A DeGiorgi type conjecture for minimal solutions to a nonlinear Stokes equation

Abstract

We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation $$ \begin{cases} -\Delta u+\nabla W(u)=\nabla p&\text{in }\mathbb{R}^d,\\ \nabla\cdot u=0&\text{in }\mathbb{R}^d, \end{cases} $$ which are periodic in the $d-1$ last variables (living on the torus $\mathbb{T}^{d-1}$) and globally minimize the corresponding energy in $\Omega=\mathbb{R}\times \mathbb{T}^{d-1}$, i.e., $$ E(u)=\int_{\Omega} \frac12 |\nabla u|^2+W(u)\, dx, \quad \nabla\cdot u=0. $$ Namely, we determine a class of nonlinear potentials $W\geq 0$ such that any global minimizer $u$ of $E$ connecting two zeros of $W$ as $x_1\to\pm\infty$ is one-dimensional, i.e., $u$ depends only on the $x_1$ variable. In particular, this class includes in dimension $d=2$ the nonlinearities $W=w^2$ with $w$ being an harmonic function or a solution to the wave equation, while in dimension $d\geq 3$, this class contains a perturbation of the Ginzburg-Landau potential as well as potentials $W$ having $d+1$ wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations relying on the notion of entropy (coming from scalar conservation laws). We also study the problem of the existence of global minimizers of $E$ for general potentials $W$ providing in particular compactness results for uniformly finite energy maps $u$ in $\Omega$ connecting two wells of $W$ as $x_1\to\pm\infty$.