由负幂次的阿尔特 - 菲利普斯方程产生的退化单相自由边界问题
A Degenerate One-Phase Free Boundary Problem Arising From the Alt-Phillips Equation for Negative Powers
AI总结：
研究形如$w\Delta w = h(\nabla w)$的退化单相自由边界问题，通过规范变换与阿尔特 - 菲利普斯方程相关，证明了粘性解等性质，还得出当$\gamma$趋于$2$时自由边界收敛到极小曲面的结论。
AI中文摘要：
我们研究一类形如$w\Delta w = h(\nabla w)$的退化单相自由边界问题的粘性解。假设存在一个星形区域$D$，使得在$D$中$h < 0$，在$\partial D$上$h = 0$，在$\bar{D}^{c}$中$h > 0$。这类问题源于对$\Delta u = f(u)$进行规范变换，当$f$类似于$-\gamma u^{-(\gamma + 1)}$时出现，此即负幂次势的阿尔特 - 菲利普斯方程，此时$h(\rho) = c(|\rho|^2 - 1)$。我们证明了粘性解的存在性、利普希茨正则性以及自由边界在平坦点处的正则性。此外，还表明当$\gamma$趋于$2$时，自由边界收敛到一个极小曲面。
英文摘要：
We study viscosity solutions for a class of degenerate one-phase free boundary problems of the form $wΔw = h(\nabla w)$. We assume the existence of a star-shaped domain $D$ such that $h < 0$ in $D$, $h = 0$ on $\partial D$, and $h > 0$ in $\bar{D}^{c}$. This class of degenerate one-phase free boundary problems arises when a canonical transformation is performed to a semilinear equation $Δu = f(u)$, and $f$ behaves like $-γu^{-(γ+ 1)}$ for some $γ\in (0,2)$. In this case, known as the Alt-Phillips equation for negative power potentials, $h(ρ) = c(|ρ|^2 - 1)$. We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as $γ$ converges to $2$, the free boundary converges to a minimal surface.
