Speaker
Description
Many equations arising in mathematics and the sciences have a remarkable smoothing effect. The heat equation provides a striking example: even when starting from very irregular initial data, its solutions become smooth instantaneously. Classical regularity theory seeks to explain this phenomenon for broad classes of elliptic and parabolic equations. In many important problems, however, the regularizing mechanism is only partially present. It may operate only in certain regions or above a distinguished scale, as in problems arising from homogenization and numerical schemes. In this talk, I will revisit some results from the classical theory and present recent work on multiscale Schauder estimates.
Applications include higher-order regularity for nonlocal equations with rough kernels, further time regularity for parabolic equations, free boundary problems, and problems whose elliptic structure degenerates below a scale-dependent threshold.