The Boundary Behavior of a Solution to the Dirichlet Problem for the p -Laplacian with Weight Uniformly Degenerate on a
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
THE BOUNDARY BEHAVIOR OF A SOLUTION TO THE DIRICHLET PROBLEM FOR THE p-LAPLACIAN WITH WEIGHT UNIFORMLY DEGENERATE ON A PART OF DOMAIN WITH RESPECT TO SMALL PARAMETER Yu. A. Alkhutov
∗
A. G. and N. G. Stoletov Vladimir State University 87, Gor’kogo St., Vladimir 600000, Russia [email protected]
M. D. Surnachev Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences 4, Miusskaya sq., Moscow 125047, Russia [email protected]
UDC 517.9
We consider the Dirichlet problem for the p-Laplacian with weight and continuous boundary function in a domain D divided into two parts by the hyperplane Σ. The weight is equal to 1 in some part of the domain D and coincides with a small parameter ε in the other. We estimate the modulus of continuity for the solution at a boundary point x0 ∈ ∂D ∩ Σ with a constant independent of ε. Bibliography: 22 titles. Dedicated to the 80th anniversary of Vasilii Vasil’evich Zhikov
1
Introduction
In a bounded domain D ⊂ Rn , n 2, we consider the equation Lu = div ωε (x)|∇u|p−2 ∇u = 0, p = const > 1.
(1.1)
We assume that the domain D is divided into the parts D (1) = D ∩ {xn > 0} and D (2) = D ∩ {xn < 0} by the hyperplane Σ = {xn = 0} and ε, xn > 0, ε ∈ (0, 1]. (1.2) ωε (x) = ωε (xn ) = 1, xn < 0, Below, W 1,p (D) denotes the Sobolev space of functions that, together with all generalized first order derivatives, belong to Lp (D) and W01,p (D) is the closure of the set C0∞ (D) of compactly ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 3-17. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0183
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supported and infinitely differentiable functions in D in the W 1,p (D)-norm. We say that a function u ∈ W 1,p (D) is a solution to Equation (1.1) in D if the integral identity ωε |∇u|p−2 ∇u · ∇ϕ dx = 0 (1.3) D
holds for any test function ϕ ∈ W01,p (D). A function u ∈ W 1,p (D) is called a supersolution to Equation (1.1) in D if for all nonnegative ϕ ∈ W01,p (D) ωε |∇u|p−2 ∇u · ∇ϕ dx 0. (1.4) D
We consider the Dirichlet problem Lu = 0 in D,
u ∈ W 1,p (D), h ∈ W 1,p (D), (u − h) ∈ W01,p (D).
(1.5)
The solution to this problem coincides with the minimizer of the variational problem |∇ψ|p F (ψ), F (ψ) = ωε (x) dx. min p ψ−h∈W01,p (D) D
This paper is devoted to the study of boundary properties of solutions to the Dirichlet problem (1.6) Luf = 0 in D, uf |∂D = f, where f is continuous on ∂D. A solution to the problem (1.6) is defined as follows. Using the Tietze–Uryson theorem, we extend the boundary function f by continuity to Rn , preserving the same notation. We consider a sequence of infinitely differentiable functions fk in Rn , uniformly converging to f in D. We solve the Dirichlet problem ◦
Luk = 0 in D,
uk ∈ Wp1 (D), (uk − fk ) ∈Wp1 (D).
By the maximum principle, the sequence uk converges uniformly in D to a function u that belongs to the space W 1,p (D ) in an arbitrary subdomain D D and sati
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