The regular free boundary in the thin obstacle problem for degenerate parabolic equations
HTML articles powered by AMS MathViewer
- by A. Banerjee, D. Danielli, N. Garofalo and A. Petrosyan
- St. Petersburg Math. J. 32, 449-480
- DOI: https://doi.org/10.1090/spmj/1656
- Published electronically: May 11, 2021
- PDF | Request permission
Abstract:
This paper is devoted to the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called regular points in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $(\partial _t - \Delta _x)^s$ for $s \in (0,1)$. The regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. The approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. By using several results from the elliptic theory, including the epiperimetric inequality, the optimal regularity is established for solutions as well as the $H^{1+\gamma ,\frac {1+\gamma }{2}}$ regularity of the free boundary near such regular points.References
- Ioannis Athanasopoulous, Regularity of the solution of an evolution problem with inequalities on the boundary, Comm. Partial Differential Equations 7 (1982), no. 12, 1453–1465. MR 679950, DOI 10.1080/03605308208820258
- Ioannis Athanasopoulos, Luis Caffarelli, and Emmanouil Milakis, On the regularity of the non-dynamic parabolic fractional obstacle problem, J. Differential Equations 265 (2018), no. 6, 2614–2647. MR 3804726, DOI 10.1016/j.jde.2018.04.043
- Ioannis Athanasopoulos, Luis Caffarelli, and Emmanouil Milakis, Parabolic obstacle problems, quasi-convexity and regularity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 2, 781–825. MR 3964414
- I. Athanasopoulos, L. A. Caffarelli, and S. Salsa, The structure of the free boundary for lower dimensional obstacle problems, Amer. J. Math. 130 (2008), no. 2, 485–498. MR 2405165, DOI 10.1353/ajm.2008.0016
- A. A. Arkhipova and N. N. Ural′tseva, Regularity of the solution of a problem with a two-sided limit on a boundary for elliptic and parabolic equations, Trudy Mat. Inst. Steklov. 179 (1988), 5–22, 241 (Russian). Translated in Proc. Steklov Inst. Math. 1989, no. 2, 1–19; Boundary value problems of mathematical physics, 13 (Russian). MR 964910
- A. Audrito and S. Terracini, On the nodal set of solutions to a class of nonlocal parabolic reaction-diffusion equations, arXiv:1807.10135, 2018.
- Agnid Banerjee and Nicola Garofalo, Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations, Adv. Math. 336 (2018), 149–241. MR 3846151, DOI 10.1016/j.aim.2018.07.021
- A. Banerjee, D. Danielli, N. Garofalo, and A. Petrosyan, The structure of the singular set in the thin obstacle problem for degenerate parabolic equations, arXiv:1902.07457, 2019.
- Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213. MR 1005611, DOI 10.2307/1971480
- L. Caffarelli, D. De Silva, and O. Savin, The two membranes problem for different operators, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 4, 899–932. MR 3661864, DOI 10.1016/j.anihpc.2016.05.006
- Luis A. Caffarelli, Sandro Salsa, and Luis Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), no. 2, 425–461. MR 2367025, DOI 10.1007/s00222-007-0086-6
- Filippo Chiarenza and Raul Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova 73 (1985), 179–190. MR 799906
- Donatella Danielli, Nicola Garofalo, Arshak Petrosyan, and Tung To, Optimal regularity and the free boundary in the parabolic Signorini problem, Mem. Amer. Math. Soc. 249 (2017), no. 1181, v+103. MR 3709717, DOI 10.1090/memo/1181
- G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques, No. 21, Dunod, Paris, 1972 (French). MR 0464857
- M. Focardi and E. Spadaro, The local structure of the free boundary in the fractional obstacle problem, arXiv:1903.05909, 2019.
- Matteo Focardi and Emanuele Spadaro, On the measure and the structure of the free boundary of the lower dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 (2018), no. 1, 125–184. MR 3840912, DOI 10.1007/s00205-018-1242-4
- Matteo Focardi and Emanuele Spadaro, An epiperimetric inequality for the thin obstacle problem, Adv. Differential Equations 21 (2016), no. 1-2, 153–200. MR 3449333
- Nicola Garofalo and Arshak Petrosyan, Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem, Invent. Math. 177 (2009), no. 2, 415–461. MR 2511747, DOI 10.1007/s00222-009-0188-4
- Nicola Garofalo, Arshak Petrosyan, and Mariana Smit Vega Garcia, An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients, J. Math. Pures Appl. (9) 105 (2016), no. 6, 745–787 (English, with English and French summaries). MR 3491531, DOI 10.1016/j.matpur.2015.11.013
- Nicola Garofalo, Arshak Petrosyan, Camelia A. Pop, and Mariana Smit Vega Garcia, Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 3, 533–570. MR 3633735, DOI 10.1016/j.anihpc.2016.03.001
- Nicola Garofalo and Xavier Ros-Oton, Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian, Rev. Mat. Iberoam. 35 (2019), no. 5, 1309–1365. MR 4018099, DOI 10.4171/rmi/1087
- Nicola Garofalo and Mariana Smit Vega Garcia, New monotonicity formulas and the optimal regularity in the Signorini problem with variable coefficients, Adv. Math. 262 (2014), 682–750. MR 3228440, DOI 10.1016/j.aim.2014.05.021
- H. Koch, A. Rüland, and W. Shi, Higher regularity for the fractional thin obstacle problem, arXiv:1605.06662, 2016.
- Aleš Nekvinda, Characterization of traces of the weighted Sobolev space $W^{1,p}(\Omega ,d^\epsilon _M)$ on $M$, Czechoslovak Math. J. 43(118) (1993), no. 4, 695–711. MR 1258430, DOI 10.21136/CMJ.1993.128436
- K. Nyström and O. Sande, Extension properties and boundary estimates for a fractional heat operator, Nonlinear Anal. 140 (2016), 29–37. MR 3492726, DOI 10.1016/j.na.2016.02.027
- Arshak Petrosyan and Camelia A. Pop, Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift, J. Funct. Anal. 268 (2015), no. 2, 417–472. MR 3283160, DOI 10.1016/j.jfa.2014.10.009
- Arshak Petrosyan and Andrew Zeller, Boundedness and continuity of the time derivative in the parabolic Signorini problem, Math. Res. Lett. 26 (2019), no. 1, 281–292. MR 3963984, DOI 10.4310/MRL.2019.v26.n1.a13
- W. Shi, An epiperimetric inequality approach to the parabolic Signorini problem, arXiv:1810.11791, 2018.
- Yannick Sire, Susanna Terracini, and Giorgio Tortone, On the nodal set of solutions to degenerate or singular elliptic equations with an application to $s$-harmonic functions, J. Math. Pures Appl. (9) 143 (2020), 376–441 (English, with English and French summaries). MR 4163134, DOI 10.1016/j.matpur.2020.01.010
- Pablo Raúl Stinga and José L. Torrea, Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation, SIAM J. Math. Anal. 49 (2017), no. 5, 3893–3924. MR 3709888, DOI 10.1137/16M1104317
- Y. Sire, S. Terracini, and S. Vita, Liouville type theorems and regularity of solutions to degenerate or singular problems Part I. Even solutions, arXiv:1904.02143, 2019.
- N. N. Ural′tseva, Hölder continuity of gradients of solutions of parabolic equations with boundary conditions of Signorini type, Dokl. Akad. Nauk SSSR 280 (1985), no. 3, 563–565 (Russian). MR 775926
Bibliographic Information
- A. Banerjee
- Affiliation: TIFR Centre for Applicable Mathematics, Bangalore, 560065, India
- MR Author ID: 1006299
- Email: agnidban@gmail.com
- D. Danielli
- Affiliation: Department of Mathematics, Purdue University, 47907 West Lafayette, Indiana
- MR Author ID: 324114
- Email: danielli@math.purdue.edu
- N. Garofalo
- Affiliation: Dipartimento di Ingegneria Civile, Edile e Ambientale (DICEA) Università di Padova 35131 Padova, Italy
- MR Author ID: 71535
- Email: rembrandt54@gmail.com
- A. Petrosyan
- Affiliation: Department of Mathematics, Purdue University, 47907 West Lafayette, Indiana
- MR Author ID: 654444
- Email: arshak@purdue.edu
- Received by editor(s): June 16, 2019
- Published electronically: May 11, 2021
- Additional Notes: The first author was supported in part by SERB Matrix grant MTR/2018/000267. The third author was supported in part by a Progetto SID (Investimento Strategico di Dipartimento) “Non-local operators in geometry and in free boundary problems, and their connection with the applied sciences,” University of Padova, 2017. The fourth author was supported in part by NSF Grant DMS-1800527.
- © Copyright 2021 American Mathematical Society
- Journal: St. Petersburg Math. J. 32, 449-480
- MSC (2020): Primary 35R35
- DOI: https://doi.org/10.1090/spmj/1656
- MathSciNet review: 4099095
Dedicated: Dedicated to Nina, with affection and deep admiration. Her pioneering ideas have left a permanent mark in PDEs, and inspired scores of mathematicians.