Symmetry and Overdetermined Boundary Value Problems
Symmetry and Overdetermined Boundary Value Problems
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DOI：
10.1515/form.1991.3.143
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发表时间：
1991
影响因子：
3.9
通讯作者：
R. Molzon
中科院分区：
文献类型：
--
作者：
R. Molzon
博赫纳恒等式的极化版本用于获得常曲率流形上超定边值问题的对称结果。该恒等式还用于证明拉普拉斯算子的狄利克雷特征值的 Rellich 恒等式。最后，使用Alexandrov提出的反射论证来获得常曲率流形上超定边值问题的对称性结果。 1980年数学学科分类（1985年修订）：53C20。 l 引言 在本文中，我们考虑常曲率空间上拉普拉斯算子的一些超定边值问题。假设 M 是常曲率流形，Ω c M 是具有 C 边界的域。设 u e C(·) 是一个函数，对于某个给定的径向对称函数 /，Au=f，并且另外假设 u 满足 δ Ω 上的边界条件 w = 0 和 3Ω 上的 (d/dn)(u) = k（常数）。然后我们想证明 Ω 是 M 中的一个公制球。给定的径向对称函数 / 和流形 M 决定了所使用的方法所获得的结果。 C. Berenstein 和 M. Karlowitz 提供的一个例子 [3] 将有助于说明这一点。在标准球体 S" 上，他们构造了一个具有平滑边界和平滑函数 u 的域 Ω c S"，使得 Ω 上的 Au = —i，5 Ω 上的 w = 0 以及 du 上的 (djdn)(u) = k。此外，域 Ω 不是径向对称的。另一方面，如果 Ω 包含在半球中，并且我们考虑方程 Au=f，其中 / = cos r 且 r 是距固定点的测地距离或 / = — l，则实际上 Ω 是径向对称的。而且，证明这两个结果所采用的方法也有很大不同。我们使用的第一种技术涉及从博赫纳恒等式的极化版本获得的积分公式。第二种技术是 Alexandrov [1] 提出的反射论证。
A polarized version of Bochner's identity is used to obtain symmetry results for an overdetermined boundary value problem on constant curvature manifolds. The identity is also used to prove a Rellich identity for Dirichlet eigenvalues of the Laplacian. Finally, a reflection argument which was developed by Alexandrov is used to obtain symmetry results for overdetermined boundary value problems on constant curvature manifolds. 1980 Mathematics Subject Classification (1985 Revision): 53C20. l Introduction In this paper we consider some overdetermined boundary value problems for the Laplacian on spaces of constant curvature. Suppose M is a manifold of constant curvature and Ω c M is a domain with C boundary. Let u e C( ) be a function such that Au=ffor some given radially Symmetrie function / and suppose in addition u satisfies the boundary conditions w = 0 on δ Ω and (d/dn)(u) = k (constant) on 3Ω. We want to then show that Ω is a metric ball in M. The given radially Symmetrie function,/, and the manifold M dictate the result one obtains s well s the method used. An example due to C. Berenstein and M. Karlowitz, [3], will help illustrate this point. On the Standard sphere, S", they construct a domain Ω c S" with smooth boundary and a smooth function u such that Au = —i on Ω, w = 0 on 5 Ω and (djdn)(u) = k on du. Furthermore the domain Ω is not radially Symmetrie. On the other hand if Ω is contained in the hemisphere and we consider the equation Au=f where/ = cos r and r is the geodesic distance from a fixed point or/ = — l, then in fact Ω is radially Symmetrie. Moreover, the methods used to prove these two results are quite different. The first technique we use involves an integral formula obtained from a polarized version of Bochner's identity. The second technique is a reflection argument due to Alexandrov [1].