scholarly journals Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions

1986 ◽  
Vol 105 (3) ◽  
pp. 415-441 ◽  
Author(s):  
Joel A. Smoller ◽  
Arthur G. Wasserman
Keyword(s):  
Symmetry Breaking ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Semilinear Elliptic

Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone

1992 ◽  
Vol 122 (1-2) ◽  
pp. 137-160
Author(s):  
Chie-Ping Chu ◽  
Hwai-Chiuan Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Semilinear Elliptic Equations ◽  
Spherically Symmetric ◽  
Semilinear Elliptic ◽  
Symmetry Properties

SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

Symmetry breaking for semilinear elliptic equations on sectorial domains in ℝ2

1991 ◽  
Vol 118 (3-4) ◽  
pp. 327-353
Author(s):  
Song-Sun Lin
Keyword(s):  
Symmetry Breaking ◽  
Poisson Equation ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Complete Picture ◽  
Fredholm Alternative ◽  
Alternative Theory ◽  
Semilinear Elliptic ◽  
The Poisson Equation ◽  
Sectorial Domains

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.

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