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Bernstein and De Giorgi type problems: new results via a geometric approach. (English) Zbl 1180.35251
In the present paper there are established various symmetry properties of stable solutions \(u\in C^1(\mathbb R^N)\cap (\{\nabla u\not=0\})\) of possible degenerate or singular elliptic equations of the form \(\text{div}(a(|\nabla u(x)|)\nabla u(x))+f(u(x))=0\) in \(\mathbb R^N\), with \(N=2,3\). The nonlinear function \(f\) is assumed to be a locally Lipschitz mapping and the function \(a\) is positive in \((0,\infty)\) and satisfying \(a(t)+ta'(t)>0\) for any \(t\in (0,\infty)\). The setting considered in the present paper includes the quasilinear framework corresponding to \(a(t)=t^{p-2}\) (for \(p>1\)) and the mean curvature case that corresponds to \(a(t)=1/\sqrt{1+t^2}\).
The main results are the following: (i) a new proof of a conjecture of De Giorgi for phase transitions in \(\mathbb R^2\) and \(\mathbb R^3\); (ii) a new proof of the Bernstein problem on the flatness of minimal area graphs in \(\mathbb R^3\); (iii) a one-dimensional result in the half-space; (iv) the one-dimensional symmetry for 1-Laplacian type operators. The proofs combine several interesting techniques such as a Liouville type property, the Poincaré formula, and level set analysis tools.

35J70 Degenerate elliptic equations
35J20 Variational methods for second-order elliptic equations
35J62 Quasilinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35J93 Quasilinear elliptic equations with mean curvature operator
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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