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Global well-posedness for fractional Sobolev-Galpern type equations
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 |
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.
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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