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Flat level set regularity of \(p\)-Laplace phase transitions. (English) Zbl 1138.35029

Mem. Am. Math. Soc. 858, 144 p. (2006).
The booklet deals with a functional \({\mathcal F}_\Omega(u):=\int_\Omega \frac1p| \nabla u| ^p+h_0(u)\,dx\) where \(\Omega\subset\mathbb R^N\) and \(h_0\) is a \(C^1\)-function of a double-well type, e.g. \(h_0(u)=(1-u^2)^p\). Attention is focused to minimizers of \({\mathcal F}_\Omega\), which models steady-states of physical systems with two phases divided by a diffuse interface with surface tension. A Harnack-type inequality is proved for level sets of local minimizers. Roughly speaking, such results state that, if the zero level set of a minimizer is trapped in a rectangle whose height is small enough, then it can be trapped in a rectangle with even smaller height in a smaller neighbourhood. Moreover, in some cases when \(N\leq8\), it is proved that level sets are hyperplanes, which was conjectured by DeGiorgi and known so far only for \(p=2\) and \(N\leq3\).

MSC:

35J70 Degenerate elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
49J10 Existence theories for free problems in two or more independent variables
58E30 Variational principles in infinite-dimensional spaces
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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