Regularity theory for elliptic and parabolic free boundary problems
Final Report Abstract
During the funding period I have worked on three projects. The main results obtained are: • Establishing an epiperimetric inequality for the parabolic Signorini problem, which gives a quantitative decay estimate of the solution towards the blow-ups at regular and singular free boundary points. An unexpected result is that at singular points, the decay rate is different from the elliptic counterpart, and we speculate one can improve the decay estimate. • We study the well-posedness of the subsonic jet problem in the case of the nonzero vorticity. One of our key observations is that the compressible Euler system for the flows with nonzero vorticity enjoys a variational structure. Using this variational structure, we established the existence of jet for the subsonic Euler flows with nonzero vorticity. Further fine properties for the jet are analyzed. • Together with a master student at University of Heidelberg, we study the minimum configuration of an energy functional consisting of Riesz type repulsive self-interaction and attractive interaction with a fixed charged domain Ω+ . We establish the existence and uniqueness for the minimizer, screening property for the charge potential w.r.t the minimizer, and prove that outside Ω+ the potential solves an obstacle problem for the fractional Laplacian.
Publications
- (2020) An epiperimetric inequality approach to the parabolic Signorini problem. Discrete & Continuous Dynamical Systems – A 40 (3) 1813–1846
Shi, Wenhui
(See online at https://doi.org/10.3934/dcds.2020095)
