We consider a biharmonic problem with Navier type boundary conditions , on a family of truncated sectors in of radius , and opening angle , when is close to . The family of right-hand sides is assumed to depend smoothly on in . The main result is that converges to when with respect to the -norm. We can also show that the -topology is optimal for such a convergence result.
In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.
The main purpose of this work is to obtain a Harnack inequality and estimates for the Green function for the general class of degenerate elliptic operators described below.
We prove Harnack's inequality for non-negative solutions of some degenerate elliptic operators in divergence form with the lower order term coefficients satisfying a Kato type contition.
There is a long history of studying nonlinear boundary value problems for elliptic differential equations in a domain with sufficiently smooth boundary. In this paper, we show that the gradient of the solution of such a problem is continuous when a directional derivative is prescribed on the boundary of a Lipschitz domain for a large class of nonlinear equations under weak conditions on the data of the problem. The class of equations includes linear equations with fairly rough coefficients as well...