A minimum problem associated with scalar Ginzburg-Landau equation and free boundary

Abstract

Let $N>2$, $p\in \left(\frac{2N}{N+2},+\infty \right)$, and $\Omega$ be an open bounded domain in ${\mathbb{R}}^N$. We consider the minimum problem $$\mathcal{J}(u):={\int }_{\Omega }\left(\frac{1}{p}|\nabla u{|}^p+{\lambda }_1{\left(1-(u^{+}{)}^2\right)}^2+{\lambda }_2{u}^{+}\right)\text{d}x\to \text{min}$$ over a certain class $\mathcal{K}$, where ${\lambda }_1\ge 0$ and ${\lambda }_2\in \mathbb{R}$ are constants, and $u^{+}:=\max \{u,0\}$. The corresponding Euler-Lagrange equation is related to the Ginzburg-Landau equation and involves a subcritical exponent when ${\lambda }_1>0$. For ${\lambda }_1\ge 0$ and ${\lambda }_2\in \mathbb{R}$, we prove the existence, non-negativity, and uniform boundedness of minimizers of $\mathcal{J}(u)$. Then, we show that any minimizer is locally $C^{1,\alpha }$-continuous with some $\alpha \in (0,1)$ and admits the optimal growth $\frac{p}{p-1}$ near the free boundary. Finally, under the additional assumption that ${\lambda }_2>0$, we establish non-degeneracy for minimizers near the free boundary and show that there exists at least one minimizer for which the corresponding free boundary has finite ($N-1$)-dimensional Hausdorff measure.

Submitted to journal in 2024