Regularity for very weak solutions to elliptic equations of p-Laplacian type

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Abstract

We study the regularity problem with non-homogeneous terms of p-Laplacian type, which is a still unsolved problem for nonlinear elliptic equations. The main results of this work are obtained by three steps. First, we use the Hodge decomposition theorem to construct a suitable test function that satisfies the solution definition. Second, by combining the solution definition with the Hodge decomposition theorem, we establish a properly formulated inverse Hölder inequality to enhance the integrability of the very weak solutions. Finally, through an iterative process, we show that the considered very weak solutions can be improved to classical weak solutions.

Keywords

regularity theory / generalized weak solutions / Hodge-type decomposition / elliptic equations / p-Laplacian class

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References

[1] L. Diening, P. Kaplicky, S. Schwarzacher, BMO estimates for the p-laplacian, Nonlinear Anal. Theory Methods Appl., 75 (2012), 637-650. https://doi.org/10.1016/j.na.2011.08.065

[2] T. Iwaniec, p-harmonic tensors and quasiregular mappings, Ann. Math., 136 (1992), 589-624. https://doi.org/10.2307/2946602

[3] G. Mingione, Nonlinear aspects of calderon-zygmund theory, Jahresber. Dtsch. Math. Ver., 112 (2010), 159-191. https://doi.org/10.1365/s13291-010-0004-5

[4] J. Serrin, Isolated singularities of solutions of quasi-linear equations, Acta Math., 113 (1965), 219-240. https://doi.org/10.1007/BF02391778

[5] M. Bulicek, S. Schwarzacher, Existence of very weak solutions to elliptic systems of p-laplacian type, Calc. Var. Partial Differ. Equations, 55 (2016), 52. https://doi.org/10.1007/s00526-016-0986-7

[6] S. Chen, L. Guo, Existence of very weak solutions for a class of p-laplacian elliptic equations, J. Guangxi Univ. (Nat. Sci. Ed.), 48 (2023), 236-245. https://doi.org/10.13624/j.cnki.issn.1001-7445.2023.0236

[7] T. Iwaniec, C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math., 454 (1994), 143-161. https://doi.org/10.1515/crll.1994.454.143

[8] J. Kinnunen, S. Zhou, A note on very weak p-harmonic mappings, Electron. J. Differ. Equations, 1997 (1997), 1-4.

[9] L. Greco, A. Verde, A regularity property of p-harmonic functions, Ann. Acad. Sci. Fenn. Math., 25 (2000), 317-323.

[10] J. Kinnunen, J. L. Lewis, Very weak solutions of parabolic systems of p-laplacian type, Ark. Mat., 40 (2002), 105-132. https://doi.org/10.1007/BF02384505

[11] B. Stroffolini, A stability result for p-harmonic systems with discontinuous coefficients, Electron. J. Differ. Equations, 2001 (2001), 1-7.

[12] S. Zhou, Very weak solutions of p-laplacian type equations with VMO coefficients, J. Partial Differ. Equations, 14 (2001), 12-20.

[13] S. Baasandorj, S. Byun, W. Kim, Self-improving properties of very weak solutions to double phase systems, Trans. Am. Math. Soc., 376 (2023), 8733-8768. https://doi.org/10.1090/tran/9039

[14] S. Byun, M. Lim, Gradient estimates of very weak solutions to general quasilinear elliptic equations, J. Funct. Anal., 283 (2012), 109668 https://doi.org/10.1016/j.jfa.2022.109668

[15] M. Carozza, A. P. di Napoli, On very weak solutions of a class of nonlinear elliptic systems, Comment. Math. Univ. Carolin., 41 (2000), 493-508.

[16] G. Du, J. Han, Global higher integrability for very weak solutions to nonlinear subelliptic equations, Boundary Value Probl., 2017 (2017), 93. https://doi.org/10.1186/s13661-017-0825-6

[17] J. Merker, J. Rakotoson, Very weak solutions of poisson's equation with singular data under neumann boundary conditions, Calc. Var. Partial Differ. Equations, 52 (2015), 705-726. https://doi.org/10.1007/s00526-014-0730-0

[18] K. Adimurthi, S. Byun, J. Oh, Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents, Nonlinear Anal., 194 (2020), 111370. https://doi.org/10.1016/j.na.2018.10.014

[19] V. Bogelein, Q. Li, Very weak solutions of degenerate parabolic systems with non-standard p(x,t)-growth, Nonlinear Anal. Theory Methods Appl., 98 (2014), 190-225. https://doi.org/10.1016/j.na.2013.12.009

[20] J. Rakotoson, Regularity of a very weak solution for parabolic equations and applications, Adv. Differ. Equations, 16 (2011), 867-894. https://doi.org/10.57262/ade/1355703179

[21] J. L. Lewis, On the very weak solutions of certain elliptic systems, Commun. Partial Differ. Equations, 18 (1993), 1515-1537. https://doi.org/10.1080/03605309308820984

[22] T. Iwaniec, L. Migliaccio, L. Nania, C. Sbordone, Integrability and removability results for quasiregular mappings in high dimensions, Math. Scand., 75 (1994), 263-279.

[23] N. G. Meyers, A. Elcrat, Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J., 42 (1975), 121-136. https://doi.org/10.1215/S0012-7094-75-04211-8