A boundary estimate for singular parabolic diffusion equations. (English) Zbl 1401.35206
Summary: 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.
MSC:
| 35K67 | Singular parabolic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
parabolic \(p\)-Laplacian; boundary estimates; continuity; elliptic \(p\)-capacity; Wiener-type integralReferences:
| [1] | Avelin, B; Kuusi, T; Parviainen, M, Variational parabolic capacity, Discrete Contin. Dyn. Syst., 35, 5665-5688, (2015) · Zbl 1342.35141 · doi:10.3934/dcds.2015.35.5665 |
| [2] | Biroli, M; Mosco, U, Wiener estimates at boundary points for parabolic equations, Ann. Mat. Pura Appl. (4), 141, 353-367, (1985) · Zbl 0601.35015 · doi:10.1007/BF01763181 |
| [3] | Biroli, M; Mosco, U, Wiener estimates for parabolic obstacle problems, Nonlinear Anal., 11, 1005-1027, (1987) · Zbl 0662.35053 · doi:10.1016/0362-546X(87)90081-2 |
| [4] | Björn, A; Björn, J; Gianazza, U; Parviainen, M, Boundary regularity for degenerate and singular parabolic equations, Calc. Var. Part. Differ. Equ., 52, 797-827, (2015) · Zbl 1333.35087 · doi:10.1007/s00526-014-0734-9 |
| [5] | Bonforte, M; Iagar, RG; Vázquez, JL, Local smoothing effects, positivity, and Harnack inequalities for the fast \(p\)-Laplacian equation, Adv. Math., 224, 2151-2215, (2010) · Zbl 1198.35136 · doi:10.1016/j.aim.2010.01.023 |
| [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 1990 |
| [7] | DiBenedetto, E.: Degenerate Parabolic Equations, Universitext. Springer, New York (1993) · Zbl 0794.35090 · doi:10.1007/978-1-4612-0895-2 |
| [8] | DiBenedetto, E., Gianazza, U., Vespri, V.: Harnack’s Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics. Springer, New York (2012) · Zbl 1237.35004 · doi:10.1007/978-1-4614-1584-8 |
| [9] | Fornaro, S; Vespri, V, Harnack estimates for non-negative weak solutions of a class of singular parabolic equations, Manuscr. Math., 141, 85-103, (2013) · Zbl 1264.35065 · doi:10.1007/s00229-012-0562-1 |
| [10] | Frehse, J, Capacity methods in the theory of partial differential equations, Jahresber. Deutsch. Math.-Verein., 84, 1-44, (1982) · Zbl 0486.35002 |
| [11] | Gariepy, RF; Ziemer, WP, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal., 67, 25-39, (1977) · Zbl 0389.35023 · doi:10.1007/BF00280825 |
| [12] | Kilpeläinen, T; Lindqvist, P, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal., 27, 661-683, (1996) · Zbl 0857.35071 · doi:10.1137/0527036 |
| [13] | Kinnunen, J; Korte, R; Kuusi, T; Parviainen, M, Nonlinear parabolic capacity and polar sets of superparabolic functions, Math. Ann., 355, 1349-1381, (2013) · Zbl 1276.35111 · doi:10.1007/s00208-012-0825-x |
| [14] | Korte, R; Kuusi, T; Parviainen, M, A connection between a general class of superparabolic functions and supersolutions, J. Evol. Equ., 10, 1-20, (2010) · Zbl 1239.35076 · doi:10.1007/s00028-009-0037-3 |
| [15] | Kuusi, T, Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 7, 673-716, (2008) · Zbl 1178.35100 |
| [16] | Kuusi, T, Lower semicontinuity of weak supersolutions to nonlinear parabolic equations, Differ. Integral Equ., 22, 1211-1222, (2009) · Zbl 1240.35220 |
| [17] | Lewis, JL, Uniformly fat sets, Trans. Am. Math. Soc., 308, 177-196, (1988) · Zbl 0668.31002 · doi:10.1090/S0002-9947-1988-0946438-4 |
| [18] | Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations, SURV 51. AMS, Providence (1997) · Zbl 0882.35001 · doi:10.1090/surv/051 |
| [19] | Marchi, S, Boundary regularity for parabolic quasiminima, Ann. Mat. Pura Appl. (4), 166, 17-26, (1994) · Zbl 0816.35052 · doi:10.1007/BF01765627 |
| [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) · Zbl 0166.37901 |
| [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) · Zbl 1198.35136 |
| [22] | Maz’ya, VG, On the continuity at a boundary point of solutions of quasi-linear elliptic equations, Vestnik Leningrad Univ. Math., 3, 225-242, (1976) |
| [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) · Zbl 1142.35459 |
| [24] | Ziemer, WP, Behavior at the boundary of solutions of quasilinear parabolic equations, J. Differ. Equ., 35, 291-305, (1980) · Zbl 0451.35037 · doi:10.1016/0022-0396(80)90030-3 |
| [25] | Ziemer, WP, Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Am. Math. Soc., 271, 733-748, (1982) · Zbl 0506.35053 · doi:10.1090/S0002-9947-1982-0654859-7 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
