非正则核非局部方程的高阶Hölder正则性
Higher Hölder regularity for nonlocal equations with irregular kernel
摘要：
研究了一类非线性非局部椭圆型方程的局部弱解的高阶Hölder正则性，其核满足一个温和的连续性假设.我们的主要结果的一个有趣的特点是，所获得的规律性是更好的比人们可能会期望考虑相应的结果时，当地椭圆型方程的散度型连续系数。因此，在某种意义上，我们的结果可以被认为是纯非局部类型的，遵循近年来观察到的各种纯非局部现象的趋势。我们的做法可以概括如下。首先，我们使用某些检验函数，涉及离散分数阶导数，以获得更高的Hölder正则性齐次方程驱动的本地平移不变内核，而全球的内核的行为是允许的更一般。这使我们能够在一般情况下通过近似论证推导出所需的正则性。
英文摘要：
We study the higher Hölder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for local elliptic equations in divergence form with continuous coefficients. Therefore, in some sense our result can be considered to be of purely nonlocal type, following the trend of various such purely nonlocal phenomena observed in recent years. Our approach can be summarized as follows. First, we use certain test functions that involve discrete fractional derivatives in order to obtain higher Hölder regularity for homogeneous equations driven by a locally translation invariant kernel, while the global behaviour of the kernel is allowed to be more general. This enables us to deduce the desired regularity in the general case by an approximation argument.
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