Extended Harnack inequalities with exceptional sets and a boundary Harnack principle
Extended Harnack inequalities with exceptional sets and a boundary Harnack principle
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具有例外集和边界 Harnack 原理的扩展 Harnack 不等式
DOI：
10.1007/s11854-014-0028-3
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
H. Aikawa
中科院分区：
文献类型：
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作者：
Hiroak Aikawa;H. Aikawa;H. Aikawa and T. Itoh;H. Aikawa
Harnack不等式是正调和函数最基本的不等式之一，并已推广到一般椭圆型方程和抛物型方程的正解。本文给出了一个不同的推广；即，我们推广了Harnack链而不是方程。更准确地说，我们允许一个很小的例外集，但得到了类似的Harnack不等式。一套特殊设备的大小是通过容量来衡量的。即使对于经典调和函数，其结果也是新的。推广的Harnack不等式包含了关于正调和函数的边界行为的信息。它给出了一个非常恶劣的区域的边界Harnack原理，该区域的边界是由一个连续模比Hölder连续差的函数的图形局部给出的。
Harnack’s inequality is one of the most fundamental inequalities for positive harmonic functions and has been extended to positive solutions of general elliptic equations and parabolic equations. This article gives a different generalization; namely, we generalize Harnack chains rather than equations. More precisely, we allow a small exceptional set and yet obtain a similar Harnack inequality. The size of an exceptional set is measured by capacity. The results are new even for classical harmonic functions. Our extended Harnack inequality includes information about the boundary behavior of positive harmonic functions. It yields a boundary Harnack principle for a very nasty domain whose boundary is given locally by the graph of a function with modulus of continuity worse than Hölder continuity.