Existence and geometric properties of solutions of a free boundary problem in potential theory.
Existence and geometric properties of solutions of a free boundary problem in potential theory.
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势论中自由边界问题解的存在性和几何性质。
DOI：
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发表时间：
1995
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通讯作者：
Björn Gustafsson
中科院分区：
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作者：
H. Shahgholian;Björn Gustafsson
设0≤g， h∈L∞(RN)（N≥2）为两个给定的密度函数，其中至少有一个在紧集外离零有界，且g （Holder）连续。我们证明了对于任何紧支持的正测度μ，它足够集中（例如具有足够高的（N−1）维密度），存在一个有界开集Ω∧RN，使得测度hLN⌊Ω + gHN−1⌊∂Ω与μ在Ω外的牛顿势一致，并得到了∂Ω的一些正则性，以及Ω几何上的几个结果。如果h和g是常数，那么对于任何x∈∂Ω，∂Ω at x的向内法向射线（如果存在）与μ的闭合凸包相交。由AMS-TEX 12排版比约恩·古斯塔夫松和亨里克·沙赫霍利安
Let 0 ≤ g, h ∈ L∞(RN )(N ≥ 2) be two given density functions, at least one of them bounded away from zero outside a compact set and g (Holder) continuous. We prove that for any compactly supported positive measure μ which is sufficiently concentrated (e.g. has sufficiently high (N − 1)-dimensional density) there exists a bounded open set Ω ⊂ RN such that the Newtonian potential of the measure hLN ⌊Ω + gHN−1⌊∂Ω agrees with that of μ outside Ω. Some regularity of ∂Ω is obtained, as well as several results on the geometry of Ω. Example: if h and g are constant then, for any x ∈ ∂Ω, the inward normal ray of ∂Ω at x (if it exists) intersects the closed convex hull of μ. Typeset by AMS-TEX 1 2 BJORN GUSTAFSSON AND HENRIK SHAHGHOLIAN