| Abstract: |
| We present a flexible framework for proving regularity estimates for solutions of partial differential equations that does not rely on scaling invariance. This makes it particularly suited to problems where classical blow-up methods are unavailable.
The approach has been used in the study of Holder regularity for parabolic equations driven by integro-differential operators, as well as in the analysis of free boundary regularity for the Hele-Shaw problem. It also applies to establish $C^{1,\alpha}$ estimates for solutions of nonlocal equations with rough kernels. As a new application, we consider the logarithmic Laplacian, a zero-order operator for which rescaling produces a non-integrable tail, creating a serious obstruction to standard compactness and blow-up arguments. In this setting, we obtain optimal Schauder-type logarithmic Holder estimates. |
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