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On the \(L^ q\) norm of functions having equidistributed gradients. (English) Zbl 0884.49011

The paper deals with the following maximization problem \[ \max|x|_{L^q(G)} \] where \(G\subset R^n\) is any bounded open set with given measure, \(\text{meas}(G)=V\), and \(u\) is a weakly differentiable function on \(G\) subjected to the constraints \[ u=0\quad\text{on }\partial G\qquad |Du|*=h \] where \(*\) denotes the decreasing rearrangement operator, \(h\in L^p(0,V)\) and \[ p=\begin{cases} 1\quad &n=1\;\vee\;q\leq{n\over n-1}\\ {qn\over q+n} &n\geq 2\;\wedge\;{n\over n-1}<q<+\infty\end{cases},\qquad p>n\quad\text{otherwise}. \] In particular he gives a representation result and studies some properties of \(|Du|\) for a spherically symmetric solution \(u\) of the problem.
A list of references on previous connected results is also presented.

MSC:

49J52 Nonsmooth analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46N20 Applications of functional analysis to differential and integral equations
35B99 Qualitative properties of solutions to partial differential equations
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References:

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