# American Institute of Mathematical Sciences

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

## Higher order parabolic boundary Harnack inequality in C1 and Ck, 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 Ck, 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.

Citation: Teo Kukuljan. Higher order parabolic boundary Harnack inequality in C1 and Ck, 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.

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##### 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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