SYMMETRY-BREAKING FOR SEMILINEAR ELLIPTIC-EQUATIONS ON SECTORIAL DOMAINS IN R2

Abstract

We first study the Poisson equation DELTA-u = f in OMEGA-omega, u + b partial u/partial v = 0 on GAMMA-0 and partial u/partial v = 0 on GAMMA-1, where OMEGA-omega = {(r cos theta, r sin theta): 0 < r < 1, theta element-of (0, omega)} is a sector in R2, omega element-of (0, 2-pi), GAMMA-0 = {(cos theta, sin theta): theta element-of (0, omega)} and GAMMA-1 = partial-OMEGA-omega -GAMMA-0, b and lambda are in R1. We obtain Schauder-type estimates and Fredholm alternative theory for the problem. We then study the symmetry breaking problem for the Gel'fand equation DELTA-u + lambda-e(u) = 0 in OMEGA-omega and obtain a complete picture about the relationships among three parameters lambda, b, and omega in the problem.