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On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation

Year 2021, , 93 - 103, 01.03.2021
https://doi.org/10.33205/cma.824336

Abstract

We give a proof of H ̈older continuity for bounded local weak solutions to the equation

ut =\sum_{i=1}^N (|u_{x_i}|^{p_i−2} u_{x_i} )_{x_i} , in Ω × [0, T], with Ω ⊂⊂ R^N

under the condition 2 < pi < p(1 + 2/N) for each i = 1, .., N, being p the harmonic mean of the pi's, via
recently discovered intrinsic Harnack estimates. Moreover we establish equivalent forms of these Harnack
estimates within the proper intrinsic geometry.

Supporting Institution

Università degli Studi di Firenze

Thanks

Prof. Francesco Altomare.

References

  • S. Antontsev, S. Shmarev: Evolution PDEs with nonstandard growth conditions, Atlantis Studies in Differential Equations 4, Atlantis Press, Paris (2015).
  • L. Boccardo, P. Marcellini: L∞-Regularity for Variational Problems with Sharp Non Standard Growth Conditions, Bollettino della Unione Matematica Italiana, 7 (4-A), 219-226, 1990.
  • P. Bousquet, L. Brasco: Lipschitz regularity for orthotropic functionals with nonstandard growth conditions, Rev. Mat. Iberoam, Electronically published on April 7, 2020.
  • S. Ciani, V. Vespri: A new short proof of regularity for local weak solutions for a certain class of singular parabolic equations, Rend. Mat. Appl., 41 (7), 251-264, 2020.
  • S. Ciani, V. Vespri: An Introduction to Barenblatt Solutions for Anisotropic p-Laplace Equations, Anomalies in partial differential equations Springer Indam Series. Cicognani, Del Santo, Parmeggiani and Reissig Editors. In press
  • S. Ciani, S. Mosconi and V. Vespri: Parabolic Harnack estimates for anisotropic slow diffusion, (https://arxiv.org/pdf/2012.09685.pdf).
  • E. DiBenedetto: Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York (1993).
  • E. DiBenedetto, U. Gianazza and V. Vespri: Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Mathematica, 200 (2), 181-209, 2008.
  • E. DiBenedetto, U. Gianazza and V. Vespri: Alternative forms of the Harnack inequality for non-negative solutions to certain degenerate and singular parabolic equations, Rendiconti Lincei-Matematica e Applicazioni, 20 (4), 369-377, 2009.
  • E. DiBenedetto, U. Gianazza and V. Vespri: Harnack’s Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, Springer-Verlag, New York (2012).
  • E. DiBenedetto, U. Gianazza and V. Vespri: Remarks on Local Boundedness and Local Holder Continuity of Local Weak Solutions to Anisotropic p-Laplacian Type Equations, Journal of Elliptic and Parabolic Equations 2 (1-2), 157-169, 2016.
  • F. G. Düzgün, P. Marcellini and V. Vespri: Space expansion for a solution of an anisotropic p-Laplacian equation by using a parabolic approach, Riv. Mat. Univ. Parma, 5 (1), 2014.
  • F. G. Düzgün, S. Mosconi and V. Vespri: Anisotropic Sobolev embeddings and the speed of propagation for parabolic equations, Journal of Evolution Equations, 19 (3), 845-882, 2019.
  • M. Eleuteri, P. Marcellini and E. Mascolo: Regularity for scalar integrals without structure conditions, Advances in Calculus of Variations, 2018.
  • M. Giaquinta: Growth conditions and regularity, a counterexample, Manuscripta Mathematica, 59 (2), 245-248, 1987.
  • J. Haškovec, C. Schmeiser: A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems, Monatshefte für Mathematik 158 (1), 71-79, 2009.
  • I. M. Kolodii: The boundedness of generalized solutions of elliptic differential equations, Moscow Univ. Math. Bull., 25, 31–37, 1970.
  • A. G. Korolev: Boundedness of generalized solutions of elliptic differential equations, Russian Math. Surveys, 38, 186–187, 1983.
  • P. Marcellini: Un example de solution discontinue d’un probleme variationnel dans ce cas scalaire, preprint, Istituto Matematico “U. Dini”, Universitá di Firenze, 88, 1987.
  • P. Marcellini: Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Archive for Rational Mechanics and Analysis, 105 (3), 267-284, 1989.
  • P. Marcellini: Regularity and existence of solutions of elliptic equations with (p, q)-growth conditions, Journal of Differential Equations, 90 (1), 1-30, 1991.
  • P. Marcellini: Regularity under general and p, q-growth conditions, Dicrete Contin. Dyn. Syst. Ser., 13, 2009–2031, 2020.
  • Y. Mingqi, L. Xiting: Boundedness of solutions of parabolic equations with anisotropic growth conditions, Canadian Journal of Mathematics, 49 (4), 798-809, 1997.
  • J. Moser: A Harnack inequality for parabolic differential equations, Communications on Pure and Applied Mathematics, 17 (1), 101-134, 1964.
  • B. Pini: Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rendiconti del Seminario Matematico della Universita di Padova, 23, 422-434, 1954.
  • M. Ruzicka: Electrorheological fluids: modeling and mathematical theory, Springer Science and Business Media, 2000.
  • I. I. Skrypnik: Removability of an isolated singularity for anisotropic elliptic equations with absorption, Sbornik: Mathematics, 199 (7), 1033-1050, 2008.
  • N. N. Ural’tseva, A. B. Urdaletova: The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations, Vest. Leningr. Univ. Math, 16, 263-270, 1984.
Year 2021, , 93 - 103, 01.03.2021
https://doi.org/10.33205/cma.824336

Abstract

References

  • S. Antontsev, S. Shmarev: Evolution PDEs with nonstandard growth conditions, Atlantis Studies in Differential Equations 4, Atlantis Press, Paris (2015).
  • L. Boccardo, P. Marcellini: L∞-Regularity for Variational Problems with Sharp Non Standard Growth Conditions, Bollettino della Unione Matematica Italiana, 7 (4-A), 219-226, 1990.
  • P. Bousquet, L. Brasco: Lipschitz regularity for orthotropic functionals with nonstandard growth conditions, Rev. Mat. Iberoam, Electronically published on April 7, 2020.
  • S. Ciani, V. Vespri: A new short proof of regularity for local weak solutions for a certain class of singular parabolic equations, Rend. Mat. Appl., 41 (7), 251-264, 2020.
  • S. Ciani, V. Vespri: An Introduction to Barenblatt Solutions for Anisotropic p-Laplace Equations, Anomalies in partial differential equations Springer Indam Series. Cicognani, Del Santo, Parmeggiani and Reissig Editors. In press
  • S. Ciani, S. Mosconi and V. Vespri: Parabolic Harnack estimates for anisotropic slow diffusion, (https://arxiv.org/pdf/2012.09685.pdf).
  • E. DiBenedetto: Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York (1993).
  • E. DiBenedetto, U. Gianazza and V. Vespri: Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Mathematica, 200 (2), 181-209, 2008.
  • E. DiBenedetto, U. Gianazza and V. Vespri: Alternative forms of the Harnack inequality for non-negative solutions to certain degenerate and singular parabolic equations, Rendiconti Lincei-Matematica e Applicazioni, 20 (4), 369-377, 2009.
  • E. DiBenedetto, U. Gianazza and V. Vespri: Harnack’s Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, Springer-Verlag, New York (2012).
  • E. DiBenedetto, U. Gianazza and V. Vespri: Remarks on Local Boundedness and Local Holder Continuity of Local Weak Solutions to Anisotropic p-Laplacian Type Equations, Journal of Elliptic and Parabolic Equations 2 (1-2), 157-169, 2016.
  • F. G. Düzgün, P. Marcellini and V. Vespri: Space expansion for a solution of an anisotropic p-Laplacian equation by using a parabolic approach, Riv. Mat. Univ. Parma, 5 (1), 2014.
  • F. G. Düzgün, S. Mosconi and V. Vespri: Anisotropic Sobolev embeddings and the speed of propagation for parabolic equations, Journal of Evolution Equations, 19 (3), 845-882, 2019.
  • M. Eleuteri, P. Marcellini and E. Mascolo: Regularity for scalar integrals without structure conditions, Advances in Calculus of Variations, 2018.
  • M. Giaquinta: Growth conditions and regularity, a counterexample, Manuscripta Mathematica, 59 (2), 245-248, 1987.
  • J. Haškovec, C. Schmeiser: A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems, Monatshefte für Mathematik 158 (1), 71-79, 2009.
  • I. M. Kolodii: The boundedness of generalized solutions of elliptic differential equations, Moscow Univ. Math. Bull., 25, 31–37, 1970.
  • A. G. Korolev: Boundedness of generalized solutions of elliptic differential equations, Russian Math. Surveys, 38, 186–187, 1983.
  • P. Marcellini: Un example de solution discontinue d’un probleme variationnel dans ce cas scalaire, preprint, Istituto Matematico “U. Dini”, Universitá di Firenze, 88, 1987.
  • P. Marcellini: Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Archive for Rational Mechanics and Analysis, 105 (3), 267-284, 1989.
  • P. Marcellini: Regularity and existence of solutions of elliptic equations with (p, q)-growth conditions, Journal of Differential Equations, 90 (1), 1-30, 1991.
  • P. Marcellini: Regularity under general and p, q-growth conditions, Dicrete Contin. Dyn. Syst. Ser., 13, 2009–2031, 2020.
  • Y. Mingqi, L. Xiting: Boundedness of solutions of parabolic equations with anisotropic growth conditions, Canadian Journal of Mathematics, 49 (4), 798-809, 1997.
  • J. Moser: A Harnack inequality for parabolic differential equations, Communications on Pure and Applied Mathematics, 17 (1), 101-134, 1964.
  • B. Pini: Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rendiconti del Seminario Matematico della Universita di Padova, 23, 422-434, 1954.
  • M. Ruzicka: Electrorheological fluids: modeling and mathematical theory, Springer Science and Business Media, 2000.
  • I. I. Skrypnik: Removability of an isolated singularity for anisotropic elliptic equations with absorption, Sbornik: Mathematics, 199 (7), 1033-1050, 2008.
  • N. N. Ural’tseva, A. B. Urdaletova: The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations, Vest. Leningr. Univ. Math, 16, 263-270, 1984.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Simone Ciani 0000-0001-7065-4163

Vincenzo Vesprı This is me 0000-0002-2684-8646

Publication Date March 1, 2021
Published in Issue Year 2021

Cite

APA Ciani, S., & Vesprı, V. (2021). On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation. Constructive Mathematical Analysis, 4(1), 93-103. https://doi.org/10.33205/cma.824336
AMA Ciani S, Vesprı V. On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation. CMA. March 2021;4(1):93-103. doi:10.33205/cma.824336
Chicago Ciani, Simone, and Vincenzo Vesprı. “On Hölder Continuity and Equivalent Formulation of Intrinsic Harnack Estimates for an Anisotropic Parabolic Degenerate Prototype Equation”. Constructive Mathematical Analysis 4, no. 1 (March 2021): 93-103. https://doi.org/10.33205/cma.824336.
EndNote Ciani S, Vesprı V (March 1, 2021) On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation. Constructive Mathematical Analysis 4 1 93–103.
IEEE S. Ciani and V. Vesprı, “On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation”, CMA, vol. 4, no. 1, pp. 93–103, 2021, doi: 10.33205/cma.824336.
ISNAD Ciani, Simone - Vesprı, Vincenzo. “On Hölder Continuity and Equivalent Formulation of Intrinsic Harnack Estimates for an Anisotropic Parabolic Degenerate Prototype Equation”. Constructive Mathematical Analysis 4/1 (March 2021), 93-103. https://doi.org/10.33205/cma.824336.
JAMA Ciani S, Vesprı V. On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation. CMA. 2021;4:93–103.
MLA Ciani, Simone and Vincenzo Vesprı. “On Hölder Continuity and Equivalent Formulation of Intrinsic Harnack Estimates for an Anisotropic Parabolic Degenerate Prototype Equation”. Constructive Mathematical Analysis, vol. 4, no. 1, 2021, pp. 93-103, doi:10.33205/cma.824336.
Vancouver Ciani S, Vesprı V. On Hölder continuity and equivalent formulation of intrinsic Harnack estimates for an anisotropic parabolic degenerate prototype equation. CMA. 2021;4(1):93-103.