Behaviour of strong solutions to the degenerate oblique derivative problem for elliptic quasi-linear equations in a neighbourhood of a boundary conical point

Abstract

We study the behaviour of strong solutions to the degenerate oblique derivative problem for quasi-linear second-order elliptic equations in a neighbourhood of the boundary conical point of a bounded domain.

Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities

“…He [7] and Trudinger [8] have obtained local gradient bound estimate and local Hölder gradient estimate of strong solutions in any sub-domain with a C 2 boundary portion of the domain. The results obtained in [1] are a generalization and improvement of results of [9] on the case of the oblique boundary condition. We consider the oblique derivative problem for the elliptic second-order linear equation:…”

“…The precise exponents of the solution's decrease rate depend on these exact constants. For details we refer to [1][2][3]. The existence of the smallest positive eigenvalue of problem (EVP ) for n D 3 was proved in [4].…”

“…In [1] we have investigated the behaviour of strong solutions to the oblique derivative problem for the general second-order quasi-linear elliptic equation in a neighbourhood of a conical boundary point of an n-dimensional bounded domain, n 2. In the case of the linear equation we refer to [4,5].…”

“…This is a brief description of results of [1][2][3] with comments and improvements, which were presented at the International conference on differential equations dedicated to the 110th anniversary of Ya. B. Lopatynsky.…”