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Zbl 0653.35027
Díaz, J.I.
Applications of symmetric rearrangement to certain nonlinear elliptic equations with a free boundary.
(English)
[A] Nonlinear differential equations, Lect. 7th Congr., Granada/Spain 1984, Res. Notes Math. 132, 155-181 (1985).

[For the entire collection see Zbl 0638.00015.] \par 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 $$Lu=\sum\sp{N}\sb{i,j=1}(\partial /\partial x\sb j)(a\sb{ij}(x)(\partial u/\partial x\sb i))+\sum\sp{N}\sb{j=1\quad}(\partial /\partial x\sb j)(b\sb j(x)u)+c(x)u.$$ (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.
[J.H.Tian]
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35R35 Free boundary problems for PDE

Keywords: symmetric rearrangement; nonlinear elliptic equations; isothermal chemical reactions; stationary solutions; free boundary

Citations: Zbl 0638.00015

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