The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions
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Інститут прикладної математики і механіки НАН України
1999
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| Назва видання: | Нелинейные граничные задачи |
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| Цитувати: | The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions / Yu.A. Alkhutov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 3-7. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1692642020-06-10T01:26:10Z The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions Alkhutov, Yu.A. 1999 Article The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions / Yu.A. Alkhutov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 3-7. — Бібліогр.: 14 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169264 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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English |
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Alkhutov, Yu.A. |
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Alkhutov, Yu.A. The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions Нелинейные граничные задачи |
| author_facet |
Alkhutov, Yu.A. |
| author_sort |
Alkhutov, Yu.A. |
| title |
The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions |
| title_short |
The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions |
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The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions |
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The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions |
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The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions |
| title_sort |
wiener test for quasilinear elliptic equations with non – standardgrowth conditions |
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Інститут прикладної математики і механіки НАН України |
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1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/169264 |
| citation_txt |
The Wiener test for quasilinear elliptic equations with non – standardgrowth conditions / Yu.A. Alkhutov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 3-7. — Бібліогр.: 14 назв. — англ. |
| series |
Нелинейные граничные задачи |
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2025-07-15T04:01:26Z |
| last_indexed |
2025-07-15T04:01:26Z |
| _version_ |
1837684056751341568 |
| fulltext |
THE WIENER TEST FOR QUASILINEAR ELLIPTIC
EQUATIONS WITH NON - STANDARD GROWTH CONDITIONS
c© Yu.A.Alkhutov
Let D be a bounded domain in Rn. Consider in D the quasilinear partial dirrefential
equation
Lu =
n∑
i=1
∂
∂xi
(
|∇u|p(x)−2 ∂u
∂xi
)
= 0 (1)
where p(x) is a measurable in D function and
1 < p1 ≤ p(x) ≤ p2 < ∞.
For strict definition of solution for equation (1) we introduce some classes of functions.
Let
V (D) =
{
ψ(x) : ψ ∈ W 1
1 (D) , |∇ψ|p ∈ L1 (D)
}
,
where by W 1
1 (D) denote the classical Sobolev space with the norm
‖ u ‖W 1
1 (D)=
∫
D
(|u|+ |∇u|) dx.
Under the class V0 (D) we shall understood the subset of V (D) such that for any
u ∈ V0 (D) exists a sequence of functions uj ∈ V (D) with compact supports in D
satisfying relations
lim
j→∞
‖uj − u‖W 1
1 (D) = 0, lim
j→∞
∫
D
|∇uj |p dx =
∫
D
|∇u|p dx. (2)
This research was partially supported by Russin Foundation for Basic Research under grant No 96
- 01 -00443
A function u ∈ V (D) we shall call a solution of equation (1) if for every test function
ψ ∈ V0 (D) realized the integral identity
n∑
i=1
∫
D
|∇u|p(x)−2
uxi
ψxi
dx = 0.
The solution of class Vloc (D) may be define analogously.
The important question about density of smooth functions in V (D) was investigated
by Zhikov [1-4]. He proved that under assumption
|p(x)− p(y)| ≤ const
ln 1
|x−y|
, |x− y| ≤ 1/2. (3)
for any u ∈ V (D) exists a sequence uj ∈ C∞ (D) such that (2) holds.
The condition (3) is exact for this assertion. As showes the countrexample [3] the
previous is false for p(x) having the modulus of continuity |ln t|ε−1 for any ε ∈ (0, 1).
From countrexample [3] it follows that the solution of equation (1) may be not Hölder
continuous in D without condition (3). This result stimulated investigation of Hölder
continuity for solutions (1).
The next result was obtained by Xianling Fan [5] and author of present paper [6] by
different methods.
Theorem 1. If condition (3) satisfies then any solution u ∈ Vloc (D) of the equation
(1) is Hölder continuous in any compact subset of D.
The proof in [6] based on Trudinger’s weak Harnack inequality [7]. We shall formulate
it for supersolutions of equation (1): such functions u that Lu ≤ 0 in generalized sence.
Further will be make use of standard notation Bx0
r for open ball with radius r and
center x0, p0 = p(x0).
Theorem 2. (Weak Harnack inequality.) Let u ∈ V (Bx0
4r ) be a nonnegative bounded
supersolution of equaiton (1) in Bx0
4r and condition (3) satisfies. If p0 ≤ n and q > 0
such that q(n− p0) < n(p0 − 1) then for sufficiently small r ≤ r0(n, p)
∫
B
x0
2r
uq dx
1/q
≤ c(n, p, q, M)rn/q
(
inf
B
x0
r
u + r
)
,
where M = sup
B
x0
4r
u.
The weak Harnack inequality for supersolutions allowes to investigate a boundary
behavior of solutions of the Dirichlet problem. We shall return to this question just a
little later.
Now consider equation (1) with piecewise continuous function p(x).
Theorem 3. Let D1 and D2 are open subsets of D with common Lipschitz bound-
ary Σ and D = D1 ∪ D1. If condition (3) satisfies in every Di, i = 1, 2, and p(x)
have nonzero jump on Σ then any solution u ∈ Vloc (D) of the equation (1) is Hölder
continuous in any compact subset of D.
The piecewise constant exponents p(x) was investigated by Acerbi and Fusco [8].
Consider a question about the continuity at a boundary point x0 ∈ ∂D of solutions
of the equation (1). At first using the construction of Kondratiev and Landis [9] define
Wiener’s generalized solution of the problem
Luf = 0 in D, uf |∂D = f (4)
with continuous on the boundary ∂D function f .
The construction is based on the maximum principle and the solvability of the Dirich-
let problem
Lu = 0 in D, (u− h) ∈ V0 (D) , h ∈ V (D) . (5)
The function u ∈ V (D) satisfying the equation (1) in the sence of integral identity and
the boundary condition (u− h) ∈ V0 (D) is called the solution of the Dirichlet problem
(5). Unique solvability of this problem follows from the results of Zhikov [4]. The proof
based on the fact that integral identity is the Euler equation for the corresponding
variational problem.
Before to formulate a maximum principle we introduce the next notion. We shall say
that the function v ∈ V (D) is nonnegative in the sence of V (D) (notation: v ≥ 0) on
compact subset E ⊂ D, if for function u = inf(v, 0) exists a sequence uj ∈ V (D) such
that uj = 0 in a neghborhood of D ∩E and holds (2). If u, v ∈ V (D) and u− v ≥ 0 on
E in the sence of V (D) we shall say that u ≥ v on E in the sence of V (D).
Maximum principle. If u and v are two solutions belonging to V (D) of the equa-
tion (1) in D and u ≥ v on ∂D in the sence of V (D), then u ≥ v almost everywhere in
D.
For construction of the Wiener solution for the Dirichlet problem (4) we shall continue
the boundary function f on Rn continuously. Continued function as before denote by
f . By {fj} denote a sequence of infinitely differentiable functions such that restrictions
of {fj} on D converges uniformly to f in D. Let us solve the Dirichlet problem
Luj = 0 in D, (uj − fj) ∈ V0 (D) .
By maximum principle the sequence {uj} converges uniformly in compact subsets of
the domain D to some function uf . This function does not depend on the methods of
approximation and continuation of f and is called the Wiener solution of the Dirichlet
problem (4). It is not difficult to show that uf ∈ Vloc (D) satisfies equation (1). If
h ∈ V (D) ∩ C
(
D
)
then the Wiener solution uf ∈ Vloc (D) of the problem (4) with the
boundary function f = h|∂D coincides with the solution of the problem (5).
Definition 1. The boundary point x0 ∈ ∂D is called regular if for any continuous
on ∂D function f the Wiener solution uf of the problem (4) is continuous at x0.
The criterion of regularity of a boundary point for Laplace equation was proved by
Wiener [10]. This criterion is characterized by so call Wiener test. In the fundamental
work Littman, Stampacchia, and Weinberger [11] showed that the same Wiener test
identifies the regular boundary points whenever a uniformly elliptic linear operator with
bounded measurable coefficients. The sufficient condition of regularity of the boundary
point for p - Laplace equation (equation (1) with p = const) was estsblished by Maz’ya
[12]. He also received the estimate of modulus of continuity for solution near a regular
boundary point. Later Gariepy and Ziemer [13] extended this result to a very general
equation. For these equations some necessary condition of reqularity close to sufficient
one was proved by Skrypnik [13]. The criterion of regularity of a boundary point for p
- Laplace equation was obtained by Kilpeläinen and Malý [14].
Let us define a notion of Vp - capasity. Further we assume that p(x) = p(x0) = p0 in
Rn \D.
Definition 2. Let E be a compact subset of Br. The number
Cp (E, Br) = inf
∫
Br
|∇ψ|p(x)
dx,
where ψ runs through all ψ ∈ V0 (Rr) with ψ ≥ 1 on E in the sence of V (Rr) is called
Vp - capasity of the set E with respect to Br.
Put
γ
V
(t) = Cp
(
B̄x0
t \D,Bx0
2t
)
tp0−n.
Theorem 4. If condition (3) satisfies and p0 ≤ n then for regularity of a boundary
point x0 ∈ ∂D it is necessary and sufficiently to have
∫
0
[γ
V
(t)]1/(p0−1)
t−1 dt = ∞. (6)
Let us give the estimate of modulus of continuity for solution (5) near a boundary point
x0 ∈ ∂D.
Theorem 5. Let condition (3) satisfies and uf be the Wiener solution of the Dirichlet
problem (5). Then for ρ ≤ ρ0(n, p), r ≤ ρ/4
osc
D∩B
x0
r
uf ≤ c osc
D∩B
x0
ρ
f + c osc
∂D
f exp
(−θ
ρ∫
r
[γ
V
(t)]1/(p0−1)
t−1 dt
)
,
if p0 ≤ n, or
osc
D∩B
x0
r
uf ≤ c osc
D∩B
x0
ρ
f + c osc
∂D
f (r/ρ)1−n/p0 ,
if p0 > n. Here c and θ are positive constants dependent only on n,p and max
∂D
|f |.
Let us formulate a geometric conditions of regularity of a boundary point. We shall
assume that x0 ∈ ∂D is coincides with the origin O and the exterior of D in the
neighborhood of O contain the domain
0 < xn < a,
n−1∑
i=j+1
x2
i < g2(xn), |xi| < a, i = 1, ..., j
,
where g(t) is a continuous increasing function such that tα < g(t) < t.
Theorem 6. The condition (6) is satisfied if
∫
0
(
g(t)
t
)n−1−j−p0
p0−1
t−1 dt = ∞,
for p0 < n− 1− j, and if ∫
0
|ln g(t)|−1
t−1 dt = ∞,
for p0 = n− 1− j. In the case p0 > n− 1− j condition (6) is always satisfied.
Earlier the analogous result for p - Laplace equation was proved in [12].
Theorem 7. Let condition (3) satisfies and f be a Hölder continuous at x0 ∈ ∂D.
If the exterior of D contain a cone with the vertex at x0 then the generalized by Wiener
solution of the Dirichlet problem (5) is Hölder continuous at x0.
All results of the present paper are correct for equations
n∑
i,j=1
∂
∂xi
(
aij(x)|∇u|p(x)−2 ∂u
∂xj
)
= 0,
where aij(x) are measurable and bounded in D functions such that for x ∈ D, ξ ∈ Rn
λ−1 |ξ|2 ≤
n∑
i,j=1
aij(x)ξiξj ≤ λ |ξ|2 , λ = const > 0.
References
1. Zhikov V., Averaging of functionals of the calculus of variations and elasticity theory, Izvestiya
Akad. nauk SSSR. ser. mat. 50 (1986), no. 4, 4675 - 710.
2. Zhikov V., On Lavrent’ev effect, Dokl. Ross. Akad. nauk 345 (1995), no. 1, 675 - 710.
3. Zhikov V., On Lavrentiev’s Phenomenon, Russian Journal of Math. Physics 3 (1995), no. 3, 249 -
269.
4. Zhikov V., On Some Variational Problems, Russian Journal of Math. Physics 5 (1996), no. 1, 105
- 116.
5. Xianling Fan., A class of De Giorgi Type and Hölder Continuity of Minimizers of Variationals
with m(x) – Growth Condition, Lanzhou University, China (1995).
6. Alkhutov Yu.A., Harnack inequality and Hölder continuity of solutions of non - linear elliptic
equations with non - standard growth condition, Differentsial’nye Uravneniya (1997), no. 12 (to
appear).
7. Trudinger N. S., On Harnack type inequalities and their application to quasilinear elliptic equations,
Comm. Pure Appl. Math. 20 (1967), 721 - 747.
8. Acerbi E., Fusco N., A transmission problem in the calculus of variations, Calc. Var. 2 (1994), 1 -
16.
9. Kondratiev V.A., Landis E.M., The qualitative theory of linear partial differential equations of
the second order, Sovremennie problemy matematiki. Fundamental’nie napravleniya, VINITI, 32
(1988), 99 - 215.
10. Wiener N., Certain notions in potential theory, J. Math. Phys. 3 (1924), 24 - 51.
11. Maz’ya V.G., On the continuity at a boundary point of solutions of quasi - linear elliptic equations,
Vestnik Leningrad Univ. 3 (1976), 225 - 242.
12. Gariepy R., Ziemer W.P., A regularity condition at the boundary for solutions of quasilinear elliptic
equations, Arch. Rational Mech. Anal. 67 (1977), 25 - 30.
13. Skrypnik I.V., Methods of investigation of nonlinear elliptic boundary value problems, Moscow,
Nauka, 1990.
14. Kilpeläinen T., Malý J., The Wiener test and potential estimates for quasilinear elliptic equations,
Acta Math. 172 (1994), 137 - 161.
Vladimir State Pedagogical State University, Department of Math.,
prospect Stroiteley 11, Vladimir, 600024, Russia
E-mail adress: alkhutov@vgpu.elcom.ru
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