Higher order parabolic boundary Harnack inequality in <i>C</i><sup>1</sup> and <i>C</i><sup><i>k</i>, <i>α</i></sup> domains

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June 2022, 42(6): 2667-2698. doi: 10.3934/dcds.2021207

Higher order parabolic boundary Harnack inequality in C¹ and C^{k, α} domains

Universitat de Barcelona, Departament de Matematiques i Informàtica, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain, Springfield, MO 65801-2604, USA

Received September 2021 Published June 2022 Early access January 2022

Fund Project: The author has received funding from the European Research Council (ERC) under the Grant Agreement No 801867, and from the Swiss National Science Foundation project 200021_178795

We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in C¹ and C^{k, α} domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary.

As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.

Keywords: Parabolic equations, boundary Harnack, free boundary, boundary regularity, higher regularity.

Mathematics Subject Classification: Primary: 35K10; Secondary: 35R35.

Citation: Teo Kukuljan. Higher order parabolic boundary Harnack inequality in C¹ and C^{k, α} domains. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2667-2698. doi: 10.3934/dcds.2021207

References:

[1]	N. Abatangelo and X. Ros-Oton, Obstacle problems for integro-differential operators: Higher regularity of free boundaries, Adv. Math., 360 (2020), 106931, 61 pp. doi: 10.1016/j.aim.2019.106931.
[2]	A. Banerjee and N. Garofalo, A Parabolic analogue of the higher-order comparison theorem of De Silva and Savin, J. Differential Equations, 260 (2016), 1801-1829. doi: 10.1016/j.jde.2015.09.044.
[3]	A. Baneerje, M. Smit Vega Garcia and A. Zeller, Higher regularity of the free boundary in the parabolic Signorini problem, Calc. Var. Partial Differential Equations, 56 (2017), 7, 26 pp. doi: 10.1007/s00526-016-1103-7.
[4]	G. Barles, A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time, C. R. Acad. Sci. Paris, 343 (2006), 173-178. doi: 10.1016/j.crma.2006.06.022.
[5]	L. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184. doi: 10.1007/BF02392236.
[6]	D. De Silva and O. Savin, A note on higher regularity boundary Harnack inequality, Disc. Cont. Dyn. Syst., 35 (2015), 6155-6163. doi: 10.3934/dcds.2015.35.6155.
[7]	D. dos Prazeres and J. V. da Silva, Schauder type estimates for "flat" viscosity solutions to non-convex fully nonlinear parabolic equations and applications, Potential Anal., 50 (2019), 149-170. doi: 10.1007/s11118-017-9677-z.
[8]	X. Fernández-Real and X. Ros-Oton, Regularity Theory for Elliptic PDE, Forthcoming book (2020), Available at the Webpage of the Authors.
[9]	A. Figalli, Regularity of Interfaces in Phase Transitions Via Obstacle Problems, In Proceedings of the ICM 2018.
[10]	A. Figalli and H. Shahgholian, A general class of free boundary problems for fully nonlinear parabolic equations, Ann. Mat. Pura Appl., 194 (2015), 1123-1134. doi: 10.1007/s10231-014-0413-7.
[11]	Y. Jhaveri and R. Neumayer, Higher regularity of the free boundary in the obstacle problem for the fractional Laplacian, Adv. Math, 311 (2017), 748-795.
[12]	D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 373-391.
[13]	N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics vol.12, American Mathematical Society 1996. doi: 10.1090/gsm/012.
[14]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd. 1996. doi: 10.1142/3302.
[15]	G. M. Lieberman, Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352. doi: 10.2140/pjm.1985.117.329.
[16]	E. Lindgren and R. Monneau, Pointwise estimates for the heat equation. Application to the free boundary of the obstacle problem with Dini coefficients, Indiana Univ. Math. J., 62 (2013), 171-199. doi: 10.1512/iumj.2013.62.4837.
[17]	E. Lindgren and R. Monneau, Pointwise regularity of the free boundary for the parabolic obstacle problem, Calc. Var. Partial Differential Equations, 54 (2015), 299-347. doi: 10.1007/s00526-014-0787-9.

show all references

References:

[1]	N. Abatangelo and X. Ros-Oton, Obstacle problems for integro-differential operators: Higher regularity of free boundaries, Adv. Math., 360 (2020), 106931, 61 pp. doi: 10.1016/j.aim.2019.106931.
[2]	A. Banerjee and N. Garofalo, A Parabolic analogue of the higher-order comparison theorem of De Silva and Savin, J. Differential Equations, 260 (2016), 1801-1829. doi: 10.1016/j.jde.2015.09.044.
[3]	A. Baneerje, M. Smit Vega Garcia and A. Zeller, Higher regularity of the free boundary in the parabolic Signorini problem, Calc. Var. Partial Differential Equations, 56 (2017), 7, 26 pp. doi: 10.1007/s00526-016-1103-7.
[4]	G. Barles, A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time, C. R. Acad. Sci. Paris, 343 (2006), 173-178. doi: 10.1016/j.crma.2006.06.022.
[5]	L. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184. doi: 10.1007/BF02392236.
[6]	D. De Silva and O. Savin, A note on higher regularity boundary Harnack inequality, Disc. Cont. Dyn. Syst., 35 (2015), 6155-6163. doi: 10.3934/dcds.2015.35.6155.
[7]	D. dos Prazeres and J. V. da Silva, Schauder type estimates for "flat" viscosity solutions to non-convex fully nonlinear parabolic equations and applications, Potential Anal., 50 (2019), 149-170. doi: 10.1007/s11118-017-9677-z.
[8]	X. Fernández-Real and X. Ros-Oton, Regularity Theory for Elliptic PDE, Forthcoming book (2020), Available at the Webpage of the Authors.
[9]	A. Figalli, Regularity of Interfaces in Phase Transitions Via Obstacle Problems, In Proceedings of the ICM 2018.
[10]	A. Figalli and H. Shahgholian, A general class of free boundary problems for fully nonlinear parabolic equations, Ann. Mat. Pura Appl., 194 (2015), 1123-1134. doi: 10.1007/s10231-014-0413-7.
[11]	Y. Jhaveri and R. Neumayer, Higher regularity of the free boundary in the obstacle problem for the fractional Laplacian, Adv. Math, 311 (2017), 748-795.
[12]	D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 373-391.
[13]	N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics vol.12, American Mathematical Society 1996. doi: 10.1090/gsm/012.
[14]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd. 1996. doi: 10.1142/3302.
[15]	G. M. Lieberman, Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352. doi: 10.2140/pjm.1985.117.329.
[16]	E. Lindgren and R. Monneau, Pointwise estimates for the heat equation. Application to the free boundary of the obstacle problem with Dini coefficients, Indiana Univ. Math. J., 62 (2013), 171-199. doi: 10.1512/iumj.2013.62.4837.
[17]	E. Lindgren and R. Monneau, Pointwise regularity of the free boundary for the parabolic obstacle problem, Calc. Var. Partial Differential Equations, 54 (2015), 299-347. doi: 10.1007/s00526-014-0787-9.

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