Friedland-Hayman 不等式与 Caffarelli 收缩定理
The Friedland-Hayman inequality and Caffarelli's contraction theorem
AI总结：
本文针对凸锥分区下的 Neumann-Dirichlet 齐次调和函数增长率问题，给出 Friedland-Hayman 型不等式的新证明，并刻画等号情形以推进两相自由边界正则性研究。
摘要：
Friedland-Hayman 不等式是一个尖锐不等式，涉及齐次调和函数的增长率；这些函数在将欧几里得空间分成两部分的互补锥上满足 Dirichlet 边界条件。本文证明了一个类似不等式：把一个凸锥分成两部分，在凸锥边界上施加 Neumann 条件，在界面上施加 Dirichlet 条件。该类似不等式此前已由我们与 Sarah Raynor 共同证明。这里我们给出一个新证明，使我们能够刻画等号成立的情形。按照 Alt、Caffarelli 和 Friedman 提出的两相自由边界理论，这一改进有望为自由边界问题带来进一步正则性。
英文摘要：
The Friedland-Hayman inequality is a sharp inequality concerning the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this paper, we prove an analogous inequality in which one divides a convex cone into two parts, placing Neumann conditions on the boundary of the convex cone, and Dirichlet conditions on the interface. This analogous inequality was already proved by us jointly with Sarah Raynor. Here we present a new proof that permits us to characterize the case of equality. In keeping with the two-phase free boundary theory introduced by Alt, Caffarelli, and Friedman, such an improvement can be expected to yield further regularity in free boundary problems.
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