On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points
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irk-123456789-1692682020-06-10T01:26:03Z On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points Borsuk, M.V. Dobrowolski, M. 1999 Article On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points / M.V. Borsuk, M. Dobrowolski // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 29-34. — Бібліогр.: 8 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169268 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
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Article |
| author |
Borsuk, M.V. Dobrowolski, M. |
| spellingShingle |
Borsuk, M.V. Dobrowolski, M. On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points Нелинейные граничные задачи |
| author_facet |
Borsuk, M.V. Dobrowolski, M. |
| author_sort |
Borsuk, M.V. |
| title |
On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points |
| title_short |
On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points |
| title_full |
On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points |
| title_fullStr |
On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points |
| title_full_unstemmed |
On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points |
| title_sort |
on the behavior of solutions of the dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/169268 |
| citation_txt |
On the behavior of solutions of the Dirichlet problem for a class of degenerate elliptic equations in the neighborhood of conical boundary points / M.V. Borsuk, M. Dobrowolski // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 29-34. — Бібліогр.: 8 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT borsukmv onthebehaviorofsolutionsofthedirichletproblemforaclassofdegenerateellipticequationsintheneighborhoodofconicalboundarypoints AT dobrowolskim onthebehaviorofsolutionsofthedirichletproblemforaclassofdegenerateellipticequationsintheneighborhoodofconicalboundarypoints |
| first_indexed |
2025-07-15T04:01:37Z |
| last_indexed |
2025-07-15T04:01:37Z |
| _version_ |
1837684069005000704 |
| fulltext |
ON THE BEHAVIOR OF SOLUTIONS OF THE
DIRICHLET PROBLEM FOR A CLASS OF
DEGENERATE ELLIPTIC EQUATIONS IN THE
NEIGHBORHOOD OF CONICAL BOUNDARY POINTS
c© M.V.Borsuk*, M. Dobrowolski
1. Introduction: Preliminaries.
Let G ⊂ Rn be a bounded domain with bounary ∂G, which is smooth for x ∈ ∂G\{0}.
For x ∈ Rn, we denote the spherical coordinates by (r, ω) = (r, ω1, . . . , ωn−1) with
r = |x|, ω ∈ Sn−1. It is assumed that G coincides in the neighborhood of 0 with a cone
with opening Ω ⊂ Sn−1. More precisely, using the notation
Gb
a = G ∩ {(r, ω) : a < r < b, ω ∈ Ω} ⊂ Rn for a ≥ 0, b > 0,
we assume that Gd
0 is identical to the corresponding cone for a
d > 0. We denote also Γb
a = ∂G ∩ {(r, ω) : a < r < b, ω ∈ ∂Ω} ⊂ Rn for a ≥ 0, b > 0,
— lateral surface of Gb
a; Ωρ = G ∩ {|x| = ρ}; Gε = G \Gε
0, Γε = ∂G \ Γε
0, ∀ε > 0.
We consider the Dirichlet problem (DP):{
Lmu := −div (|∇u|m−2∇u) = −a0(x)u|u|q−1 + f(x) in G,
u(x) = 0 on ∂G \ {0},
where 1 < m < ∞, q > 0 and a0(x) ≥ 0, f(x) are measurable functions in G.
Let Lp(G) and W k,p(G), p ≥ 1, be the usual Lebesgue and Sobolev spaces. W 1,p
0 (G)
denotes the space of functions in W 1,p(G) that vanish on ∂G in the sense of traces. We
define the wight space V k
p,α(G) as the space of functions with finite norm
‖u‖V k
p,α(G) =
(
∫
G
∑
|β|≤k
rp(|β|−k+ α
2 )|Dβu|pdx
) 1
p
, p ≥ 1,
where k ≥ 0 is integer.
Definition. A function u(x) is called a generalized solution of (DP), if
u ∈ W 1,m(Gε) ∩ Lq+1(Gε) ∀ε > 0 satisfies
∫
G
{|∇u|m−2uxiηxi + a0(x)u|u|q−1η − f(x)η} dx = 0 (II)
for any η ∈ W 1,m(G) ∩ Lq+1(G) having a compact in G support, and u(x) = 0 x ∈
Γε ∀ε > 0 in the sense of traces.
*The paper was written while the first author was visiting the University of Würzburg. Financial
support of the DAAD is gratefully acknowledged.
A function u(x) is called a weak solution of (DP), if u ∈ W 1,m
0 (G) ∩Lq+1(G) and
satisfies (II) for all η ∈ W 1,m
0 (G) ∩ Lq+1(G).
The properties of generalized solutions of (DP) in the neighborhood of isolated sin-
gularities are studied by many authors (see e.g. [8] and the literature cited there). The
behavior of solutions near conical boundary points are treated only in special cases: in
[7], [4] for a0(x) ≡ 0, in [1] for bounded solutions, in [5], [2] for m = 2. The purpose of
the present article is to extend the results of [5], [2] to the more general quasilinear case
m 6= 2.
The first eigenvalue problem which characterizes the singular behavior of solutions
of (DP) can be derived by inserting in Lmv = 0 functions of the form v = rλφ(ω) which
leads to
D(λ, φ) = 0 in Ω, φ = 0 on ∂Ω, (EVP1)
where D(λ, φ) = −divω{(λ2φ2 + |∇ωφ|2)m−2
2 ∇ωφ}−
−λ{λ(m− 1) + n−m}(λ2φ2 + |∇ωφ|2)m−2
2 φ.
For this eigenvalue problem, there exists a solution (λ0, φ) such that
λ0 > max
{
0, m−n
m−1
}
, φ > 0 in Ω, φ2 + |∇ωφ|2 > 0 in Ω, (see [7], [4]); λ0 = λ0(Ω)
is the smallest positive eigenvalue of (EVP1).
For technical reasons, we introduce a further eigenvalue problem,
−divω(|∇ωψ|m−2∇ωψ) = µ|ψ|m−2ψ in Ω, ψ = 0 on ∂Ω. (EVP2)
From the direct method of the calculus of variations we know that there exists a
solution (µ, ψ) of (EVP2) with µ > 0 and ψ > 0 in Ω. From the corresponding variational
principle we derive the so–called Wirtinger’s inequality∫
Ω
|ψ|m dω ≤ 1
µ(m)
∫
Ω
|∇ωψ|m dω ∀ψ ∈ W 1,m
0 (Ω)
with a sharp constant 1
µ(m) . We set µ0 = µ(n); µ0 = µ0(Ω) is the smallest positive
eigenvalue of (EVP2) under m = n.
2. Integral estimates of solutions.
The aim of this section is to present integral estimates for the solutions of (DP).
Moreover, the weak comparison principle is not used in the proof, so that it may also
apply to the case of elliptic systems.
Theorem 2.1. Let a0(x) ∈ L m
m−1−q
(G), if 0 < q < m − 1 and 0 < a0 ≤ a0(x) ≤ a1
(a0, a1 − const.), if q ≥ m − 1. Let f(x) ∈ V 0
m
m−1 ,2(G). Then the weak solution of the
problem (DP) u(x) ∈ V 1
m,0(G) and it holds the inequality∫
G
(|∇u|m + r−m|u|m + a0(x)|u|1+q
)
dx ≤ c(n,G)
∫
G
|rf | m
m−1 dx.
Proof. The assertion arises from (II) with η(x) = u(x)Θ
(
|x|
ε
)
∀ε > 0, where Θ(t) be nonnegative infinitely differentiable function and such that
Θ(t) = 0 for t < 1, Θ(t) = 1 for t > 2.
Corollary 2.2. Let m > n. Under suppositions of Theorem 2.1 a weak solution u(x)
of (DP) is the Hölder-continuous in G.
Theorem 2.3. Let m = n and let the following condition be satisfied:∫
Gρ
0
|rf |n/(n−1) dx ≤ cρκ . Let χ0 = 2
√
µ0
(1+µ0)(n−2)/n . Then for any weak solution of (DP) is
satisfied the bound
∫
Gρ
0
|∇u|n dx ≤
≤ c(n, µ0, Ω)
(ρ/d)χ0 , if χ0 < κ,
(ρ/d)χ0 ln
n
n−1 (d/ρ), if χ0 = κ, ρ ∈ (0, d),
(ρ/d)κ, if χ0 > κ.
Remark. It is well known that if m = n = 2, then µ0 = λ2
0 = π2
ω2
0
, where ω0 is the
quantity of the angle with the vertex 0. In this case the assertion of Theorem was
proved in [2] (see Theorem 2.2 in it).
The proof of the theorem will be carried out due to following lemma.
Lemma 2.4. Let 1 < m ≤ n. For any function u ∈ W 1,m
0 (G) we have∫
Ω
{
ρuur + n−m
2 u2
} |∇u|m−2 dω ≤ ρ2
χ
∫
Ω
|∇u|m dω,
where χ = m−n+
√
4µ0+(n−m)2
(1+µ0)(m−2)/m .
Proof. Lemma is proved by anology with inequality (42) [3] and with the help of the
Young inequality.
Remark. For m = n = 2 the constant χ is sharp.
Proof of the theorem 2.3. Let V (ρ) =
∫
Gρ
0
|∇u|ndx. ¿From (DP) and Lemma 2.4 it follows
that V (ρ) satisfies the differential inequality V (ρ) ≤ ρ
χ0
V ′(ρ)+cρκ
n−1
n V
1
n (ρ). In view of
Theorem 2.1 as an initial condition for the differential inequality we can use V0 = V (d) ≤∫
G
|∇u|n dx ≤ c
∫
G
|rf |n/(n−1) dx. On putting W (ρ) = V
n−1
n (ρ), we obtain differential
inequality for W (ρ) :
{
W (ρ) ≤ n
n−1
ρ
χ0
W ′(ρ) + cρκ
n−1
n , 0 < ρ < d
W (d) = V
n−1
n
0
Solwing the Cauchy problem for the corresponding equation, we get
W ∗(ρ) =
(ρ
d
)χ0
n−1
n
(
V
n
n−1
0 +
+ κχ0
n−1
n ln d
ρ , if χ0 = κ,
d
n−1
n
(κ−χ0)−ρ
n−1
n
(κ−χ0)
κ−χ0
, if χ0 6= κ
.
It is well known that the solution of differential inequality can be estimated by the
solution W ∗(ρ) of the corresponding equation: W (ρ) ≤ W ∗(ρ) and hence we obtain
finally the required estimate. The theorem 2.3 is proved.
Lemma 2.5. Let q > m− 1, 0 < a0 ≤ a0(x) ≤ a1 (a0, a1 - const). Let |f(x)| ≤ k1|x|β ,
x ∈ Gd
0, where β > −1 if m > n, β > −m if m ≤ n. Then for any generalized solution
u(x) of (DP) are hold the inequalities
||u||p;Gρ
ρ/2
≤ c(a0, m, n, p, q, k1)ρ
n
p− m
q−m+1 ∀p > m,
∫
Gρ
ρ/2
(|∇u|m + |u|1+q
)
dx ≤ c(a0,m, n, q, k1)ρn− (1+q)m
1+q−m .
Proof. Desired inequalities are obtain from (II) with η(x) =
= |u|tsgnuζs(|x|) under suitably chosen numbers t ≥ 1, s > 0 and the cut-off function
ζ(r).
Corollary 2.6. Let q > mn
n−m − 1, 1 < m < n and the hypothesis of the Lemma 2.5
about the functions a0(x), f(x) are hold. Then for any generalized solution u(x) of (DP)
the inequality ∫
Gρ
0
(|∇u|m + r−m|u|m + |u|1+q
)
dx ≤ c(a0, n, m, q, k1, d),
∀ρ ∈ (0, d) is valid. Moreover, if β > −n
s with some s > n
m and λ0 < β+m
m−1 , then u(x)
is the Hölder-continuous in G.
Proof. The properties of u(x) are proved in Theorem 7.1 of chapt. IV and Theorem 2.2
of chapt. IX [6] in virtue Lemma 2.5.
3. A solvability property of the operator D from (EVP1).
In order to construct a barrier function which can be used in the weak comparison
principle, a solvability property of the operator D associated to the eigenvalue problem
(EVP1) is proved.
Theorem 3.1. For 0 ≤ λ < λ0 there exists a solution φ of the problem
D(λ, φ) = 1 in Ω, φ = 0 on ∂Ω, (3.1)
with φ > 0 in Ω.
This theorem will be proved in a sequence of lemmas. In the proofs of these lemmas
we frequently use the fact that every solution (λ, φ) of (3.1) corresponds to a solution
of Lm(rλφ) = r(λ−1)(m−1)−1 in Gd
0, which, by local regularity of the Pseudo–Laplace
equation, implies that φ ∈ Cβ(Ω) ∩W 1+ε,m
0 (Ω) for β, ε > 0.
Lemma 3.2. The problem (3.1) is solvable for all 0 ≤ λ < λ0.
Proof. We prove that Fredholm’s alternative holds for (3.1) in the sense that, if (3.1)
is not solvable, then λ is an eigenvalue of D. For this purpose, we choose a sufficiently
large α ∈ R such that the problem D(λ, φ) + α|φ|m−2φ = g in Ω, φ = 0 on ∂Ω is
uniquely solvable for all g ∈ H−1,m′
(Ω), 1
m + 1
m′ = 1 and denote the solution operator
by φ = Φg. By the regularity of D, Φ: Cβ(Ω) → Cβ(Ω) is a compact operator for a
β > 0. Moreover, Φ is homogeneous of degree 1
m−1 . The problem D(λ, φ) = f in Ω,
φ = 0 on ∂Ω is then equivalent to
φ− αFφ = Φf, (3.2)
where Fφ = Φ(|φ|m−2φ) is compact and homogeneous of degree 1. The operator Id−αF
is studied on the unit ball B1 = {φ ∈ Cβ(Ω) : ||φ||Cβ ≤ 1}. If 0 /∈ (Id− αF )(∂B1) then
K. Borsuk’s theorem states that (3.2) is solvable for sufficiently small f. Since (3.2) is
equivalent to D(λ, φ) = f and D(λ, ·) is homogeneous of degree m − 1 we can solve
D(λ, φ) = f for all f.
Lemma 3.3. Let (λ, φ) be a solution of (3.1). Then φ(ω) 6= 0 for all ω ∈ Ω.
Proof. Let K = {(r, ω) : 1 < r < 2, ω ∈ Ω}. If (λ, φ) is a solution of (3.1) then
v = rλφ(ω) solves Lmv = r(λ−1)(m−1)−1 in K, v = 0 on (1, 2)×∂Ω, v = crφ for r = 1, 2.
Assume that φ(ω0) = 0 for ω0 ∈ Ω. We apply the weak comparison principle on the
domain K using the function v. It follows that every solution of Lmu = f in K, u = v
on ∂K with f ∈ C∞0 (K) satisfies u(r, ω0) ≤ 0 which is a contradiction.
Lemma 3.4. For sufficiently small λ ≥ 0 the solution of (3.1) is unique and satisfies
φ > 0 in Ω.
Proof. The operator D(0, ·) is strictly monotone on W 1,m
0 (Ω). Hence, problem (3.1) is
uniquely solvable and the comparison principle implies φ > 0 in Ω. Since D(λ, ·) is
continuous in λ the conclusion also holds for sufficiently small λ ≥ 0.
Lemma 3.5. There exists a constant c = c(λ1) such that ||φ||1,m ≤ c for all solutions
(λ, φ) of (3.1) satisfying 0 ≤ λ ≤ λ1 < λ0.
Proof. Assuming the converse we obtain a sequence (λi, φi) solving (3.1) with λi → λ,
||φi||1,m →∞. For the normalized functions φ̃i = φi
||φi||1,m
we obtain that D(λi, φ̃i) → 0
in W−1,m′
(Ω) and, by regularity, ||φ̃i||1+ε,m ≤ c. Hence, we can extract a subsequence
{φ̃ik
} such that φ̃ik
→ φ in W 1,m
0 (Ω) and D(λ, φ) = 0 with ||φ||1,m = 1. This contradicts
the fact that there is no eigenvalue of D in the interval [0, λ1].
Proof of Theorem 3.1. Lemma 3.5 implies a kind of continuity of solutions (λ, φ) in the
following sense. If λi → λ with 0 ≤ λi, λ < λ0, then there exists a subsequence {φik
}
such that φik
→ φ in Cα(Ω), where (λ, φ) is a solution of (3.1). Hence, by Lemmas 3.3
and 3.4 there exists a solution (λ, φ) with φ > 0 in Ω for all 0 ≤ λ < λ0.
4. Estimates of solutions for singular f .
In [7], [4] it is proved that the weak solution u ∈ W 1,m
0 (G) of (DP) can be bounded
by |u(x)| ≤ crλ0 , if a0(x) ≡ 0 and the condition |f(x)| ≤ crβ , β > (λ0 − 1)(m− 1)− 1
is satisfied. The proof of this is based on the weak comparison principle for (DP). Here
we shall obtain the estimates of modulus of the solutions of (DP). Let d > 0 be a small
fixed number. We also suppose that |f(x)| ≤ k1|x|β , β > −n
s with some s > n
m .
Observe that a function v = rαφ(ω) is a weak solution v ∈ W 1,m
0 , if φ(ω) is suffi-
ciently smooth and α > m−n
m . Since Lmv ∼ rα(m−1)−m and the right-hand side of (DP)
−a0(x)v|v|q−1 + f(x) ∼ rαq + rβ , hence we obtain that rα(m−1)−m ∼ rαq + rβ . This
arguments suggest the following theorems to us.
Theorem 4.1. Let 1 < m ≤ n, q > 0 be given. Let 0 ≤ a0(x) ≤ a1 = const. Let u(x)
be a weak solution of (DP). Then the following assertions are hold:
1) if λ0 < β+m
m−1 , then |u(x)| ≤ c0|x|λ0 , x ∈ Gd
0;
2) if λ0 > β+m
m−1 , then |u(x)| ≤ c0|x|
β+m
m−1 , x ∈ Gd
0;
Proof. The operator Lm,qu := Lmu + a0(x)u|u|q−1 satisfies the weak comparison prin-
ciple which states that Lm,qu ≤ Lm,qv in Gd
0, u ≤ v on ∂Gd
0 ⇒ u ≤ v in Gd
0. Then the
first assertion follows from the Theorem 2 [4]. To proof the second assertion we choose
the barrier function v(x) = A|x|λφ(ω) where A > 0, λ = β+m
m−1 < λ0 and (λ, φ) is the
solution of (3.1).
Theorem 4.2. Let 1 < m < n, q > m − 1 be given. Let 0 < a0 ≤ a0(x) ≤ a1, (a0, a1
— const). Let u(x) be any generalized solution of (DP). If λ0 < β+m
m−1 , q > mn
n−m − 1,
then |u(x)| ≤ c0|x|λ0 , x ∈ Gd
0.
Proof. Firstly, from the Lemma 2.5 with p →∞ we get the bound |u(x)| ≤ c|x| m
m−1−q .
Then we construct the barrier function and use the method by contradiction (just
similarly as in [5] ) in view of the Corollary 2.6.
References
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(Russian).
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E-mail: borsuk@moskit.art.olzstyn.pl
E-mail: dobro@mathematik.uni-wuerzburg.de
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