Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. (English) Zbl 0942.35059

In the bounded domain \(\Omega\subset \mathbb{R}^n, n>1\), with smooth boundary \(\partial\Omega\) there is considered the Dirichlet problem: \[ Lu=\sum^n_{i,j=1}a_{ij}(x) D_{x_ix_j}u=f(x) \quad\text{a.e. in } \Omega,\quad u=0 \quad\text{on } \partial\Omega, \] where \(L\) is a uniformly elliptic operator with coefficients \(a_{ij}(x)\) belonging to \(\text{VMO}\cap L^\infty (\Omega)\) (VMO is the Sarason’s class of functions with vanishing mean oscillation), \(f(x)\) is an arbitrary function from the Morrey space \(L^{p,\lambda}(\Omega), 1<p<\infty, 0<\lambda<n\). It is proved the well-posedness of the mentioned problem in the space \(W^{2,p,\lambda}(\Omega)\cap W_0^{1,p}(\Omega)\).


35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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