G. Berikelashvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON A NONLOCAL GENERALIZATION OF THE DIRICHLET PROBLEM

Let Ω = {(x_{1}, x_{2}) : 0 < x_{k} < 1, k = 1, 2} be a unit square with a boundary Γ, and let Γ_{1} = {(0, x_{2}) : 0 < x_{2} < 1}, Γ_{∗} = Γ\Γ_{1}, ξ ∈ (0; 1], ε ∈ (0; 1).

Consider the nonlocal boundary value problem

Lu = f(x), x ∈ Ω, u(x) = 0, x ∈ Γ_{∗}, l(u) = 0, 0 < x_{2} < 1,  (1)

where

Lu := ∑_{i,j=1}^{2} \frac{∂}{∂x_{i}} (a_{ij} \frac{∂u}{∂x_{j}} ) − a_{0}u, l(u) := ∫_{0}^{ξ} β(x)u(x) dx_{1}, β(x) := εx_{1}^{ε−1}/ξ^{ε}

and the constant coefficients satisfying the following conditions

∑_{i,j=1}^{2} a_{ij}t_{i}t_{j} ≥ ν(t_{1}^{2} + t_{2}^{2}), ν > 0, a_{0} ≥ 0.

By L_{2}(Ω, ρ) we denote the weighted Lebesgue space; ρ(x) := (x_{1}/ξ)^{ε} for x_{1} < ξ, ρ(x) := 1 for x_{1} ≥ ξ.

We denote by W_{2}^{1}(Ω, ρ) the weighted Sobolev space with the norm

∥u∥_{W_{2}^{1}(Ω,ρ)} = (∥u∥_{L_{2}(Ω,ρ)}^{2} + ∥\frac{∂u}{∂x_{1}} ∥_{L_{2}(Ω,ρ)}^{2} + ∥\frac{∂u}{∂x_{2}} ∥_{L_{2}(Ω,ρ)}^{2})^{1/2}.

Define the subspace of the space W_{2}^{1}(Ω, ρ) which can be obtained by closing the set

C^{∗}^{∞}(\overline{Ω}) = \{u ∈ C^{∞}(\overline{Ω}) : supp u ∩ Γ_{∗} = ∅, l(u) = 0, 0 < x_{2} < 1\}

with the norm ∥ · ∥_{W_{2}^{1}(Ω,ρ)}. Denote it by W^{∗} _{2}^{1}(Ω, ρ).

Let the right-hand side f(x) in equation (1) be a linear continuous func-

tional on W^{∗} _{2}^{1}(Ω, ρ) which can be represented as

f = f_{0} + ∂f_{1}/∂x_{1} + ∂f_{2}/∂x_{2}, f_{k}(x) ∈ L_{2}(Ω, ρ), k = 0, 1, 2.

Theorem 1. Problem (1) has a unique solution from W^{∗} _{2}^{1}(Ω, ρ).

By passing to the limit ξ → 0 the nonlocal condition l(u) = 0 transforms to u(0, x_{2}) = 0, while Theorem 1 to the well-known theorem on the existence and uniqueness of a solution of the Dirichlet problem. In this sense, the nonlocal problem (1) can be regarded as a generalization of the Dirichlet boundary value problem.