Semilinear and Lp – Dirichlet Problem for Elliptic Operators with Functional Calculus of Dirac Operators

Abstract

We show results about existence multiplicity and estimates of the semilinear elliptic problems with mixed Dirichlet – Neumann boundary conditions. We establish the solvability of the Dirichlet problem on Lipschitz domains with small Lipschitz constants for elliptic divergence and non-divergence type operators on Lebesgue spaces. We discuss Neumann, Dirichlet and regularity problems for the divergence from elliptic equations in the half – space of L2 for small complex perturbation of coefficient matrix. We establish quadratic estimates for the Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix.