\(W^{2,2}\)-solvability of the Dirichlet problem for a class of elliptic equations with discontinuous coefficients
Summary: We study the Dirichlet problem for the second order elliptic equation \[ -\sum\limits_{i,j=1}^Na_{ij}(x)\frac{\partial^2u}{\partial x_i\partial x_j}=f(x) \] in a bounded regular domain \(\Omega\subset\mathbb{R}^N,\;N>2\). We assume that \(f\in L^{2}\) and that the coefficients \(a_{ij}\) are measurable and bounded functions with the first derivatives in the Marcinkiewicz class weak-\(L^{N}\) and having a sufficiently small distance to \(L^{\infty}\). Under these assumptions we prove the solvability of the problem in \(W^{2,2}\cap W_0^{1,2^\ast}\), where \(2^\ast=\frac{2N}{N-2}\). An higher integrability result for the gradient of the solution is achieved when \(f\in L^p,\;p>2\).
scientific article; zbMATH DE number 6903550
scientific article