Valdinoci, Enrico; Sciunzi, Berardino; Savin, Vasile Ovidiu 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 \(N\leq3\). Reviewer: Tomáš Roubíček (Praha) Cited in 1 ReviewCited in 21 Documents 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 Keywords:Ginzburg-Landau-Allen-Cahn model; De Giorgi conjecture; sliding methods; minimizers; Harnack-type inequality PDFBibTeX XMLCite \textit{E. Valdinoci} et al., Flat level set regularity of \(p\)-Laplace phase transitions. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1138.35029) Full Text: DOI Link