A Minimum Problem Associated With Scalar Ginzburg–Landau Equation and Free Boundary

Abstract

Let $N\ge 2$, $p\in \left(\frac{2N}{N+2},,,+,\infty \right)$, and $\Omega$ be an open bounded domain in $R^N$ with smooth boundary. We consider the minimum problem $J\left(u\right):={\int }_{\Omega }\left(\frac{1}{p},{|\nabla u|}^p,+,{\lambda }_1,{\left(1,-,{\left(u^{+}\right)}^2\right)}^2,+,{\lambda }_2,u^{+}\right)dx\to \min$ over a certain class $K$, where ${\lambda }_1\ge 0$ and ${\lambda }_2\in R$ 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 ${\lambda }_2\in R$, we prove the existence, non-negativity, and uniform boundedness of minimizers of $J\left(u\right)$. 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.