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To a nonlocal generalization of the Dirichlet problem. (English) Zbl 1108.35040

Denote by \( \Omega = \{(x _{1}, x _{2}):0 < x _{k}<1,k=1,2\}\) and for \( \xi \in (0,1]\) by \( \rho \) the weight function \( \rho =(x _{1}/\xi ) ^{\varepsilon }\) if \( x _{1} \leq \xi \), respectivley \( \rho (x)=1\) if \( x _{1} > \xi \). Further, let \( {\mathcal L} \) be the operator \( {\mathcal L} u= \sum _{i,j=1 } ^{2 } (\partial / \partial x _{i}) ( a _{ij} \partial u/ \partial x _{j}) - a _{0} u\) and consider the non-local operator \( \ell (u)( x _{2}) = \int_{ 0}^{ \xi } (\varepsilon x _{1} ^{\varepsilon -1 }/ \xi ^{\varepsilon })u(x)\,d x _{1}\).
The main aim of the paper is to study the problem \( {\mathcal L} u=f\), \(u(x)=0 \) on \( x \in \partial \Omega \setminus \{(0, x _{2}): 0 < x _{2} < 1\}\), \( \ell (u)( x _{2})=0\), for \( x _{2}\in (0,1)\). The right hand side \(f\) and the solution \(u\) of the problem are assumed to lie in weighted Sobolev spaces asociated with the weight \( \rho \) and the coefficients of the operator satisfy a number of conditions, among which we mention \( a _{1j}\in {\mathcal L} ^{\infty } ( \Omega )\), \( a _{21}=\)const, \(0 \leq x _{1} ^{\varepsilon }(\partial ( x ^{1- \varepsilon } a _{22})/ \partial x _{1} \in {\mathcal L} ^{\infty } ((0, \xi ) \times(0,1)\), and \( \sum _{i,j } ^{ } a _{ij} t _{i} t _{j} \geq c( t _{1} ^{2} + t _{2} ^{2})\), a.e. in \( \Omega \), \(c >0\).

MSC:

35J25 Boundary value problems for second-order elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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References:

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