A necessary condition in a De Giorgi type conjecture for elliptic systems in infinite strips

Abstract : Given a bounded Lipschitz domain $\omega\subset\mathbb{R}^{d-1}$ and a lower semicontinuous function $W:\mathbb{R}^N\to\mathbb{R}_+\cup\{+\infty\}$ that vanishes on a finite set and that is bounded from below by a positive constant at infinity, we show that every map $u:\mathbb{R}\times\omega\to\mathbb{R}^N$ with \[ \int_{\mathbb{R}\times\omega}\big(\lvert\nabla u\rvert^2+W(u)\big)\mathop{}\mathopen{}\mathrm{d} x_1\mathop{}\mathopen{}\mathrm{d}x'<+\infty\] has a limit $u^\pm\in\{W=0\}$ as $x_1\to\pm\infty$. The convergence holds in $L^2(\omega)$ and almost everywhere in $\omega$. We also prove a similar result for more general potentials $W$ in the case where the considered maps $u$ are divergence-free in $\mathbb{R}\times\omega$ with $\omega$ being the $(d-1)$-torus and $N=d$.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-02265221
Contributor : Radu Ignat <>
Submitted on : Thursday, August 8, 2019 - 6:57:25 PM
Last modification on : Sunday, August 11, 2019 - 1:09:06 AM

Links full text

Identifiers

  • HAL Id : hal-02265221, version 1
  • ARXIV : 1905.11162

Collections

Citation

Radu Ignat, Antonin Monteil. A necessary condition in a De Giorgi type conjecture for elliptic systems in infinite strips. 2019. ⟨hal-02265221⟩

Share

Metrics

Record views

21