Unique Solvability of the Homogeneous Dirichlet Problem

ThisDirichlet problem chapterHomogeneous Dirichlet problem isSecond order elliptic differential operator devotedElliptic differential operator to the study of the homogeneous Dirichlet problem for a second order, uniformly elliptic differential operator with vanishing mean oscillation (VMO) coefficients inExistence and uniqueness theorem theExistence theorem frameworkUniqueness theorem of Sobolev spaces of \(L^{p}\) style. We proveVMO (vanishing mean oscillation) anVanishing mean oscillation (VMO) existence and uniqueness theorem for the Dirichlet problem (Theorem 15.1). Our proof is based on some interior and boundary a priori estimates for the solutions of problem (15.2) (Theorems 12.1 and 12.2). Both the interior and boundary a priori estimates are consequences of explicit representation formulas (13.1) and (14.1) for the solutions of problem (15.2) (Theorems 13.1 and 14.1) and also of the \(L^{p}\) -boundedness of Calderón–Zygmund singular integral operators and boundary commutators appearing in those representation formulas (Theorems 14.2 and 14.5).