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Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights

Part of: Elliptic equations and systems Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  28 April 2023

Changxing Miao  and
Zhiwen Zhao
Changxing Miao
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, China Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China (miao_changxing@iapcm.ac.cn)
Zhiwen Zhao
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, China (zwzhao365@163.com)
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Abstract

The main purpose of this paper is to capture the asymptotic behaviour for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the degenerate or singular point, especially covering the weighted p-Laplace equations and weighted fast diffusion equations. As a consequence, we also establish the local Hölder estimates for their solutions in the presence of single power-type weights.

Keywords

local regularityanisotropic weightsweighted Poincaré inequalityweighted p-Laplace equationsweighted fast diffusion equations

MSC classification

Primary: 35B65: Smoothness and regularity of solutions 35J92: Quasilinear elliptic equations with $p$-Laplacian 35K57: Reaction-diffusion equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 2 , May 2023 , pp. 391 - 436
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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References

Akagi, G., Rates of convergence to non-degenerate asymptotic profiles for fast diffusion via energy methods, Arch. Ration. Mech. Anal. 247(no. 2) (2023), .CrossRefGoogle Scholar
Badiale, M. and Tarantello, G., A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 163(4) (2002), 259–293.CrossRefGoogle Scholar
Bahouri, H., Chemin, J. -Y. and Gallagher, I., Refined Hardy inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5(3) (2006), 375–391.Google Scholar
Berryman, J. B. and Holland, C. J., Stability of the separable solution for fast diffusion, Arch. Ration. Mech. Anal. 74 (1980), 379–388.CrossRefGoogle Scholar
Bonforte, M. and Figalli, A., Sharp extinction rates for fast diffusion equations on generic bounded domains, Comm. Pure Appl. Math. 74(4) (2021), 744–789.CrossRefGoogle Scholar
Bonforte, M., Grillo, G. and Vázquez, J. L., Behaviour near extinction for the fast diffusion equation on bounded domains, J. Math. Pures Appl. (9) 97(1) (2012), 1–38.CrossRefGoogle Scholar
Bonforte, M. and Vázquez, J. L., Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations, Adv. Math. 223(2) (2010), 529–578.CrossRefGoogle Scholar
Cabré, X. and Ros-Oton, X., Sobolev and isoperimetric inequalities with monomial weights, J. Differential Equations 255(11) (2013), 4312–4336.CrossRefGoogle Scholar
Cabré, X., Ros-Oton, X. and Serra, J., Sharp isoperimetric inequalities via the ABP method, J. Eur. Math. Soc. (JEMS) 18(12) (2016), 2971–2998.CrossRefGoogle Scholar
Caffarelli, L., Kohn, R. and Nirenberg, L., First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259–275.Google Scholar
Chanillo, S. and Wheeden, R. L., Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math. 107(5) (1985), 1191–1226.CrossRefGoogle Scholar
Chen, Y. Z. and DiBenedetto, E., On the local behavior of solutions of singular parabolic equations, Arch. Ration. Mech. Anal. 103(4) (1988), 319–345.Google Scholar
Chiarenza, F. and Serapioni, R. P., A Harnack inequality for degenerate parabolic equations, Comm. Partial Differential Equations 9(8) (1984), 719–749.CrossRefGoogle Scholar
Chua, S. -K. and Wheeden, R. L., Estimates of best constants for weighted Poincaré inequalities on convex domains, Proc. Lond. Math. Soc. 93(1) (2006), 197–226.CrossRefGoogle Scholar
Daskalopoulos, P. and Kenig, C., Degenerate diffusions. Initial value problems and local regularity theory, EMS Tracts in Mathematics, Volume 1 (European Mathematical Society (EMS), Zürich, 2007).CrossRefGoogle Scholar
De Giorgi, E., Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43.Google Scholar
DiBenedetto, E., Gianazza, U. and Vespri, V., Harnack’s inequality for degenerate and singular parabolic equations, Springer Monographs in Mathematics (Springer, New York, 2012).CrossRefGoogle Scholar
DiBenedetto, E. and Kwong, Y. C., Harnack estimates and extinction profile for weak solution of certain singular parabolic equations, Trans. Amer. Math. Soc. 330(2) (1992), 783–811.CrossRefGoogle Scholar
DiBenedetto, E., Kwong, Y. C. and Vespri, V., Local space-analyticity of solutions of certain singular parabolic equations, Indiana Univ. Math. J. 40(2) (1991), 741–765.CrossRefGoogle Scholar
Dong, H. J., Li, Y. Y. and Yang, Z. L., Optimal gradient estimates of solutions to the insulated conductivity problem in dimension greater than two, arXiv:2110.11313.Google Scholar
Dong, H. J., Li, Y. Y. and Yang, Z. L., Gradient estimates for the insulated conductivity problem: the non-umbilical case, arXiv:2203.10081.Google Scholar
Fabes, E. B., Kenig, C. E. and Serapioni, R. P., The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7(1) (1982), 77–116.CrossRefGoogle Scholar
Feireisl, E. and Simondon, F., Convergence for semilinear degenerate parabolic equations in several space dimension, J. Dynam. Differential Equations 12 (2000), 647–673.CrossRefGoogle Scholar
Grafakos, L., Classical Fourier analysis, Graduate Texts in Mathematics, 3rd edn, Volume 249 (Springer, New York, 2014).CrossRefGoogle Scholar
Gutiérrez, C. E. and Wheeden, R. L., Harnack’s inequality for degenerate parabolic equations, Comm. Partial Differential Equations 16(4–5) (1991), 745–770.CrossRefGoogle Scholar
Heinonen, J., Kilpeläinen, T. and Martio, O., Nonlinear potential theory of degenerate elliptic equations, Unabridged Republication of the 1993 Original (Dover Publications, Inc., Mineola, NY, 2006).Google Scholar
Jin, T. L., and Xiong, J. G., Optimal boundary regularity for fast diffusion equations in bounded domains, Amer. J. Math. 145(no. 1) (2023), 151–219.CrossRefGoogle Scholar
Jin, T. L. and Xiong, J. G., Regularity of solutions to the Dirichlet problem for fast diffusion equations, arXiv:2201.10091.Google Scholar
Kwong, Y. C., Interior and boundary regularity of solutions to a plasma type equation, Proc. Amer. Math. Soc. 104(2) (1988), 472–478.CrossRefGoogle Scholar
Li, Y. Y., and Yan, X. K., Anisotropic Caffarelli–Kohn–Nirenberg type inequalities, Adv. Math. 419 (2023), .CrossRefGoogle Scholar
Lin, C. S., Interpolation inequalities with weights, Comm. Partial Differential Equations 11(14) (1986), 1515–1538.Google Scholar
Lindqvist, P., Notes on the stationary p-Laplace equation, SpringerBriefs in Mathematics, (Springer, Cham, 2019).CrossRefGoogle Scholar
Nguyen, H. and Squassina, M., Fractional Caffarelli–Kohn–Nirenberg inequalities, J. Funct. Anal. 274(9) (2018), 2661–2672.CrossRefGoogle Scholar
Nguyen, H. and Squassina, M., On Hardy and Caffarelli–Kohn–Nirenberg inequalities, J. Anal. Math. 139(2) (2019), 773–797.CrossRefGoogle Scholar
Sacks, P. E., Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7(4) (1983), 387–409.CrossRefGoogle Scholar
Serrin, J., Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302.CrossRefGoogle Scholar
Surnachev, M., A Harnack inequality for weighted degenerate parabolic equations, J. Differential Equations 248(8) (2010), 2092–2129.CrossRefGoogle Scholar
Trudinger, N. S., On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747.CrossRefGoogle Scholar
Vázquez, J. L., The porous medium equation: mathematical theory, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, Oxford, 2007).Google Scholar