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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)-Lu+f(u)=ginΩ;u=honΩ

where Ω is a regular bounded open set of R N, L is a linear elliptic second order operator

Lu= i,j=1 N (/x j )(a ij (x)(u/x i ))+ j=1 N (/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
MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35R35Free boundary problems for PDE