On optimization problems with prescribed rearrangements. (English) Zbl 0678.49003

Alvino, Angelo; Trombetti, Guido; Lions, Pierre-Louis

Nonlinear Anal., Theory Methods Appl. 13, No. 2, 185-220 (1989).

The goal of the paper is to review some results concerning functions with prescribed rearrangement and related optimization problems. The first part is devoted to the study of the set of weak limits for sequences of equimeasurable nonnegative functions in \(L^ p(\Omega)\) with \(\Omega \subseteq R^ n\). Let \(C(f_ 0)=\{f\geq 0:\) \(f^*=f^*_ 0\}\) where \(f_ 0\in L^ p(\Omega)\) and \(f^*,f^*_ 0\) are the rearrangements of \(f,f_ 0\). If \(K(f_ 0)\) consists of all weak limits of sequences in \(C(f_ 0)\) and \[ f\prec g\quad \Leftrightarrow \quad \int^{t}_{0}f^*(s)ds\leq \int^{t}_{0}g^*(s)ds,\quad \int^{| \Omega |}_{0}f^*(s)ds=\int^{| \Omega |}_{0}g^*(s)ds \] then (i) \(K(f_ 0)=\{f\geq 0:\) \(f\prec f_ 0\}\) and (ii) \(K(f_ 0)\) is a closed weakly compact, convex set in \(L^ p(\Omega)\) and \(C(f_ 0)\) is the set of extreme points of \(K(f_ 0).\)
The results (i) and (ii) are partially known and cited with references.
In the remainder of the paper the above geometric description is applied to optimization results in \(C(f_ 0)\). For example the authors consider the following problem \[ \max \{\int_{\Omega}| u|^ qdx:\quad u\in W_ 0^{1,p}(\Omega),\quad | \nabla u| \leq f\quad with\quad f^*\quad fixed\}, \] where \(1\leq q\leq np/(n-p)\) if \(p<n\), \(1\leq q<+\infty\) if \(p\geq n\). They prove that the maximum is achieved for a ball and u,f spherically symmetric.

Reviewer: A.Alvino

MSC:

49J20	Existence theories for optimal control problems involving partial differential equations
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

rearrangement; equimeasurable nonnegative functions

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