不连续介质中的空洞问题
Cavity problems in discontinuous media
摘要：
研究了有界可测椭圆介质$a_{ij}(X)$的空化型方程$\text{div}(a_{ij}(X) \nabla u) \sim δ_0(u)$. De Giorgi-Nash-Moser理论保证解在其正性集$\{u>0\}$内是$α$ -Hölder连续的，对于某个指数$α$严格小于1。尽管如此，本文证明的关键，主要结果提供了一个尖锐的Lipschitz正则性估计，这样的解决方案沿其自由边界，$\partial \{u>0 \}$。这样一个尖锐的估计意味着几何测量约束的自由边界。特别地，我们证明了非重合$\{u>0\}$集具有均匀正密度，自由边界具有有限的$(n- ς)$ -Hausdorff测度，对于一个普适数$0< ς\le 1$。
英文摘要：
We study cavitation type equations, $\text{div}(a_{ij}(X) \nabla u) \sim δ_0(u)$, for bounded, measurable elliptic media $a_{ij}(X)$. De Giorgi-Nash-Moser theory assures that solutions are $α$-Hölder continuous within its set of positivity, $\{u>0\}$, for some exponent $α$ strictly less than one. Notwithstanding, the key, main result proven in this paper provides a sharp Lipschitz regularity estimate for such solutions along their free boundaries, $\partial \{u>0 \}$. Such a sharp estimate implies geometric-measure constrains for the free boundary. In particular, we show that the non-coincidence $\{u>0\}$ set has uniform positive density and that the free boundary has finite $(n- ς)$-Hausdorff measure, for a universal number $0< ς\le 1$.
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