Regularity results for generalized elliptic problems in bounded domains

Abstract

Three different types of problems will be studied in this thesis. The three problems are the G-Laplace equation in a convex domain, a quasilinear equation with p-growth condition in a quasiconvex domain and generalized steady Stokes system in a Reifenberg flat domain. In each problem, we focus on a gradient estimate of a weak solution. At first, we prove local boundedness of of the gradient for the homogeneous G-Laplace equation in a convex domain under vanishing Neumann boundary condition. G is a Young function which is a non-decreasing convex function such that G(0) = 0 and lim_{t→+∞}G(t)/t = +∞. In this problem, one of our interests is a convex domain, since Lipschitz regularity of a solution to even the Laplace equation cannot be obtained in a Lipschitz domain. Next, we derive Calderón-Zygmund type estimate for the solution to quasilinear equation with p-growth condition in a quasiconvex domain, which is locally trapped by two convex domains. As far as the domain is concerned, our regularity assumption on the boundary is weaker than any other one reported in this direction. In addition, we extend our result in Lebesgue spaces to Orlicz spaces. In last chapter, We prove the global weighted L^q-estimates for the gradient of the weak solution and an associated pressure under the assumptions that the coefficients have small BMO (bounded mean oscillation) semi-norms and the domain is sufficiently flat in the Reifenberg sense. On the other hand, a given weight is assumed to belong to a Muckenhoupt class. Our result generalizes the global W 1,q estimate for a solution with respect to the Lebesgue measure for the Stokes system in a Lipschitz domain.