Abstract
This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for the gradient. The results extend to the parabolic systems as well. The higher integrability estimates provide a useful tool in several applications.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Acerbi, E., Mingione, G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136(2), 285–320 (2007)
Ancona, A.: On strong barriers and an inequality of Hardy for domains in R n. J. Lond. Math. Soc. (2) 34(2), 274–290 (1986)
Arkhipova, A.A.: L p-estimates for the gradients of solutions of initial boundary value problems to quasilinear parabolic systems (Russian). St. Petersburg State Univ., Problems Math. Anal. 13, 5–18 (1992)
Arkhipova, A.A.: Reverse Hölder inequalities with boundary integrals and L p-estimates for solutions of nonlinear elliptic and parabolic boundary-value problems. In: Nonlinear Evolution Equations, Amer. Math. Soc. Transl. Ser. 2, vol. 164, pp. 15–42. Amer. Math. Soc., Providence, RI (1995)
Bojarski, B.V.: Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients (Russian). Mat. Sb. N.S. 43(85), 451–503 (1957)
DiBenedetto, E.: Degenerate Parabolic Equations. Universitext. Springer, New York (1993)
Duzaar, F., Mingione, G.: The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. Partial Differential Equations 20(3), 235–256 (2004)
Duzaar, F., Mingione, G.: Second order parabolic systems, optimal regularity, and singular sets of solutions. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(6), 705–751 (2005)
Elcrat, A., Meyers, N.G.: Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions. Duke Math. J. 42, 121–136 (1975)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Gehring, F.W.: The L p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)
Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)
Giaquinta, M., Modica, G.: Regularity results for some classes of higher order nonlinear elliptic systems. J. Reine Angew. Math. 311/312, 145–169 (1979)
Giaquinta, M., Struwe, M.: On the partial regularity of weak solutions of nonlinear parabolic systems. Math. Z. 179(4), 437–451 (1982)
Granlund, S.: An L p-estimate for the gradient of extremals. Math. Scand. 50(1), 66–72 (1982)
Hedberg, L.I.: Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem. Acta Math. 147(3–4), 237–264 (1981)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. Oxford University Press, New York (1993)
Kilpeläinen, T., Koskela, P.: Global integrability of the gradients of solutions to partial differential equations. Nonlinear Anal. 23(7), 899–909 (1994)
Kinnunen, J., Lewis, J.L.: Higher integrability for parabolic systems of p-Laplacian type. Duke Math. J. 102(2), 253–271 (2000)
Lewis, J.L.: Uniformly fat sets. Trans. Amer. Math. Soc. 308(1), 177–196 (1988)
Malý, J., Ziemer, W.P.: Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51. American Mathematical Society, Providence, RI (1997)
Maz’ja, V.G.: Sobolev Spaces. Springer Series in Soviet Mathematics. Springer, Berlin (1985)
Mikkonen, P.: On the Wolff potential and quasilinear elliptic equations involving measures. Ann. Acad. Sci. Fenn. Math. Diss. 104, 1–71 (1996)
Parviainen, M.: Global higher integrability for parabolic quasiminimizers in nonsmooth domains. Calc. Var. Partial Differential Equations 31(1), 75–98 (2008)
Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series. vol. 43. Princeton University Press, Princeton (1993)
Stredulinsky, E.W.: Higher integrability from reverse Hölder inequalities. Indiana Univ. Math. J. 29(3), 407–413 (1980)
Zygmund, A.: On the differentiability of multiple integrals. Fund. Math. 23, 143–149 (1934)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Parviainen, M. Global gradient estimates for degenerate parabolic equations in nonsmooth domains. Annali di Matematica 188, 333–358 (2009). https://doi.org/10.1007/s10231-008-0079-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-008-0079-0
Keywords
- Boundary value problem
- Gehring lemma
- Global higher integrability
- Initial value problem
- Reverse Hölder inequality