doi: 10.3934/dcds.2021207
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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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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L. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184.  doi: 10.1007/BF02392236.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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X. Fernández-Real and X. Ros-Oton, Regularity Theory for Elliptic PDE, Forthcoming book (2020), Available at the Webpage of the Authors. Google Scholar

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A. Figalli, Regularity of Interfaces in Phase Transitions Via Obstacle Problems, In Proceedings of the ICM 2018. Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[12]

D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 373-391.   Google Scholar

[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.  Google Scholar

[14]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd. 1996. doi: 10.1142/3302.  Google Scholar

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G. M. Lieberman, Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352.  doi: 10.2140/pjm.1985.117.329.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math., 139 (1977), 155-184.  doi: 10.1007/BF02392236.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

X. Fernández-Real and X. Ros-Oton, Regularity Theory for Elliptic PDE, Forthcoming book (2020), Available at the Webpage of the Authors. Google Scholar

[9]

A. Figalli, Regularity of Interfaces in Phase Transitions Via Obstacle Problems, In Proceedings of the ICM 2018. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 373-391.   Google Scholar

[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.  Google Scholar

[14]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Pte. Ltd. 1996. doi: 10.1142/3302.  Google Scholar

[15]

G. M. Lieberman, Regularized distance and its applications, Pacific J. Math., 117 (1985), 329-352.  doi: 10.2140/pjm.1985.117.329.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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