An elliptic boundary-value problem with a discontinuous nonlinearity

SynopsisWe study the boundary-value problem, for(λ/k,ψ),Here ∆ denotes the Laplacian,His the Heaviside step function and one of A or k is a given positive constant. We defineand usually omit the subscript. Throughout we are interested in solutions with ψ>0 inΩ and hence with λ/=0.In the special case Ω = B(0, R), denoting the explicit exact solutions by ℑe, the following statements are true, (a) The set Aψ, issimply-connected, (b) Along ℑe, the diameter of Aψtendsto zero when the area of Aψ, tends to zero.For doubly-symmetrised solutions in domains Ω such as rectangles, it is shown that the statements (a) and (b) above remain true.