Regularity of a free boundary with application to the Pompeiu problem
Regularity of a free boundary with application to the Pompeiu problem
复制标题
自由边界的正则性及其在庞贝问题中的应用
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
H. Shahgholian
中科院分区：
文献类型：
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作者：
L. Caffarelli;Lavi Karp;H. Shahgholian
在单位球B（0; 1），设u和（N中的一个区域）解如下超定问题：B（0; 1）中的u =; 0 2 @;u =jruj =0 in B（0; 1）n，其中表示特征函数，方程在分布意义下满足.如果<$的补集在原点处不产生尖点奇点，则我们证明@<$在原点的某个小邻域内是解析的。这个结果可以修改以得到更一般的散度型算子。作为这一点的一个应用，我们得到了一个没有庞培性质的区域的边界的正则性，只要它的补没有尖点奇点。
In the unit ball B(0; 1), let u and › (a domain in N ) solve the following overdetermined problem: ¢u = ´› in B(0; 1); 02 @› ;u =jruj =0 inB(0; 1)n ›; where ´› denotes the characteristic function, and the equation is satisfled in the sense of distributions. If the complement of › does not develop cusp singularities at the origin then we prove @› is analytic in some small neighborhood of the origin. The result can be modifled to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.