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Applications of symmetric rearrangement to certain nonlinear elliptic equations with a free boundary. (English) Zbl 0653.35027
Nonlinear differential equations, Lect. 7th Congr., Granada/Spain 1984, Res. Notes Math. 132, 155-181 (1985).
[For the entire collection see Zbl 0638.00015.]
Consider a nonlinear elliptic equation of the type \[ (1)\quad - Lu+f(u)=g\quad in\quad \Omega;\quad u=h\quad on\quad \partial \Omega \] where \(\Omega\) is a regular bounded open set of R N, L is a linear elliptic second order operator \[ Lu=\sum^{N}_{i,j=1}(\partial /\partial x_ j)(a_{ij}(x)(\partial u/\partial x_ i))+\sum^{N}_{j=1\quad}(\partial /\partial x_ j)(b_ j(x)u)+c(x)u. \] (1) appears in many different contexts: in the study of isothermal chemical reactions, of stationary solutions of many nonlinear evolution equations, and others. Many authors considered the existence and properties of a free boundary F(u) for solutions of (1). The author in this paper obtains some qualitative properties in F(u) using symmetric rearrangement of a function in the sense of Hardy and Littlewood.
Reviewer: J.H.Tian

35J65 Nonlinear boundary value problems for linear elliptic equations
35R35 Free boundary problems for PDEs