Symmetry breaking for semilinear elliptic equations on sectorial domains in ℝ2

SynopsisWe first study the Poisson equation Δu =fin Ώω,and, where Ωω= {(rcos θ,rsin θ): 0<r<1, θ ∈(0,ω)} is a sector in ℝ2, ω ∈ (0, 2π), Г0= {(cos θ, sin θ): θ ∈ (0, ω)} and Г1= ∂Ωω− Г0,band λ 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 Δu+ λeu= 0 in Ωωand obtain a complete picture about the relationships among three parameters λ,b, and ω in the problem.