Boundary Behavior of p-Laplace Type Equations

Sammanfattning: This thesis consists of six scientific papers, an introduction and a summary. All six papers concern the boundary behavior of non-negative solutions to partial differential equations.Paper I concerns solutions to certain p-Laplace type operators with variable coefficients. Suppose that u is a non-negative solution that vanishes on a part ? of an Ahlfors regular NTA-domain. We prove among other things that the gradient Du of u has non-tangential limits almost everywhere on the boundary piece ?, and that log|Du| is a BMO function on the boundary.  Furthermore, for Ahlfors regular NTA-domains that are uniformly (N,?,r0)-approximable by Lipschitz graph domains we prove a boundary Harnack inequality provided that ? is small enough. Paper II concerns solutions to a p-Laplace type operator with lower order terms in ?-Reifenberg flat domains. We prove that the ratio of two non-negative solutions vanishing on a part of the boundary is Hölder continuous provided that ? is small enough. Furthermore we solve the Martin boundary problem provided ? is small enough.In Paper III we prove that the boundary type Riesz measure associated to an A-capacitary function in a Reifenberg flat domain with vanishing constant is asymptotically optimal doubling.Paper IV concerns the boundary behavior of solutions to certain parabolic equations of p-Laplace type in Lipschitz cylinders. Among other things, we prove an intrinsic Carleson type estimate for the degenerate case and a weak intrinsic Carleson type estimate in the singular supercritical case.In Paper V we are concerned with equations of p-Laplace type structured on Hörmander vector fields. We prove that the boundary type Riesz measure associated to a non-negative solution that vanishes on a part ? of an X-NTA-domain, is doubling on ?.Paper VI concerns a one-phase free boundary problem for linear elliptic equations of non-divergence type. Assume that we know that the positivity set is an NTA-domain and that the free boundary is a graph. Furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction. We prove that the graph giving the free boundary is Lipschitz continuous.