Cavity problems in discontinuous media
Cavity problems in discontinuous media
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不连续介质中的空腔问题
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
E. Teixeira
中科院分区：
文献类型：
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作者：
D. Prazeres;E. Teixeira
我们研究空化型方程，$$ ext {div}(a_{ij}(X)
abla u) sim delta _0(u)$$div(aij(X)∇u)∼δ0(u)，对于有界、可测量的椭圆媒体 $$a_{ij}(X)$$aij(X)。 De Giorgi–Nash–Moser 理论确保解在其正性集合 $${u>0}$${u>0} 内是 $$alpha $$α-Hölder 连续的，对于某些指数 $$alpha $$α 严格小于 1。尽管如此，本文证明的关键、主要结果为此类解沿其自由边界 $$partial {u>0 }$$∂{u>0} 提供了尖锐的 Lipschitz 正则性估计。如此精确的估计意味着自由边界的几何测量约束。特别是，我们证明非重合 $${u>0}$${u>0} 集具有均匀的正密度，并且自由边界具有有限的 $$(n- varsigma )$$(n-ς)-Hausdorff 测度，对于通用数 $$0< varsigma le 1$$0
We study cavitation type equations, $$ ext {div}(a_{ij}(X)
abla u) sim delta _0(u)$$div(aij(X)∇u)∼δ0(u), for bounded, measurable elliptic media $$a_{ij}(X)$$aij(X). De Giorgi–Nash–Moser theory assures that solutions are $$alpha $$α-Hölder continuous within its set of positivity, $${u>0}$${u>0}, for some exponent $$alpha $$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, $$partial {u>0 }$$∂{u>0}. Such a sharp estimate implies geometric-measure constrains for the free boundary. In particular, we show that the non-coincidence $${u>0}$${u>0} set has uniform positive density and that the free boundary has finite $$(n- varsigma )$$(n-ς)-Hausdorff measure, for a universal number $$0< varsigma le 1$$0