奇异半线性椭圆边值问题与边界Harnack原理
On a singular semilinear elliptic boundary value problem and the boundary Harnack principle
摘要：
考虑D; u中的奇异边值问题Δu = f（u）|_dD= phi，其中1. D是R^d的有界C^2-整环，d ≥ 3 2. f：（0，1）->（0，1）是一个局部Hölder连续函数使得f（u）-> 1，当u -> 0时，速率为u^{-α}，对于（0，1）中的某个α，3。并且φ是满足某些增长假设的正连续函数。我们证明了由正调和函数下有界的解的存在性，这些解在D上是光滑的，在D-杆上是连续的。这样的解决方案，以满足边界Harnack原则。主要结果的证明采用了概率方法。
英文摘要：
We consider the singular boundary-value problem Δu = f(u) in D; u|_dD= phi, where 1. D is a bounded C^2-domain of R^d, d >= 3 2. f: (0,1) -> (0,1) is a locally Hölder continuous function such that f(u) -> 1 as u -> 0 at the rate u^{-α}, for some αin (0,1), 3. and phi is a positive continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in D-bar. Such solutions are shown to satisfy a boundary Harnack principle. Probabilistic techniques are used in proving the main results.
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