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A semilinear elliptic boundary-value problem describing small patches of vorticity in an otherwise irrotational flow. (English) Zbl 0568.35040
Nonlinear analysis, Miniconf. Canberra/Aust. 1984, Proc. Cent. Math. Anal. Aust. Natl. Univ. 8, 123-132 (1984).
[For the entire collection see Zbl 0552.00008.]
Let \(\Omega\) be a bounded convex domain in \(R^ 2\) and the boundary \(\partial \Omega\) of class \(C^ 2\) has the curvature bounded away from zero. The problem \[ (P)\quad -\Delta \psi \in \lambda H(\psi -k)\quad in\quad \Omega,\quad \psi =0\quad on\quad \partial \Omega \] is considered, H being the set-valued Heaviside stepfunction \((H(t)=0\) if \(t<0\), \(H(0)=[0,1]\) and \(H(t)=1\) for \(t>0)\). let \((\lambda_ n,k_ n,\psi_ n)\) be a sequence of the positive solutions of the problem (P) such that area of \(\{\) \(x\in \Omega:\) \(\psi_ n(x)>k_ n\}\) tends to zero. Then \((\sup_{x\in \Omega}\psi_ n(x)-k_ n)/k_ n\to 0\).
Reviewer: V.Tsalyuk

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations