A minimum problem associated with scalar Ginzburg-Landau equation and free boundary
Yuwei Hu, Jun Zheng, Leandro S. Tavares
TL;DR
This work analyzes a variational minimum problem for a nonlinear $p$-Laplacian energy augmented by a Ginzburg–Landau-type nonlinearity and a free boundary. It establishes existence, nonnegativity, and uniform bounds for minimizers, then proves local $C^{1,\alpha}$ regularity and an optimal near-boundary growth rate with exponent $\tfrac{p}{p-1}$. When $\lambda_2>0$, the paper proves non-degeneracy of minimizers near the free boundary and introduces a penalized problem to study the free boundary's geometry, showing that at least one minimizer has finite $(N-1)$-dimensional Hausdorff measure for its free boundary. The results extend obstacle/GL-type free boundary theory to a general $p$-Laplacian setting, bridging variational methods and geometric measure theory in a broad nonlinear regime.
Abstract
Let $N>2$, $p\in \left(\frac{2N}{N+2},+\infty\right)$, and $Ω$ be an open bounded domain in $\mathbb{R}^N$. We consider the minimum problem $$ \mathcal{J} (u) := \displaystyle\int_{Ω} \left(\frac{1}{p}| \nabla u| ^p+λ_1\left(1-(u^+)^2\right)^2+λ_2u^+\right)\text{d}x\rightarrow \text{min} $$ over a certain class $\mathcal{K}$, where $λ_1\geq 0$ and $ λ_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 $λ_1>0$. For $λ_1\geq 0$ and $ λ_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 $C^{1,α}$-continuous with some $α\in (0,1)$ and admits the optimal growth $\frac{p}{p-1}$ near the free boundary. Finally, under the additional assumption that $λ_2>0$, 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.