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Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. (English) Zbl 0822.35036

The authors consider elliptic equations of the form \(Lu= \sum_{i,j=1}^ n a_{ij} {{\partial^ 2 u} \over {\partial x_ i \partial x_ j}}=f\) and show: If \(f\) belongs to the Morrey space \(L^{p, \lambda} (\Omega)\), \(u\in W^{2,p} (\Omega)\) is a solution of \(Lu=f\) then the second derivatives of \(u\) belong to \(L^{p, \lambda} (\Omega')\) and \(\| D^ 2 u\|_{L^{p, \lambda} (\Omega')}\) can be estimated by \(\| u\|_{L^{p, \lambda} (\Omega')}+ \| Lu\|_{L^{p, \lambda} (\Omega'')}\), where \(\Omega' \subset \subset \Omega'' \subset \subset \Omega\), \(1<p <+\infty\), \(0<\lambda< n\). If \(n- p< \lambda<n\) then the first derivatives of \(u\) belong to \(C^{0, \alpha} (\Omega')\) with \(\alpha= 1- {n\over p}+ {\lambda\over p}\), and \(\| Du\|_{C^{0, \alpha} (\Omega')}\) is delimited like \(\| D^ 2 u\|_{L^{p, \lambda} (\Omega')}\).

MSC:

35J15 Second-order elliptic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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