On the degenerate oblique derivative problem for elliptic second-order equation in a domain with boundary conical point

Abstract

We study the behaviour of strong solutions to the degenerate oblique derivative problem for linear second-order elliptic equation in a neighbourhood of the boundary conical point of a bounded domain. IntroductionIn our article we study the behaviour of strong solutions to the degenerate oblique derivative problem for linear second-order elliptic equation in a neighbourhood of the boundary conical point of an n-dimensional, !3, bounded domain. Solonnikov et al. [1,2] have studied a similar problem for the Laplace operator in a plane angle. They proved the unique existence of solutions and obtained a priori estimates for weak solutions in the Sobolev-Kondratiev weighted spaces.Lieberman [3][4][5] derived some pointwise estimates and regularity results for solution to the linear uniformly elliptic second-order oblique derivative problem.We study our problem for general second-order linear elliptic equation. The degeneration of the problem occurs in the boundary condition. We derive a priori estimates in the Kondratiev spaces and an estimate of the type u(x) ¼ O(jxj ) with exact exponent . Similar estimates have been established in [6] for Dirichlet problem. We apply methods of [6,7].Our article is an extension of [8,9]. In [8], such problem in a plain domain with angular point and in [9] for similar problem in a three-dimensional bounded domain with the boundary conical point were considered.

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

“…• By assumption (D) and the inequality (9), using Theorem 2.3 [7] analogously to (26) and using (23), we get…”

“…• By inequality (10.2.45) [6] for α = 4 − n and Theorem 2.3 [7] with regard to notation (14), we obtain G 0…”

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

“…Theorem 2.3 ( [4,5]). There exists the smallest positive eigenvalue of problem (EVP ), which satisfies the following inequalities…”

An integro-differential inequality related to the smallest positive eigenvalue of p(x)-Laplacian Dirichlet problem

“…In this article we prove some integro-differential inequalities related to the smallest positive eigenvalue of the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. Such inequalities play very important role -they are necessary to investigate the behaviour of weak solutions of boundary value problems (Dirichlet, Neumann, Robin and mixed) for linear, weak quasilinear, and quasilinear elliptic divergence second order equations in cone-like domains (see [4,18]) and domains with boundary singularities: angular, conic points or edges (see [2,3]).…”