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 109 In order to illustrate our approach to the study of the equation (1.2), we complete the introduction with several non-trivial model examples. The corresponding proofs together with other examples have been given in [36] and [38]. n 1. Assume that the reaction term in (1.2) is non-negative. Since u = U(ω(z)) and U satisfies in G = ω(D) the equation (1.12) we see, having in mind that the Jacobian J(w) of the quasiconformal mapping ω−1(w) is non-negative almost everywhere in G, that U is a subharmonic function in G. Thus we arrive at the following generalization of the mentioned above Harmonic factorization theorem: Every solution u ∈ W 1,2 loc (D) of the semi-linear equation (1.2) with f(u) ≥ 0 can be expressed as u(z) = U(ω(z)), where ω : D → G is a K-quasiconformal mapping agreed with the matrix function A(z) and U is subharmonic in G. n 2. Let us consider the divergence form of the model Liouville– Bieberbach semi-linear equation div [A(z)∇u] = eu, z = x+ iy, (1.14) in the unit disk D = {z : \|z\| < 1}. If the matrix function A(z) has the following entries a11 = 3− 2 x2 − y2 − 2xy (x2 + y2) , a12 = −2 x2 − y2 + 2xy (x2 + y2) , a22 = 3 + 2 x2 − y2 − 2xy (x2 + y2) , it is easy to verify that the function u(x, y) = −2 log(1− x2 − y2) + log 8 realizes the blow-up solution to the equation (1.14) in the disk D. In this case the matrix function A(z) generates the well-known logarithmic spiral quasiconformal mapping ω(z) = ze2i log \|z\| which plays important role in the study of different problems of contem- porary analysis, see, e.g., [13, §13.2], [22, 32]. This function ω maps the unit disk D onto itself and transforms radial lines into spirals, infinitely winding about the origin. Since the mapping ω is volume preserving, the problem (1.14) is reduced to the well-known solvability result, see [10] and also [58], Theorem 5.3.7, for the Liouville–Bieberbach equation (1.13). 110 To the theory of semi-linear equations in the plane n 3. Let C be the complex plane and let A(z) =   1 ∓ 2ν(x)√ 1−ν2(x) ∓ 2ν(x)√ 1−ν2(x) 1+3ν2(x) 1−ν2(x)   , (1.15) where ν(x), x ∈ R, stands for an arbitrary measurable real-valued func- tion, such that \|ν(x)\| ≤ k < 1. Then the semi-linear equation div [A(z)∇u] = uq, 0 < q < 1, z ∈ C, (1.16) has in the complex plane the following solution with the “dead zone”: u(x, y) =    γ ( y ± x∫ 0 2ν(t)√ 1−ν2(t) dt ) 2 1−q , if y > ϕ(x), x ∈ R, 0 if x ≤ ϕ(x). (1.17) Here γ = ( (1− q)2 2(1 + q) ) 1 1−q , and y = ϕ(x) = ± x∫ 0 2ν(t)dt√ 1− ν2(t) , ∞ < x < +∞, stands for the corresponding free boundary parametrization. In this paper, for the sake of completeness, we will collect some basic facts from our recent research concerning semi–linear partial differential equations in the plane and give a number of new results on the topic. The paper is organized as follows. In Section 2 we give basic facts from the potential theory. In Sections 3 and 4 one can find existence theorems for the quasilinear Poisson equation (1.12) as well as for the corresponding semi-linear equation (1.2) without boundary conditions. We study the solvability of the Dirichlet problem with arbitrary continuous boundary data for the quasilinear Poisson equations (1.12) in Section 5. Section 6 is devoted to the solvability of the Dirichlet problem with continuous boundary data for semi-linear equation (1.2) and it also contains some applications. In the rest sections we discuss the boundary value problem for the Poisson equations with boundary data that are measurable with respect to the logarithmic capacity. V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 111 2. Basic facts from the potential theory For the sake of completeness we repeat the fundamental results con- cerning the potential theory in the plane, given in [35,39], and strengthen some of them. From now on, D denotes the unit disk {z ∈ C : \|z\| < 1} in the complex plane C, DR(z0) := {z ∈ C : \|z − z0\| < R} for z0 ∈ C and R ∈ (0,∞), DR := DR(0). For z and w ∈ D with z 6= w, set G(z, w) := log ∣∣∣∣ 1− zw̄ z − w ∣∣∣∣ and P (z, eit) := 1− \|z\|2 \|1− ze−it\|2 (2.1) be the Green function and Poisson kernel in D. If ϕ ∈ C(∂D) and g ∈ C(D), then a solution to the Poisson equation △f(z) = g(z) (2.2) satisfying the boundary condition f \|∂D = ϕ is given by the formula f(z) = Pϕ(z) − Gg(z) (2.3) where Pϕ(z) = 1 2π 2π∫ 0 P (z, eit)ϕ(e−it) dt , Gg(z) = ∫ D G(z, w) g(w) dm(w) , (2.4) see e.g. [45], p. 118-120. Here m(w) denotes the Lebesgue measure in C. Next, we give the representation of solutions to the Poisson equation in the form of the Newtonian (normalized antilogarithmic) potential. Given a finite Borel measure ν on C with compact support, its poten- tial is the function pν : C → [−∞,∞) defined by pν(z) = ∫ C ln \|z − w\| dν(w), (2.5) see [64], point 3.1.1. Remark 1. Note that the function pν is subharmonic by Theorem 3.1.2 and, consequently, it is locally integrable on C by Theorem 2.5.1 in [64]. Moreover, pν is harmonic outside of the support of ν. This definition can be extended to finite charges ν with compact sup- port (named also signed measures), i.e., to real valued sigma-additive 112 To the theory of semi-linear equations in the plane functions on Borel sets in C, because of ν = ν+ − ν− where ν+ and ν− are Borel measures by the well–known Jordan decomposition, see e.g. Theorem 0.1 in [52]. The key fact is the following statement, see e.g. Theorem 3.7.4 in [64]. Proposition 1. Let ν be a finite charge with compact support in C. Then △pν = 2π · ν (2.6) in the distributional sense, i.e., ∫ C pν(z)△ψ(z) dm(z) = 2π ∫ C ψ(z) d ν(z) ∀ ψ ∈ C∞ 0 (C) . (2.7) Here as usual C∞ 0 (C) denotes the class of all infinitely differentiable functions ψ : C → R with compact support in C, △ = ∂2 ∂x2 + ∂2 ∂y2 is the Laplace operator and dm(z) corresponds to the Lebesgue measure in C. Corollary 1. In particular, if for every Borel set B in C ν(B) := ∫ B g(z) dm(z) (2.8) where g : C → R is an integrable function with compact support, then △Ng = g , (2.9) where Ng(z) := 1 2π ∫ C ln \|z − w\| g(w) dm(w) , (2.10) in the distributional sense, i.e., ∫ C Ng(z)△ψ(z) dm(z) = ∫ C ψ(z) g(z) dm(z) ∀ ψ ∈ C∞ 0 (C) . (2.11) Here the function g is called a density of the charge ν and the function Ng is said to be the Newtonian potential of g. The next statement on continuity in the mean of functions ψ : C → R in Lq(C), q ∈ [1,∞), with respect to shifts is useful for the study of the Newtonian potential, see e.g. Theorem 1.4.3 in [69], cf. also Theorem III(11.2) in [67]. The one-dimensional analog of the statement can be found also in [65], Theorem 9.5. V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 113 Lemma 1. Let ψ ∈ Lq(C), q ∈ [1,∞), have a compact support. Then lim ∆z→0 ∫ C \|ψ(z +∆z)− ψ(z)\|q dm(z) = 0 . (2.12) Recall that a shift of a set E ⊂ C by a vector ∆z ∈ C is the set E +∆z := { ξ ∈ C : ξ = z + ∆z , z ∈ E } . We prefer to give a direct proof of this important statement that may be of independent interest. The proof is based on arguments by contradiction and the absolute continuity of indefinite integrals. Proof. Let us assume that there is a sequence ∆zn ∈ C, n = 1, 2, . . ., such that ∆zn → 0 as n → ∞ and, for some δ > 0 and ψn(z) := ψ(z +∆zn), n = 1, 2, . . ., In :=   ∫ C \|ψn(z)− ψ(z)\|q dm(z)   1 q ≥ δ ∀ n = 1, 2, . . . . (2.13) Denote by K the closed disk in C centered at 0 with the minimal radius R that contains the support of ψ. By the Luzin theorem, see e.g. Theorem 2.3.5 in [20], for every prescribed ε > 0, there is a compact set C ⊂ K such that g\|C is continuous and m(K \ C) < ε. With no loss of generality, we may assume that C ⊂ K∗ where K∗ is a closed disk in C centered at 0 with a radius r ∈ (0, R) and, moreover, that Cn ⊂ K, where Cn := C − ∆zn, for all n = 1, 2, . . .. Note that m(Cn) = m(C) and then m(K \ Cn) < ε and, consequently, m(K \ C∗ n) < 2ε, where C∗ n := C ∩ Cn, because K \ C∗ n = (K \ Cn) ∪ (K \ C). Next, setting Kn = K −∆zn, we see that K ∪Kn = C∗ n ∪ (K \C∗ n)∪ (Kn \ C∗ n) and that Kn \ C∗ n + ∆zn = K \ C∗ n. Hence by the triangle inequality for the norm in Lp the following estimate holds In≤4·   ∫ K\C∗ n \|ψ(z)\|qdm(z)   1 q +   ∫ C∗ n \|ψn(z)− ψ(z)\|qdm(z)   1 q ∀ n = 1, 2, . . . By construction the both terms from the right hand side can be made to be arbitrarily small, the first one for small enough ε because of absolute continuity of indefinite integrals and the second one for all large enough n after the choice of the set C. Thus, the assumption (2.13) is disproved. 114 To the theory of semi-linear equations in the plane Let (X, d) and (X ′ , d ′ ) be metric spaces with distances d and d ′ , respectively. A family F of mappings f : X → X ′ is called equicontin- uous at a point x0 ∈ X if, for every ε > 0, there is δ > 0 such that d ′ (f(x), f(x0)) < ε for all f ∈ F and x ∈ X with d(x, x0) < δ. The family F is said to be equicontinuous if F is equicontinuous at every point x0 ∈ X. Lemma 2. Let g : C → R be in Lp(C) for p > 1 with compact support. Then Ng is continuous. A collection {Ng} is equicontinuous if the collection {g} is bounded by the norm in Lp(C) with supports in a fixed disk K. Moreover, under the latter hypothesis, on each compact set S in C ‖Ng‖C ≤ M · ‖g‖p (2.14) where M is a constant depending in general on S but not on g. Proof. By the Hölder inequality with 1 q + 1 p = 1 we have that \|Ng(z)−Ng(ζ)\| ≤ ‖g‖p 2π ·   ∫ K \| ln \|z − w\| − ln \|ζ − w\| \|q dm(w)   1 q = ‖g‖p 2π ·   ∫ C \|ψζ(ξ +∆z)− ψζ(ξ) \|q dm(ξ)   1 q where ξ = ζ−w, ∆z = z− ζ, ψζ(ξ) := χK+ζ(ξ) ln \|ξ\|. Thus, the first two conclusions follow by Lemma 1 because of the function ln \|ξ\| belongs to the class Lqloc(C) for all q ∈ [1,∞). The third conclusion similarly follows through the direct estimate \|Ng(ζ)\| ≤ ‖g‖p 2π ·   ∫ K \| ln \|ζ − w\|\|q dm(w)   1 q = ‖g‖p 2π ·   ∫ C \|ψζ(ξ)\|q dm(ξ)   1 q because the latter integral is continuous in ζ ∈ C. Indeed, by the triangle inequality for the norm in Lq(C) we see that \|‖ψζ‖q − ‖ψζ∗‖q\| ≤ ‖ψζ − ψζ∗‖q =    ∫ ∆ \| ln \|ξ\|\|q dm(ξ)    1 q V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 115 where ∆ denotes the symmetric difference of the disks K+ ζ and K+ ζ∗. Thus, the statement follows from the absolute continuity of the indefinite integral. The corresponding statement on the continuity of integrals of poten- tial type in higher dimensions can be found in [69], Theorem 1.6.1. Proposition 2. There exist functions g ∈ L1(C) with compact sup- port whose potentials Ng are not continuous, furthermore, Ng /∈ L∞ loc. Proof. Indeed, let us consider the function ω(t) := 1 t2(1− ln t)α , t ∈ (0, 1] , α ∈ (1, 2) , ω(0) := ∞ and, correspondingly, g(z) := ω(\|z\|) , z ∈ D , g(z) := 0 , z ∈ C \D . Then, setting Ω(t) = t · ω(t), we see, firstly, that ∫ D \|g(w)\|dm(w) = 2π lim ρ→+0 1∫ ρ Ω(t)d t = 2π lim ρ→+0 1∫ ρ d ln t (1− ln t)α = 2π α− 1 and, secondly, that the Newtonian potential Ng at the origin is equal to lim ρ→+0 1∫ ρ Ω(t) ln t dt= lim ρ→+0     ln 1 t 1∫ t Ω(τ)dτ   1 ρ + 1∫ ρ  1 t 1∫ t Ω(τ)dτ   dt    = 1 α− 1 · lim ρ→+0   [ ln t (1− ln t)α−1 ]1 ρ − 1∫ ρ d t t(1− ln t)α−1   = 1 α− 1 · lim ρ→+0 [ 1 (1− ln t)α−1 + α− 1 2− α · (1− ln t)2−α ]1 ρ = −∞ . The following lemma on the Newtonian potentials is important for obtaining solutions of a higher regularity to the Poisson equations as well as to the corresponding semi–linear equations. 116 To the theory of semi-linear equations in the plane In this connection, recall the definition of the formal complex deriva- tives: ∂ ∂z := 1 2 { ∂ ∂x − i · ∂ ∂y } , ∂ ∂z := 1 2 { ∂ ∂x + i · ∂ ∂y } , z = x+ iy . (2.15) The elementary algebraic calculations show their relation to the Laplacian △ := ∂2 ∂x2 + ∂2 ∂y2 = 4 · ∂2 ∂z∂z = 4 · ∂2 ∂z∂z (2.16) Further we apply the theory of the well-known integral operators Tg(z) := 1 π ∫ C g(w) dm(w) z − w , Tg(z) := 1 π ∫ C g(w) dm(w) z − w defined for integrable functions with a compact support K and studied in detail. Recall the known results on them in Chapter 1 of [71], confining the case K = D, that are relevant to the proof of Theorem 2. First of all, if g ∈ L1(C), then by Theorem 1.13 the integrals Tg and Tg exist a.e. in C and belong to Lqloc(C) for all q ∈ [1, 2) and by Theorem 1.14 they have generalized derivatives by Sobolev (Tg)z = g = (Tg)z . Furthermore, if g ∈ Lp(C), p > 1, then by Theorem 1.27 and (6.27) Tg and Tg belong to Lqloc(C) for some q > 2. and, moreover, by Theorems 1.36–1.37 (Tg)z and (Tg)z also belong to Lploc(C). Finally, if g ∈ Lp(C) for p > 2, then by Theorem 1.19 Tg and Tg belong to Cαloc(C) with α = (p− 2)/p. Here, given a domain D in C, a function g : D → R is assumed to be extended onto C by zero outside of D. Lemma 3. Let D be a bounded domain in C. Suppose that g ∈ L1(D). Then Ng ∈ W 1,q loc (C) for all q ∈ [1, 2) and there exist the generalized derivatives by Sobolev ∂2Ng ∂z∂z and ∂2Ng ∂z∂z and 4 · ∂ 2Ng ∂z∂z = △Ng = 4 · ∂ 2Ng ∂z∂z = g a.e. in C (2.17) Moreover, Ng ∈ Lsloc(C) for all s ∈ [1,∞). More precisely, ‖Ng‖s ≤ ‖g‖1 · ‖ ln \|ξ\|‖s ∀ s ∈ [1,∞) , (2.18) where ‖Ng‖s is in Dr for all r ∈ (0,∞) and ‖ ln \|ξ\|‖s is in DR+r if D ⊆ DR. If g ∈ Lp(D) for some p ∈ (1, 2], then Ng ∈W 2,p loc (C) and Ng ∈W 1,γ loc (C) ∀ γ ∈ (1, q) , where q = 2p/(2 − p) > 2 . (2.19) V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 117 In addition, the collection {Ng} is locally β−Hölder equicontinuous in C for all β ∈ (0, 1 − 2/q) and the collection {N ′ g} of its first partial derivatives is strictly compact in Lγ(D) for all γ ∈ (1, q) if the collection {g} is bounded in Lp(D). Finally, if g ∈ Lp(D) for some p > 2, then Ng ∈ C1,α loc (C) with α = (p − 2)/p. Furthermore, the collection {N ′ g} is locally Hölder equi- continuous in C with the given α if {g} is bounded in Lp(D). Proof. Note that Ng is the convolution ψ ∗ g, where ψ(ξ) = ln \|ξ\|, and hence (2.18) follows, e.g. by Corollary 4.5.2 in [46]. Moreover, ∂ψ∗g ∂z = ∂ψ ∂z ∗ g and ∂ψ∗g ∂z = ∂ψ ∂z ∗ g, see e.g. (4.2.5) in [46], and by elementary calculations ∂ ∂z ln \|z − w\| = 1 2 · 1 z − w , ∂ ∂z ln \|z − w\| = 1 2 · 1 z − w . (2.20) Consequently, ∂Ng(z) ∂z = 1 4 · Tg(z) , ∂Ng(z) ∂z = 1 4 · Tg(z) . (2.21) Thus, the rest conclusions for g ∈ L1(D) follow by Theorems 1.13–1.14 in [71]. Next, if g ∈ Lp(D) with p ∈ (1, 2], then Ng ∈ W 1,γ loc (C) for all γ ∈ (1, q), where q = 2p/(2 − p) > 2, by Theorem 1.27, (1.27) in [71] and, moreover, Ng ∈ W 2,p loc (C) by Theorems 1.36–1.37 in [71]. In addition, a collection {Ng} is locally β−Hölder equicontinuous in C for all β ∈ (0, 1 − 2/q), see e.g. Lemma 2.7 in [15], and the collection {N ′ g} of its first partial derivatives is strictly compact in Lγ(D) for all γ ∈ (1, q) if the collection {g} is bounded by the norm in Lp(D), see e.g. Theorem 1.4.3 in [69] and Theorem 1.27 in [71]. Finally, if g ∈ Lp(D) for some p > 2, then Ng ∈ C1,α loc (C) with α = (p − 2)/p by Theorem 1.19 in [71]. Furthermore, by the latter theorem the collection {N ′ g} is also locally α−Hölder equicontinuous in C with α = (p − 2)/p if the collection {g} is bounded by the norm in Lp(D), p > 2. Remark 2. Note that generally speaking Ng /∈ W 2,1 loc for the case g ∈ L1(C), see e.g. example 7.5 in [25], p.141. Note also that the corre- sponding Newtonian potentials Ng in Rn, n ≥ 3, also belong to W 2,p loc if g ∈ Lp(C) for p > 1 with compact support, see e.g. [26], Theorem 9.9. As above, we assume here that g : D → R is extended by zero outside of D. 118 To the theory of semi-linear equations in the plane Corollary 2. Let D be a subdomain of D, g : D → R be in L1(D) and in Lploc(D) for some p > 1. Then Ng satisfies (2.17) a.e. in D. Moreover, Ng ∈ W 1,q loc (D) for q > 2 and Ng is locally Hölder continuous in D. Furthermore, Ng ∈ C1,α loc (D) with α = (p− 2)/p if g ∈ Lploc(D) for p > 2. In addition, the collection {Ng} is locally β−Hölder equicontinuous in D for all β ∈ (0, 1 − 2/q) and the collection {N ′ g} of its first partial derivatives is strictly compact in Lγloc(D) for all γ ∈ (1, q) if a collection {g} is bounded in L1(D) and in Lploc(D) for some p ∈ (1, 2], where q is defined in (2.19). Finally, the collection {N ′ g} is locally α−Hölder equicontinuous in D with the given α if a collection {g} is bounded in L1(D) and in Lploc(D) for p > 2. Proof. Given z0 ∈ D and 0 < R < dist (z0, ∂D), Ng = Ng1 + Ng2 with g2 := g − g1 and g1 := g · χ where χ is the characteristic function of the disk DR(z0). The first summand satisfies all desired properties by Lemma 3 and the second one is a harmonic function in DR(z0), see e.g. Theorem 3.1.2 in [64]. Thus, the first part follows. Under the proof of the rest part, it is applied the same decomposition. However, in the case we need the following 2 explicit estimates for the second summand in a smaller disk Dr(z0), r ∈ (0, R), \|Ng2(z2)−Ng2(z1)\|≤ ∣∣∣∣∣∣ z2∫ z1 ∂Ng2 ∂z dz ∣∣∣∣∣∣ + ∣∣∣∣∣∣ z2∫ z1 ∂Ng2 ∂z̄ dz̄ ∣∣∣∣∣∣ ≤ 1 2π · ‖g‖1 (R− r) · \|z2 − z1\| and, since the function Tg2 is analytic in Dr(z0) and the function Tg2 = Tg2 (for the real–valued function g2) is anti-analytic in Dr(z0), similarly \|N ′ g2(z2)−N ′ g2(z1)\| ≤ 1 4 ∣∣∣∣∣∣ z2∫ z1 ∂ Tg2 ∂z dz ∣∣∣∣∣∣ ≤ 1 4π · ‖g‖1 (R− r)2 · \|z2 − z1\| . Here we denote by N ′ g2 any of the first partial derivatives of Ng2 , see (2.15): ∂ ∂x = ∂ ∂z + ∂ ∂z̄ , ∂ ∂y = i · ( ∂ ∂z − ∂ ∂z̄ ) , z = x+ iy , take into account relation (2.20) and calculate the given integrals over the segment [z1, z2] ⊂ Dr(z0) of straight line going through z1, z2 ∈ Dr(z0). V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 119 3. On solvability of quasilinear Poisson equations In this section we study the solvability problem for quasilinear Poisson equations of the form △U = h(z)f(U). The well-known Leray–Schauder approach allows us to reduce the problem to the study of the correspond- ing linear Poisson equation from the previous section. For the sake of completeness, we recall some definitions and basic facts of the celebrated paper [56]. First of all, Leray and Schauder define a completely continuous map- ping from a metric space M1 into a metric space M2 as a continuous mapping on M1 which takes bounded subsets of M1 into relatively com- pact ones of M2, i.e. with compact closures in M2. When a continuous mapping takes M1 into a relatively compact subset of M1, it is nowadays said to be compact on M1. Then Leray and Schauder extend as follows the Brouwer degree to compact perturbations of the identity I in a Banach space B, i.e. a com- plete normed linear space. Namely, given an open bounded set Ω ⊂ B, a compact mapping F : B → B and z /∈ Φ(∂Ω), Φ := I −F , the (Leray– Schauder) topological degree deg [Φ,Ω, z] of Φ in Ω over z is constructed from the Brouwer degree by approximating the mapping F over Ω by mappings Fε with range in a finite-dimensional subspace Bε (contain- ing z) of B. It is showing that the Brouwer degrees deg [Φε,Ωε, z] of Φε := Iε − Fε, Iε := I\|Bε , in Ωε := Ω ∩Bε over z stabilize for sufficiently small positive ε to a common value defining deg [Φ,Ω, z] of Φ in Ω over z. This topological degree “algebraically counts” the number of fixed points of F (·)− z in Ω and conserves the basic properties of the Brouwer degree as additivity and homotopy invariance. Now, let a be an isolated fixed point of F . Then the local (Leray–Schauder) index of a is defined by ind [Φ, a] := deg[Φ, B(a, r), 0] for small enough r > 0. If a = 0, then we say on the index of F . In particular, if F ≡ 0, correspondingly, Φ ≡ I, then the index of F is equal to 1. The fundamental Theorem 1 in [56] can be formulated in the following way: Let B be a Banach space, and let F (·, τ) : B → B be a family of operators with τ ∈ [0, 1]. Suppose that the following hypotheses hold: (H1) F (·, τ) is completely continuous on B for each τ ∈ [0, 1] and uniformly continuous with respect to the parameter τ ∈ [0, 1] on each bounded set in B; (H2) the operator F := F (·, 0) has finite collection of fixed points whose total index is not equal to zero; 120 To the theory of semi-linear equations in the plane (H3) the collection of all fixed points of the operators F (·, τ), τ ∈ [0, 1], is bounded in B. Then the collection of all fixed points of the family of operators F (·, τ) contains a continuum along which τ takes all values in [0, 1]. In the proof of the next theorem the initial operator F (·) := F (·, 0) ≡ 0. Hence F has the only one fixed point (at the origin) and its index is equal to 1 and, thus, hypothesis (H2) will be automatically satisfied. Theorem 1. Let h : C → R be a function in the class Lp(C) for p > 1 with compact support. Suppose that a function f : R → R is continuous and lim t→∞ f(t) t = 0 . (3.1) Then there is a continuous function U : C → R in the class W 2,p loc (C) such that △U(z) = h(z) · f(U(z)) a.e. (3.2) and U = Ng where g : C → R is a function in Lp whose support is in the support of h and the upper bound of ‖g‖p depends only on ‖h‖p and on the function f . Moreover, U ∈ W 1,q loc (C) for some q > 2 and U is locally Hölder continuous. Furthermore, U ∈ C1,α loc (C) with α = (p−2)/p if p > 2. In particular, U ∈ C1,α loc (C) for all α = (0, 1) if h is bounded in Theo- rem 1. Proof. If ‖h‖p = 0 or ‖f‖C = 0, then any constant function U in C gives the desired solution of (3.2). Thus, we may assume that ‖h‖p 6= 0 and ‖f‖C 6= 0. Set f∗(s) = max \|t\|≤s \|f(t)\|, s ∈ R+ := [0,∞). Then the function f∗ : R+ → R+ is continuous and nondecreasing and, moreover, f∗(s)/s → 0 as s→ ∞ by (3.1). By Lemma 2 we obtain the family of operators F (g; τ) : Lph(C) → Lph(C), where L p h(C) consists of functions g ∈ Lp(C) with supports in the support of h, F (g; τ) := τh · f(Ng) ∀ τ ∈ [0, 1] (3.3) which satisfies all groups of hypothesis H1-H3 of Theorem 1 in [56]. In- deed: H1). First of all, F (g; τ) ∈ Lph(C) for all τ ∈ [0, 1] and g ∈ Lph(C) because by Lemma 2 the function f(Ng) is continuous and ‖F (g; τ)‖p ≤ ‖h‖p f∗ (M ‖g‖p) < ∞ ∀ τ ∈ [0, 1], V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 121 where M is the constant from the estimate (2.14). Thus, by Lemma 2 in combination with the Arzela–Ascoli theorem, see e.g. Theorem IV.6.7 in [17], the operators F (g; τ) are completely continuous for each τ ∈ [0, 1] and even uniformly continuous with respect to the parameter τ ∈ [0, 1]. H2). The index of the operator F (g; 0) is obviously equal to 1. H3). By Lemma 2 we have also for solutions of the equations g = F (g; τ): ‖g‖p ≤ ‖h‖p f∗ (M ‖g‖p) i.e., f∗(M ‖g‖p) M ‖g‖p ≥ 1 M ‖h‖p (3.4) and hence ‖g‖p should be bounded in view of condition (3.1). Thus, by Theorem 1 in [56] there is a function g ∈ Lph(D) with F (g; 1) = g, and by Lemma 3 the function U := Ng gives the desired solution of (3.2). Corollary 3. Let D be a subdomain of D, h : D → R be in L1(D) and in Lploc(D) for some p > 1. Suppose that a function f : R → R satisfies the hypothesis of Theorem 1. Then there is a weak solution u : D → R of the quasilinear Poisson equation (3.2) which is locally Hölder continuous in D. Proof. Let Dk be an expanding sequence of domains in C with Dk ⊂ D, k = 1, 2, . . ., exhausting D, i.e. ∞⋃ k=1 Dk = D. Let us extend h by zero outside of D. Set hk = hχk, where χk is a characteristic function of Dk in C, and Uk = Ngk , where gk corresponds to hk by Theorem 1. Note that the sequence ‖gk‖p, k = 1, 2, . . ., is bounded on each Dm, m = 1, 2, . . ., by Theorem 1. Hence by Lemma 2 the sequence \|Ngk \|C is also bounded on each Dm, m = 1, 2, . . .. Now, by Corollary 2 the family of functions {Ngk} is Hölder equicontinuous on each Dm, m = 1, 2, . . .. Thus, by the Arzela–Ascoli theorem, see e.g. Theorem IV.6.7 in [17], the family of functions {Ngk} is compact on each Dm, m = 1, 2, . . .. Without loss of generality, we may assume that p ∈ (1, 2]. Then by Corollary 2 the Newtonian potential {Ngk}, m = 1, 2, . . ., is in the class W 1,q loc for some q > 2 and the family {N ′ gk } is also compact on each Dm, m = 1, 2, . . . by the norm of Lq. Consequently, the sequence {Ngk} is compact on each Dm, m = 1, 2, . . . by any norm ‖ · ‖ of W 1,q, too, see e.g. Theorem 2.5.1 in [59]. Next, let us apply the so–called diagonal process. Namely, let u (1) k , k = 1, 2, . . ., be a subsequence of {Ngk} that converges uniformly and by 122 To the theory of semi-linear equations in the plane the norm ‖ · ‖ on the domain D1 to a function u : D1 → R. Of course, we may assume that ‖u(1)k − u‖C < 1/k as well as ‖u(1)k − u‖ < 1/k for all k = 1, 2, . . .. Similarly, it is defined a subsequence u (2) k of u (1) k with respect to the domain D2. Let us continue the process by induction and, finally, consider the diagonal subsequence um := u (m) m , m = 1, 2, . . . of the sequence Ngk . It is clear by the construction that um\|D converges to a func- tion u : D → R locally uniformly and in W 1,q loc (D), q > 2. Thus, u ∈ C(D) ∩W 1,q loc (D) and, consequently, u is locally Hölder continuous in D. Moreover, u is a weak solution of the equation (3.2) in the domain D. Indeed, by Corollary 1 and the definition of generalized derivatives, we have that um satisfy the relations ∫ D 〈∇um(z),∇ψ(z)〉 dm(z) + ∫ D hm(z)f(um(z))ψ(z) dm(z) = 0 ∀ψ ∈ C∞ 0 (D) and, passing to the limit asm→ ∞, we obtain the desired conclusion. 4. On solvability of semi-linear equations In this section we study the solvability problem for the semi-linear equations of the form div [A(z)∇u] = f(u). Following the paper [36], under a weak solution to this equation we understand a function u ∈ C(D) ∩W 1,2 loc (D) such that ∫ D 〈A(z)∇u(z), ∇ϕ(z)〉 dm(z) + ∫ D f(u(z))ϕ(z) dm(z) = 0 (4.1) for all ϕ ∈ C(D) ∩W 1,2 0 (D). Theorem 2. Let D be a domain in C with a finite area that is not dense in C. Suppose that A ∈ M2×2 K (D) and a continuous function f : R → R satisfies condition (3.1). Then there is a weak solution u : D → R of equation div [A(z)∇u(z)] = f(u), (4.2) which is locally Hölder continuous in D. Proof. Let us extend by definition A ≡ I (the unit matrix) outside of D. By Theorem 4.1 in [36], if u is a weak solution of (4.2), then u = U ◦ ω, V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 123 where ω := Ω\|D and Ω is a quasiconformal mapping of C onto itself, Ω(∞) = ∞, agreed with the extended A, and U is a weak solution of (3.2) with h = J , where J is the Jacobian of the mapping ω−1 : D∗ → D, D∗ := ω(D). Note that C \D contains a nondegenerate (connected) component C because of D is not dense in C, see e.g. Corollary IV.2 and the point II.4.D in [47], see also Lemma 5.1 in [48] or Lemma 6.3 in [57]. Hence C\ D∗ contains a component C∗ := Ω(C) whose boundary is a nondegenerate continuum, see again Lemma 5.1 in [48] or Lemma 6.3 in [57], and by the Riemann theorem there is a conformal mapping H of C \C∗ onto D. Setting H∗ = H\|D∗ , we see that H∗ maps D∗ into D. Moreover, the quasiconformal mapping ω∗ := H∗◦ω : D → D∗ := H∗(D∗) is also agreed with A in D. Thus, again by Theorem 4.1 in [36], u = U∗ ◦ ω∗ where U∗ is a weak solution of (3.2) with h = J∗ in D∗ ⊆ D, here J∗ is the Jacobian of the mapping ω−1 ∗ : D∗ → D. Also by Remark 4.1 in [36], inversely, if U∗ is a weak solution of (3.2) with h = J∗ in D∗, then u := U∗ ◦ ω∗ is a weak solution of (4.2) in D. The latter implication allows us to reduce the proof of Theorem 2 to Corollary 3 with the special h = J∗. Indeed, J∗ ∈ L1(D∗) because its integral is equal to the area of the domain D, see e.g Theorem 3.2 in [12] and Theorem II.B.3 in [1]. More- over, J∗ ∈ Lploc(D∗) for some p > 1 because by the Bojarski result, see [11] and [12], the first partial derivatives of the quasiconformal map- ping ω∗ := ω−1 ∗ : D∗ → D are locally integrable with a power q > 2 and J∗ = \|ω∗ w\|2 − \|ω∗ w̄\|2, see e.g. I.A(9) in [1]. Remark 3. Note that it is easy to construct a set C in C of the Cantor type which is dense in the plane C whose completion has a finite area, furthermore, an arbitrarily small area. Indeed, let us cover the plane by a collection S plates consisting of closed squares with unit sides oriented along coordinate axes x and y, z = x+ iy ∈ C, that can intersect each other only by their common sides. Let Sn, n = 1, 2, . . ., be some enumeration of the squares in S and let ε ∈ (0, 1) be arbitrary. First, let us remove narrow symmetric strips of the same width in S1 along its sides whose total area is less that ε/4. We have in the rest a central square. Then let us cut out narrow centralized horizontal and vertical corridors of the same width in the last square whose total area is less than ε/8. These corridors form a cross that splits the last square into 4 squares. In turn, we remove from these squares similar crosses of the total area ε/16 that split them on the whole into 42 squares. Repeating the procedure by induction, we remove from S1 corridors with the total 124 To the theory of semi-linear equations in the plane area ε/2 and the intersection of all mentioned squares gives a totally disconnected compactum C1 6= ∅ of the Cantor type, see e.g. 4.41(2′) in [50]. Similarly, we are able to construct such a set Cn ⊂ Sn with its com- pletion in Sn whose area is less than ε/21+n for each n = 1, 2, . . .. Then the set C := ∞⋃ n=1 Cn has the completion in C whose area is less than ε. Note that by our construction the set C is totally disconnected and, thus, its topological dimension is equal to 0, see e.g. Proposition II.4.D in [47]. It is clear that D := C \ C is a domain, see e.g. Theorem IV.4 in [47]. Finally, note that our example of the set C of the Cantor type in the plane with its topological dimension 0 is essentially different from the well-known Sierpinski cover whose topological dimension is equal to 1. 5. The Dirichlet problem with continuous data for quasilinear Poisson equations Let D be a bounded domain in C without degenerate boundary com- ponents, i.e., any connected component of the boundary of D is not degenerated to a single point. Given a continuous boundary function ϕ : ∂D → R, let us denote by Dϕ the harmonic function in D that has the continuous extension to D with ϕ as its boundary date. Such a function exists and it is unique, see e.g. Corollary 4.1.8 and Theorem 4.2.2 in [64]. Thus, the Dirichlet operator Dϕ is well defined in the given domains. We need not its explicit description for our goals. By Lemma 3 we come to the following result on the existence, reg- ularity and representation of solutions for the Dirichlet problem to the Poisson equation in arbitrary bounded domains D in C without degener- ate boundary components where we assume that the charge density g is extended by zero outside of D. Theorem 3. Let D be a bounded domain in C without degenerate boundary components, ϕ : ∂D → R be a continuous function and g : D → R belong to the class Lp(D) for p > 1. Then the function U := Ng − DN∗ g + Dϕ , N∗ g := Ng\|∂D , (5.1) is continuous in D with U \|∂D = ϕ, belongs to the class W 2,p loc (D) and sat- isfies the Poisson equation △U = g a.e. in D. Moreover, U ∈ W 1,q loc (D) for some q > 2 and U is locally Hölder continuous in D. Furthermore, U ∈ C1,α loc (D) with α = (p− 2)/p if g ∈ Lp(D) for p > 2. V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 125 Remark 4. Note also by the way that a generalized solution of the Dirichlet problem to the Poisson equation in the class C(D)∩W 1,2 loc (D) is unique at all, see e.g. Theorem 8.30 in [26], and (5.1) gives the effective representation of this unique solution. The case of quasilinear Poisson equations is reduced to the case of the linear Poisson equations again by the Leray–Schauder approach as in the last section. Theorem 4. Let D be a bounded domain in C without degenerate boundary components, ϕ : ∂D → R be a continuous function and h : D → R be a function in the class Lp(D) for p > 1. Suppose that a function f : R → R is continuous and lim t→∞ f(t) t = 0 . (5.2) Then there is a continuous function U : D → R with U \|∂D = ϕ and U \|D ∈W 2,p loc such that △U(z) = h(z) · f(U(z)) for a.e. z ∈ D . (5.3) Moreover, U ∈ W 1,q loc (D) for some q > 2 and U is locally Hölder contin- uous. Furthermore, U ∈ C1,α loc (D) with α = (p− 2)/p if p > 2. In particular, the latter statement in Theorem 4 implies that U ∈ C1,α loc (D) for all α = (0, 1) if h is bounded. Proof. If ‖h‖p = 0 or ‖f‖C = 0, then the Dirichlet operator Dϕ gives the desired solution of the Dirichlet problem for equation (5.3), see e.g. I.D.2 in [49]. Hence we may assume further that ‖h‖p 6= 0 and ‖f‖C 6= 0. Set f∗(s) = max \|t\|≤s \|f(t)\|, s ∈ R+. Then the function f∗ : R+ → R+ is continuous and nondecreasing and, moreover, f∗(s)/s → 0 as s → ∞ by (5.2). By Lemma 2 and the maximum principle for harmonic functions, we obtain the family of operators F (g; τ) : Lp(D) → Lp(D), τ ∈ [0, 1]: F (g; τ) := τh ·f(Ng−DN∗ g +Dϕ) , N ∗ g := Ng\|∂D , ∀ τ ∈ [0, 1] (5.4) which satisfies all groups of hypothesis H1-H3 of Theorem 1 in [56]. In- deed: H1). First of all, F (g; τ) ∈ Lp(D) for all τ ∈ [0, 1] and g ∈ Lp(D) because by Lemma 2 f(Ng − DN∗ g + Dϕ) is a continuous function and, moreover, ‖F (g; τ)‖p ≤ ‖h‖p f∗ ( 2M ‖g‖p + ‖ϕ‖C ) < ∞ ∀ τ ∈ [0, 1] . 126 To the theory of semi-linear equations in the plane Thus, by Lemma 2 in combination with the Arzela–Ascoli theorem, see e.g. Theorem IV.6.7 in [17], the operators F (g; τ) are completely contin- uous for each τ ∈ [0, 1] and even uniformly continuous with respect to the parameter τ ∈ [0, 1]. H2). The index of the operator F (g; 0) is obviously equal to 1. H3). By Lemma 2 and the maximum principle for harmonic func- tions, we have the estimate for solutions g ∈ Lp of the equations g = F (g; τ): ‖g‖p ≤ ‖h‖p f∗ ( 2M ‖g‖p + ‖ϕ‖C ) ≤ ‖h‖p f∗( 3M ‖g‖p) whenever ‖g‖p ≥ ‖ϕ‖C/M , i.e. then it should be f∗( 3M ‖g‖p) 3M ‖g‖p ≥ 1 3M ‖h‖p (5.5) and hence ‖g‖p should be bounded in view of condition (5.2). Thus, by Theorem 1 in [56] there is a function g ∈ Lp(D) such that g = F (g; 1) and, consequently, by Lemma 3 the function U := Ng − DN∗ g + Dϕ gives the desired solution of the Dirichlet problem for the quasilinear Poisson equation (5.3). Remark 5. As it is clear from the proof, condition (5.2) can be replaced by the following weaker condition with M from the estimate in Lemma 2 lim sup s→∞ f∗(s) s < 1 3M‖h‖p . (5.6) Theorem 4 can be applied to some physical problems. The first circle of such applications is relevant to reaction-diffusion problems. Problems of this type are discussed in [16], p. 4, and, in detail, in [3]. A nonlin- ear system is obtained for the density u and the temperature T of the reactant. Upon eliminating T the system can be reduced to the equation △u = λ · f(u) (5.7) with h(z) ≡ λ > 0 and, for isothermal reactions, f(u) = uq where q > 0 is called the order of the reaction. It turns out that the density of the reactant u may be zero in a subdomain called a dead core. A particular- ization of results in Chapter 1 of [16] shows that a dead core may exist just if and only if 0 < q < 1 and λ is large enough, see also the corre- sponding examples in [36]. In this connection, the following statements may be of independent interest. V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 127 Corollary 4. Let D be a bounded domain in C without degenerate boundary components, ϕ : ∂D → R be a continuous function and let h : D → R be a function in the class Lp(D), p > 1. Then there exists a continuous function u : D → R with u\|∂D = ϕ such that u ∈ W 2,p loc (D) and △u(z) = h(z) · uq(z) , 0 < q < 1 (5.8) a.e. in D. Moreover, u ∈ W 1,β loc (D) for some β > 2 and u is locally Hölder continuous in D. Furthermore, u ∈ C1,α loc (D) with α = (p − 2)/p if p > 2. Corollary 5. Let D be a bounded domain in C without degenerate boundary components and ϕ : ∂D → R be a continuous function. Then there is a continuous function u : D → R with u\|∂D = ϕ such that u ∈ C1,α loc (D) for all α ∈ (0, 1), u ∈W 2,p loc (D) for all p ∈ [1,∞) and △u(z) = uq(z) , 0 < q < 1 , a.e. in D . (5.9) Note also that certain mathematical models of a thermal evolution of a heated plasma lead to nonlinear equations of the type (5.7). Indeed, it is known that some of them have the form △ψ(u) = f(u) with ψ′(0) = ∞ and ψ′(u) > 0 if u 6= 0 as, for instance, ψ(u) = \|u\|q−1u under 0 < q < 1, see e.g. [16]. With the replacement of the function U = ψ(u) = \|u\|q·sign u, we have that u = \|U \|Q · signU , Q = 1/q, and, with the choice f(u) = \|u\|q2 · signu, we come to the equation △U = \|U \|q · signU = ψ(U). Corollary 6. Let D be a bounded domain in C without degenerate boundary components and ϕ : ∂D → R be a continuous function. Then there is a continuous function U : D → R with U \|∂D = ϕ such that U ∈ C1,α loc (D) for all α ∈ (0, 1), u ∈W 2,p loc (D) for all p ∈ [1,∞) and △U(z) = \|U(z)\|q−1U(z) , 0 < q < 1 , a.e. in D . (5.10) Finally, we recall that in the combustion theory, see e.g. [5], [62] and the references therein, the following model equation ∂u(z, t) ∂t = 1 δ · △u + eu , t ≥ 0, z ∈ D, (5.11) takes a special place. Here u ≥ 0 is the temperature of the medium and δ is a certain positive parameter. We restrict ourselves here by the stationary case, although our approach makes it possible to study the 128 To the theory of semi-linear equations in the plane parabolic equation (5.11), see [36]. Namely, the equation (5.3) is appeared here with h ≡ δ > 0 and the function f(u) = e−u that is bounded. Corollary 7. Let D be a bounded domain in C without degenerate boundary components and ϕ : ∂D → R be a continuous function. Then there is a continuous function U : D → R with U \|∂D = ϕ such that U ∈ C1,α loc (D) for all α ∈ (0, 1), u ∈W 2,p loc (D) for all p ∈ [1,∞) and △U(z) = δ · eU(z) , a.e. in D . (5.12) Due to the factorization theorem in [36], we extend these results to semi–linear equations describing the corresponding physical phenomena in anisotropic and inhomogeneous media in arbitrary bounded domain without degenerate boundary components, see the next section. 6. The Dirichlet problem with continuous data for semi–linear equations By the factorization theorem from [36], mentioned in Introduction, the study of semi–linear equations (4.2) in bounded domains without degenerate boundary componentss D is reduced, by means of a suitable quasiconformal change of variables, to the study of the corresponding quasilinear Poisson equations (5.3). Theorem 5. Let D be a bounded domain in C without degenerate boundary components, A ∈ M2×2 K (D), ϕ : ∂D → R be an arbitrary con- tinuous function, f : R → R be a continuous function such that lim t→∞ f(t) t = 0 . (6.1) Then there is a weak solution u : D → R of the class C(D)∩W 1,2 loc (D) to the equation div [A(z)∇u] = f(u) which is locally Hölder continuous in D and continuous in D with u\|∂D = ϕ. Proof. Let us extend by definition A ≡ I outside of D. By Theorem 4.1 in [36], if u is a weak solution of the equation, then u = U ◦ ω, where ω := Ω\|D and Ω is a quasiconformal mapping of C onto itself agreed with the extended A, and U is a weak solution of the equation (5.3) with h = J , where J is the restriction of the Jacobian of the mapping Ω−1 : C → C to the domain D∗ := Ω(D). V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 129 Inversely, by Remark 4.1 in [36], we see that if U is a weak solution of (5.3) with h = J , then u = U ◦ω is a weak solution of our equation. The latter allows us to reduce Theorem 5 to Theorem 4. Indeed, D∗ = Ω(D) is compact and by the celebrated Bojarski result, see [11] and [12], the generalized derivatives of the quasiconformal mapping Ω∗ := Ω−1 : C → C are locally integrable with some power q > 2. Note also that the Jacobian J of its restriction ω∗ := Ω∗\|D∗ is equal to \|ω∗ w\|2 − \|ω∗ w̄\|2, see e.g. I.A(9) in [1]. Consequently, J ∈ Lp(D∗) for some p > 1. Specifying the reaction term f(u) of the semi-linear equation, we ar- rive at the following statements concerning some concrete problems of the mathematical physics in inhomogeneous and anisotropic media. Corollary 8. Let D be a bounded domain in C without degenerate boundary components, A ∈ M2×2 K (D) and ϕ : ∂D → R be a continuous function. Then there is a continuous function u : D → R with u\|∂D = ϕ which is locally Hölder continuous in D and it is a weak solution in D for the equation div [A(z)∇u(z) ] = uq(z) , 0 < q < 1 . (6.2) Corollary 9. Let D be a bounded domain in C without degenerate boundary components, A ∈ M2×2 K (D) and ϕ : ∂D → R be a continuous function. Then there is a continuous function u : D → R with u\|∂D = ϕ which is locally Hölder continuous in D and it is a weak solution in D for the equation div [A(z)∇u(z) ] = \|u(z)\|q−1u(z) , 0 < q < 1 . (6.3) Corollary 10. Let D be a bounded domain in C without degenerate boundary components, A ∈ M2×2 K (D) and ϕ : ∂D → R be a continuous function. Then there is a continuous function u : D → R with u\|∂D = ϕ which is locally Hölder continuous in D and it is a weak solution in D for the equation div [A(z)∇u(z) ] = eαu(z) , α ∈ R . (6.4) Note that the statements given above remain hold if the reaction terms in equations (6.2)–(6.4) are multiplied by functions C ∈ L∞(D). The rest of the paper we are going to devote to the study of the Dirichlet problem for the Poisson equation with measurable boundary data and start with the notion of the logarithmic capacity. 130 To the theory of semi-linear equations in the plane 7. The definition and preliminary remarks on the logarithmic capacity Given a bounded Borel set E in the plane C, a mass distribution on E is a nonnegative completely additive function ν of a set defined on its Borel subsets with ν(E) = 1. The function Uν(z) := ∫ E log ∣∣∣∣ 1 z − ζ ∣∣∣∣ dν(ζ) (7.1) is called a logarithmic potential of the mass distribution ν at a point z ∈ C. A logarithmic capacity C(E) of the Borel set E is the quantity C(E) = e−V , V = inf ν Vν(E) , Vν(E) = sup z Uν(z) . (7.2) It is also well-known the following geometric characterization of the logarithmic capacity, see e.g. the point 110 in [61]: C(E) = τ(E) := lim n→∞ V 2 n(n−1) n (7.3) where Vn denotes the supremum of the product V (z1, . . . , zn) = l=1,...,n∏ k<l \|zk − zl\| (7.4) taken over all collections of points z1, . . . , zn in the set E. Following Fékete, see [21], the quantity τ(E) is called the transfinite diameter of the set E. Remark 6. Thus, we see that if C(E) = 0, then C(f(E)) = 0 for an arbitrary mapping f that is continuous by Hölder and, in particular, for quasiconformal mappings on compact sets, see e.g. Theorem II.4.3 in [55]. In order to introduce sets that are measurable with respect to logarith- mic capacity, we define, following [18], inner C∗ and outer C∗ capacities: C∗(E) : = sup F⊆E C(E), C∗(E) : = inf E⊆O C(O) (7.5) where supremum is taken over all compact sets F ⊂ C and infimum is taken over all open sets O ⊂ C. A set E ⊂ C is called measurable with respect to the logarithmic capacity if C∗(E) = C∗(E), and the common value of C∗(E) and C∗(E) is still denoted by C(E). V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 131 A function ϕ : E → C defined on a bounded set E ⊂ C is called measurable with respect to logarithmic capacity if, for all open sets O ⊆ C, the sets {z ∈ E : ϕ(z) ∈ O} are measurable with respect to logarithmic capacity. It is clear from the definition that the set E is itself measurable with respect to logarithmic capacity. Note also that sets of logarithmic capacity zero coincide with sets of the so-called absolute harmonic measure zero introduced by Nevanlinna, see Chapter V in [61]. Hence a set E is of (Hausdorff) length zero if C(E) = 0, see Theorem V.6.2 in [61]. However, there exist sets of length zero having a positive logarithmic capacity, see e.g. Theorem IV.5 in [18]. Remark 7. It is known that Borel sets and, in particular, compact and open sets are measurable with respect to logarithmic capacity, see e.g. Lemma I.1 and Theorem III.7 in [18]. Moreover, as it follows from the definition, any set E ⊂ C of finite logarithmic capacity can be repre- sented as a union of a sigma-compactum (union of countable collection of compact sets) and a set of logarithmic capacity zero. Thus, the mea- surability of functions with respect to logarithmic capacity is invariant under Hölder continuous change of variables. It is also known that the Borel sets and, in particular, compact sets are measurable with respect to all Hausdorff’s measures and, in partic- ular, with respect to measure of length, see e.g. theorem II(7.4) in [67]. Consequently, any set E ⊂ C of finite logarithmic capacity is measur- able with respect to measure of length. Thus, on such a set any function ϕ : E → C being measurable with respect to logarithmic capacity is also measurable with respect to measure of length on E. However, there exist functions that are measurable with respect to measure of length but not measurable with respect to logarithmic capacity, see e.g. Theorem IV.5 in [18]. Dealing with measurable boundary functions ϕ(ζ) with respect to the logarithmic capacity, we will use the abbreviation q.e. (quasi-everywhere) on a set E ⊂ C, if a property holds for all ζ ∈ E except its subset of zero logarithmic capacity, see [52]. 8. The Dirichlet problem with measurable data in the unit disk for the Poisson equations We start with the following analog of the known Luzin theorem on the primitive, see e.g. Theorem VII(2.3) in [67], in terms of logarithmic capacity. 132 To the theory of semi-linear equations in the plane Proposition 3. [19]. Let ϕ : [a, b] → R be a measurable function with respect to logarithmic capacity. Then there is a continuous function Φ : [a, b] → R with Φ′(x) = ϕ(x) q.e. on (a, b). Furthermore, Φ can be chosen with Φ(a) = Φ(b) = 0 and \|Φ(x)\| ≤ ε under arbitrary prescribed ε > 0 for all x ∈ [a, b]. As a consequence of Proposition 3, we obtain the following statement. Proposition 4. Let ϕ : ∂D → R be a measurable function with respect to logarithmic capacity. Then there is a continuous function Φ : ∂D → R such that Φ′(eit) = ϕ(eit) q.e. on R. The Poisson–Stieltjes integral ΛΦ(z) := 1 2π π∫ −π Pr(ϑ − t) dΦ(eit) , z = reiϑ, r < 1 , ϑ ∈ R (8.1) is well-defined for arbitrary continuous functions Φ : ∂D → R, see e.g. Section 3 in [66]. Directly by the definition of the Riemann–Stieltjes integral and the Weierstrass type theorem for harmonic functions, see e.g. Theorem I.3.1 in [29], ΛΦ is a harmonic function in the unit disk D := {z ∈ C : \|z\| < 1} because the function Pr(ϑ− t) is the real part of the analytic function Aζ(z) := ζ + z ζ − z , ζ = eit, z = reiϑ , r < 1 , ϑ and t ∈ R . (8.2) Next, by Theorem 1 in [66] we have the following useful conclusion. Proposition 5. Let ϕ : ∂D → R be a measurable function with respect to logarithmic capacity and Φ : ∂D → R be a continuous function with Φ′(eit) = ϕ(eit) q.e. on R. Then ΛΦ has the angular limit lim z→ζ ΛΦ(z) = ϕ(ζ) q.e. on ∂D . (8.3) Thus, by Lemma 3 and Proposition 5 and the known Poisson formula, see e.g. I.D.2 in [49], we come to the following result on the existence, regularity and representation of solutions for the Dirichlet problem with measurable data to the Poisson equation in the unit disk D. We assume that the charge density g is extended by zero outside of D in the next theorem. Theorem 6. Let a function ϕ : ∂D → R be measurable with respect to the logarithmic capacity and let a continuous function Φ correspond V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 133 to ϕ by Proposition 4. Suppose that a function g : D → R is in the class Lp(D) for p > 1. Then the following function in D U := Ng − PN∗ g + ΛΦ , N∗ g := Ng\|∂D , (8.4) belongs to the class W 2,p loc (D), satisfies the Poisson equation △U = g a.e. in D and has the angular limit lim z→ζ U(z) = ϕ(ζ) q.e. on ∂D . (8.5) Moreover, U ∈ W 1,q loc (D) for some q > 2 and U is locally Hölder continuous. Furthermore, U ∈ C1,α loc (D) with α = (p − 2)/p if g ∈ Lp(D) for p > 2. Remark 8. Note that by the Luzin result, see also Theorem 2 in [66], the statement of Theorem 6 is valid in terms of the length measure as well as the harmonic measure on ∂D. 9. The Dirichlet problem with measurable data in almost smooth domains We say that a Jordan curve Γ in C is almost smooth if Γ has a tangent q.e. Here it is said that a straight line L in C is tangent to Γ at a point z0 ∈ Γ if lim sup z→z0,z∈Γ dist (z, L) \|z − z0\| = 0 . (9.1) In particular, Γ is almost smooth if Γ has a tangent at all its points except a countable set. The nature of such Jordan curves Γ is complicated enough because the countable set can be everywhere dense in Γ. Given a domain D in C, kD(z, z0) denotes the quasihyperbolic dis- tance, kD(z, z0) := inf γ ∫ γ ds d(ζ, ∂D) , (9.2) introduced in the paper [24]. Here d(ζ, ∂D) denotes the Euclidean dis- tance from the point ζ ∈ D to ∂D and the infimum is taken over all rectifiable curves γ joining the points z and z0 in D. Next, it is said that a domain D satisfies the quasihyperbolic boundary condition if kD(z, z0) ≤ a ln d(z0, ∂D) d(z, ∂D) + b ∀ z ∈ D (9.3) 134 To the theory of semi-linear equations in the plane for constants a and b and a point z0 ∈ D. The latter notion was intro- duced in [23] but, before it, was first applied in [6]. Remark 9. Given a Jordan domain D in C with the almost smooth boundary satisfying the quasihyperbolic boundary condition. By the Rie- mann theorem, see e.g. Theorem II.2.1 in [29], there is a conformal map- ping f : D → D that is extended to a homeomorphism f̃ : D → D by the Caratheodory theorem, see e.g. Theorem II.3.4 in [29]. More- over, f∗ := f̃ \|∂D, as well as f−1 ∗ , is Hölder continuous by Corollary to Theorem 1 in [6]. Thus, by Remark 7 a function ϕ : ∂D → R is mea- surable with respect to logarithmic capacity if and only if the function ψ := ϕ ◦ f−1 ∗ : ∂D → R is so. Set Φ := Ψ ◦ f∗ where Ψ : ∂D → R is a continuous function corresponding to ψ by Proposition 4. Proposition 6. Let D be a Jordan domain in C with the almost smooth boundary satisfying the quasihyperbolic boundary condition. Sup- pose that ϕ : ∂D → R is measurable with respect to logarithmic capacity and Φ : ∂D → R is the continuous function corresponding to ϕ by Re- mark 9. Then the harmonic function LΦ(z) := ΛΦ◦f−1 ∗ (f(z)) has the angular limit ϕ q.e. on ∂D. Proof. Indeed, by Remark 9 and Proposition 5 there is the angular limit lim w→ξ ΛΨ(w) = ψ(ξ) q.e. on ∂D . (9.4) By the Lindelöf theorem, see e.g. Theorem II.C.2 in [49], if ∂D has a tangent at a point ζ, then arg [f̃(ζ)− f̃(z)] − arg [ζ − z] → const as z → ζ . After the change of variables ξ := f̃(ζ) and w := f̃(z), we have that arg [ξ − w]− arg [f̃−1(ξ)− f̃−1(w)] → const as w → ξ . In other words, the conformal images of sectors in D with a vertex at ξ is asymptotically the same as sectors in D with a vertex at ζ. Thus, nontangential paths in D are transformed under f̃−1 into nontangential paths in D. Recall that firstly the almost smooth Jordan curve ∂D has a tan- gent q.e., secondly by Remark 6 the mappings f∗ and f−1 ∗ are Hölder continuous, and thirdly by Remark 7 they transform sets of logarithmic capacity zero into sets of logarithmic capacity zero. Thus, (9.4) implies the desired conclusion. V. Gutlyanskĭı, O. Nesmelova, V. Ryazanov 135 Finally, by Lemma 3, Proposition 6 and the Poisson formula, we come to the following result on the existence, regularity and representation of solutions for the Dirichlet problem with measurable data to the Poisson equation in the Jordan domains. We assume here that the charge density g is extended by zero outside of D in the next theorem. Theorem 7. Let D be a Jordan domain in C with the almost smooth boundary satisfying the quasihyperbolic boundary condition, a function ϕ : ∂D → R be measurable with respect to logarithmic capacity and let a continuous function Φ correspond to ϕ by Remark 9. Suppose that a function g : D → R is in the class Lp(D) for p > 1. Then the following function in D U := Ng − DN∗ g + LΦ , N∗ g := Ng\|∂D , (9.5) belongs to the class W 2,p loc (D), satisfies the Poisson equation △U = g a.e. in D and has the angular limit lim z→ζ U(z) = ϕ(ζ) q.e. on ∂D . (9.6) Moreover, U ∈ W 1,q loc (D) for some q > 2 and U is locally Hölder continuous. Furthermore, U ∈ C1,α loc (D) with α = (p− 2)/p if g ∈ Lp(D) for p > 2. Remark 10. Note that by the Luzin result, see also Theorem 3 in [66], the statement of Theorems 7 is valid in terms of the length mea- sure on rectifiable ∂D. Indeed, by the Riesz theorem length f−1 ∗ (E) = 0 whenever E ⊂ ∂D with \|E\| = 0, see e.g. Theorem II.C.1 and Theo- rems II.D.2 in [49]. 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Veron, Local and global aspects of quasilinear degenerate elliptic equations, Quasilinear elliptic singular problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ., 2017. [74] S. K. Vodopyanov, A. Ukhlov, Sobolev spaces and (P; Q)-quasiconformal map- pings of Carnot groups // Sib. Mat. Zh., 39 (1998), No. 4, 665–682. Contact information Vladimir Gutlyanskĭı Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slavyansk, Ukraine E-Mail: vgutlyanskii@gmail.com Olga Nesmelova Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slavyansk, Ukraine E-Mail: star-o@ukr.net Vladimir Ryazanov Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slavyansk, Ukraine, Bogdan Khmelnytsky National University of Cherkasy, Cherkasy, Ukraine E-Mail: Ryazanov@nas.gov.ua, vl.ryazanov1@gmail.com

To the theory of semi-linear equations in the plane

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