scholarly journals Symmetry, degeneracy, and universality in semilinear elliptic equations. Infinitesimal symmetry-breaking

1990 ◽  
Vol 89 (2) ◽  
pp. 364-409 ◽  
Author(s):  
Joel Smoller ◽  
Arthur Wasserman
Keyword(s):  
Symmetry Breaking ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
Infinitesimal Symmetry

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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