## Global gradient estimates for degenerate parabolic equations in nonsmooth domains.(English)Zbl 1179.35080

The author studies the global regularity of the solutions of degenerate parabolic equations of the form
$u_t=\operatorname{div}A(x,t,\nabla u),$
where $$A(x,t,\nabla u)$$ satisfies the Carathédory-type conditions and the $$p$$-growth conditions. It is shown that the 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. Integrability estimates for the gradient are obtained. The results are extended to parabolic systems as well.

### MSC:

 35B45 A priori estimates in context of PDEs 35K55 Nonlinear parabolic equations 49N60 Regularity of solutions in optimal control 35K65 Degenerate parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations
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### References:

 [1] Acerbi, E.; Mingione, G., Gradient estimates for a class of parabolic systems, Duke Math. J., 136, 2, 285-320 (2007) · Zbl 1113.35105 [2] 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) · Zbl 0629.31002 [3] 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) [4] 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) · Zbl 0838.35021 [5] 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) [6] Dibenedetto, E., Degenerate Parabolic Equations (1993), New York: Universitext. Springer, New York · Zbl 0794.35090 [7] 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) · Zbl 1142.35433 [8] 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) · Zbl 1099.35042 [9] 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) · Zbl 0347.35039 [10] Evans, L. C.; Gariepy, R. F., Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (1992), Boca Raton: CRC Press, Boca Raton · Zbl 0804.28001 [11] Gehring, F. W., The L^p-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130, 265-277 (1973) · Zbl 0258.30021 [12] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105 (1983), Princeton: Princeton University Press, Princeton · Zbl 0516.49003 [13] Giaquinta, M.; Modica, G., Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Angew. Math., 311/312, 145-169 (1979) · Zbl 0409.35015 [14] Giaquinta, M.; Struwe, M., On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179, 4, 437-451 (1982) · Zbl 0469.35028 [15] Granlund, S., An L^p-estimate for the gradient of extremals, Math. Scand., 50, 1, 66-72 (1982) · Zbl 0489.49009 [16] Hedberg, L. I., Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem, Acta Math., 147, 3-4, 237-264 (1981) · Zbl 0504.35018 [17] Heinonen, J.; Kilpeläinen, T.; Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs (1993), New York: Oxford University Press, New York · Zbl 0780.31001 [18] Kilpeläinen, T.; Koskela, P., Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal., 23, 7, 899-909 (1994) · Zbl 0820.35064 [19] Kinnunen, J.; Lewis, J. L., Higher integrability for parabolic systems of p-Laplacian type, Duke Math. J., 102, 2, 253-271 (2000) · Zbl 0994.35036 [20] Lewis, J. L., Uniformly fat sets, Trans. Amer. Math. Soc., 308, 1, 177-196 (1988) · Zbl 0668.31002 [21] 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) · Zbl 0882.35001 [22] Maz’Ja, V. G., Sobolev Spaces. Springer Series in Soviet Mathematics (1985), Berlin: Springer, Berlin [23] Mikkonen, P., On the Wolff potential and quasilinear elliptic equations involving measures, Ann. Acad. Sci. Fenn. Math. Diss., 104, 1-71 (1996) · Zbl 0860.35041 [24] Parviainen, M., Global higher integrability for parabolic quasiminimizers in nonsmooth domains, Calc. Var. Partial Differential Equations, 31, 1, 75-98 (2008) · Zbl 1173.35036 [25] Stein, E. M., Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series. vol. 43 (1993), Princeton: Princeton University Press, Princeton · Zbl 0821.42001 [26] Stredulinsky, E. W., Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J., 29, 3, 407-413 (1980) · Zbl 0442.35064 [27] Zygmund, A., On the differentiability of multiple integrals, Fund. Math., 23, 143-149 (1934) · JFM 60.0219.02
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