## 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

### Keywords:

elliptic equation; Dirichlet problem; VMO-coeficients
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