A boundary estimate for singular parabolic diffusion equations
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of p-Laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic p-capacity.
KeywordsParabolic p-Laplacian Boundary estimates Continuity Elliptic p-capacity Wiener-type integral
Mathematics Subject ClassificationPrimary 35K67 35B65 Secondary 35B45 35K20
The authors thank Juha Kinnunen, who suggested this problem, during the program “Evolutionary problems” in the Fall 2013 at the Institut Mittag-Leffler, and are very grateful to Emmanuele DiBenedetto, for discussions and comments, which greatly helped to improve the final version of this manuscript. We thank the anonymous referee for his remarks.
- 6.Chen, Y.Z., DiBenedetto, E.: On the Harnack inequality for non-negative solutions of singular parabolic equations. In: Proceedings of Non-linear Diffusion; in Honour of J. Serrin, Minneapolis, May 1990Google Scholar
- 20.Maz’ya, V.G.: Regularity at the boundary of solutions of elliptic equations, and conformal mapping, Dokl. Akad. Nauk SSSR 152(6), 1297–1300 (1963). (in Russian). English transl.: Soviet Math. Dokl. 4(5), 1547–1551 (1963)Google Scholar
- 21.Maz’ya, V.G.: Behavior, near the boundary, of solutions of the Dirichlet problem for a second-order elliptic equation in divergent form, Mat. Zametki 2(2), 209–220 (1967). (in Russian). English transl.: Math. Notes 2, 610–617 (1967)Google Scholar
- 22.Maz’ya, V.G.: On the continuity at a boundary point of solutions of quasi-linear elliptic equations. Vestnik Leningrad Univ. Math. 3, 225–242 (1976)Google Scholar
- 23.Skrypnik, I.I.: Regularity of a boundary point for singular parabolic equations with measurable coefficients. Ukraïn. Mat. Zh. 56(4), 506–516 (2004). (in Russian). English transl. Ukrainian Math. J. 56(4), 614–627 (2004)Google Scholar