Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents

Abstract

We investigate the rigidity of global minimizers u≥0u\ge 0 of the Alt-Phillips functional involving negative power potentials ∫Ω(∣∇u∣2+u−γχ{u>0})dx,γ∈(0,2),\mathop{\int }\limits_{\Omega }(| \nabla u{| }^{2}+{u}^{-\gamma }{\chi }_{\left\{u\gt 0\right\}}){\rm{d}}x,\hspace{1.0em}\gamma \in \left(0,2), when the exponent γ\gamma is close to the extremes of the admissible values. In particular, we show that global minimizers in Rn{{\mathbb{R}}}^{n} are one-dimensional if γ\gamma is close to 2 and n≤7n\le 7, or if γ\gamma is close to 0 and n≤4n\le 4.

Keywords

• free boundary problems
• energy minimizers
• regularity
• 35b65
• 35r35