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The Dirichlet problem for elliptic equations in the plane. (English) Zbl 1121.35040

Let \(\Omega \) be a bounded open subset of \(R^2\) with boundary of class \(C^{1,1}\), and let \(p\in [1,\infty ]\). Consider in \(\Omega \) the differential operator \(L=-\sum a_{ij}\partial _i \partial _j +\sum a_i \partial _i +a\) and suppose that the coefficients of \(L\) satisfy the following hypotheses: \(a_{ij}=a_{ji} \in L^\infty (\Omega )\cap VMO (\Omega )\); \(\sum a_{ij}\psi _i \psi _j \geq \nu | \psi | ^2\), \(\nu >0\); \(a_i \in L^r(\Omega )\), \(r>2\), \(r\geq p\); \(a\in L^p(\Omega )\), \(a\geq 0\). The unique solvability of the Dirichlet problem \(u\in W^{2,p}(\Omega )\cap W^{1,p}_0(\Omega )\), \(Lu=f \in L^p(\Omega )\) is proved.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
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