Boundary value problems in Lipschitz domains for equations with lower order coefficients
Georgios Sakellaris · Transactions of the American Mathematical Society · 2019
We use the method of layer potentials to study the R 2 R_2 regularity problem and the D 2 D_2 Dirichlet problem for second order elliptic equations with lower order coefficients in bounded Lipschitz domains. For R 2 R_2 we establish existence and uniqueness by assuming that L \mathcal {L} is of the form L u = − div ( A ∇ u + b u ) + c ∇ u + d u \mathcal {L}u=-\text {div}(A abla u+bu)+c abla u+du , where the matrix A A is uniformly elliptic and Hölder continuous, b b is Hölder continuous, and c , d c,d belong to Lebesgue classes and satisfy either the condition d ≥ div b d\geq \ \text {div}\,b or d ≥ div c d\geq \ \text {div}\,c in the sense of distributions. In particular, A A is not assumed to be symmetric, and there is no smallness assumption on the norms of the lower order coefficients. We also show existence and uniqueness for D 2 D_2 for the adjoint equations