Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions.
The authors investigate properties of bounded critical points of the functional \[ \mathcal E (u) = \int_{\mathbb R^n} \Phi (u,Du)\, dx \] under suitable hypotheses on the function \(\Phi\). (The model function is \(\Phi(\xi,\sigma) = (\varepsilon+| \sigma| ^2)^{p/2} + F(\xi)\) with \(\varepsilon \geq 0\), \(1\leq P <\infty\), and \(\varepsilon=1\) if \(p=1\) although more general structures are considered.) An important role is played by the function \(P\), defined by \[ P(x;u) = Du \cdot \frac {\partial \Phi}{\partial \xi}(u,Du) - \Phi(u,Du). \] Using this function, the authors prove monotonicity of the energy, Liouville type theorems, and one-dimensional symmetry. In particular, they are interested in generalizations of the DeGiorgi conjecture, which states that if \(u\) is a classical entire solution of \(\Delta u =u^3-u\) with \(| u| \leq 1\) and \(\partial u/\partial x_n>0\), then the level sets of \(u\) are all hyperplanes. They succeed in showing that this conjecture is true for \(n=3\) even for a quite general \(\Phi\).
scientific article; zbMATH DE number 1942731