Nonlocal Robin Problem for Elliptic Quasilinear Second Order Equations

Abstract

We investigate the behavior of weak solutions to the nonlocal Robin problem for quasilinear elliptic divergence second order equations in a neighborhood of the boundary corner point. We find an exponent of the solution decreasing rate near the boundary corner point.

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

“…We repeat the proof of Lemma 2.3 [3] applying inequality .F W / instead of .W / m and takingˇC D˙B : (14) and (13), from (16) it follows that…”

“…In [3] we have investigated the behaviour of weak solutions for the nonlocal Robin problem with quasi-linear elliptic divergence second-order equations in a plain domain in a neighbourhood of the boundary corner point O: In the case of the linear equation we refer to [10,11]. Here, we consider a different eigenvalue problem (see .EVP 2/) and derive a new Friedrichs-Wirtinger type inequality adapted to the quasi-linear elliptic problem considered in [3].…”

“…Here, we consider a different eigenvalue problem (see .EVP 2/) and derive a new Friedrichs-Wirtinger type inequality adapted to the quasi-linear elliptic problem considered in [3]. It allows us to improve the main result of [3] (see Theorem 3.7): the exponent of weak solution behavior (see inequality (20)) in a neighborhood of an angular point O in the case B < 0 is better than that one obtained in [3].…”

“…The rest of the proof follows verbatim the proof of Theorem 4.2 [3], starting from the paragraph following inequality (5.14).…”

“…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.…”