Home Free boundary regularity in the fully nonlinear parabolic thin obstacle problem
Article
Licensed
Unlicensed Requires Authentication

Free boundary regularity in the fully nonlinear parabolic thin obstacle problem

  • Xi Hu and Lin Tang EMAIL logo
Published/Copyright: April 24, 2024

Abstract

We study the regularity of the free boundary in the fully nonlinear parabolic thin obstacle problem. Under the assumption of time semiconvexity, our main result establishes that the free boundary is a C 1 graph in x near any regular free boundary point.

MSC 2020: 35R35; 35K55

Communicated by Luis Silvestre


Award Identifier / Grant number: 11771023

Funding statement: Supported by the National Natural Science Foundation of China (No. 11771023).

References

[1] I. Athanasopoulos and L. A. Caffarelli, Optimal regularity of lower dimensional obstacle problems, J. Math. Sci. (N. Y.) 132 (2006), no. 3, 274–284. 10.1007/s10958-005-0496-1Search in Google Scholar

[2] 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. 10.1353/ajm.2008.0016Search in Google Scholar

[3] A. Banerjee, D. Danielli, N. Garofalo and A. Petrosyan, The structure of the singular set in the thin obstacle problem for degenerate parabolic equations, Calc. Var. Partial Differential Equations 60 (2021), no. 3, 91–142. 10.1007/s00526-021-01938-2Search in Google Scholar

[4] B. Barrios, A. Figalli and X. Ros-Oton, Free boundary regularity in the parabolic fractional obstacle problem, Comm. Pure Appl. Math. 71 (2018), no. 10, 2129–2159. 10.1002/cpa.21745Search in Google Scholar

[5] B. Barrios, A. Figalli and X. Ros-Oton, Global regularity for the free boundary in the obstacle problem for the fractional Laplacian, Amer. J. Math. 140 (2018), no. 2, 415–447. 10.1353/ajm.2018.0010Search in Google Scholar

[6] S.-S. Byun, K.-A. Lee, J. Oh and J. Park, Regularity results of the thin obstacle problem for the p ( x ) -Laplacian, J. Funct. Anal. 276 (2019), no. 2, 496–519. 10.1016/j.jfa.2018.06.003Search in Google Scholar

[7] L. Caffarelli, X. Ros-Oton and J. Serra, Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries, Invent. Math. 208 (2017), no. 3, 1155–1211. 10.1007/s00222-016-0703-3Search in Google Scholar

[8] L. A. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations 4 (1979), no. 9, 1067–1075. 10.1080/03605307908820119Search in Google Scholar

[9] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43, American Mathematical Society, Providence, 1995. 10.1090/coll/043Search in Google Scholar

[10] L. A. Caffarelli, S. Salsa and L. 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. 10.1007/s00222-007-0086-6Search in Google Scholar

[11] G. Chatzigeorgiou, Regularity for the fully nonlinear parabolic thin obstacle problem, Commun. Contemp. Math. 24 (2022), no. 3, Article ID 2150011. 10.1142/S0219199721500115Search in Google Scholar

[12] D. Danielli, N. Garofalo, A. Petrosyan and T. To, Optimal regularity and the free boundary in the parabolic Signorini problem, Mem. Amer. Math. Soc. 249 (2017), no. 1181, 1–103. 10.1090/memo/1181Search in Google Scholar

[13] D. De Silva and O. Savin, Boundary Harnack estimates in slit domains and applications to thin free boundary problems, Rev. Mat. Iberoam. 32 (2016), no. 3, 891–912. 10.4171/rmi/902Search in Google Scholar

[14] X. Fernández-Real, C 1 , α estimates for the fully nonlinear Signorini problem, Calc. Var. Partial Differential Equations 55 (2016), no. 4, 94–113. 10.1007/s00526-016-1034-3Search in Google Scholar

[15] X. Fernández-Real and X. Ros-Oton, Free boundary regularity for almost every solution to the Signorini problem, Arch. Ration. Mech. Anal. 240 (2021), no. 1, 419–466. 10.1007/s00205-021-01617-8Search in Google Scholar PubMed PubMed Central

[16] A. Figalli and H. Shahgholian, A general class of free boundary problems for fully nonlinear parabolic equations, Ann. Mat. Pura Appl. (4) 194 (2015), no. 4, 1123–1134. 10.1007/s10231-014-0413-7Search in Google Scholar

[17] J. Frehse, On Signorini’s problem and variational problems with thin obstacles, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 4 (1977), no. 2, 343–362. Search in Google Scholar

[18] N. Garofalo and A. Petrosyan, Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem, Invent. Math. 177 (2009), no. 2, 415–461. 10.1007/s00222-009-0188-4Search in Google Scholar

[19] N. Garofalo, A. Petrosyan and M. 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. 10.1016/j.matpur.2015.11.013Search in Google Scholar

[20] N. Guillen, Optimal regularity for the Signorini problem, Calc. Var. Partial Differential Equations 36 (2009), no. 4, 533–546. 10.1007/s00526-009-0242-5Search in Google Scholar

[21] X. Hu and L. Tang, H 1 + α estimates for the fully nonlinear parabolic thin obstacle problem, J. Differential Equations 321 (2022), 40–65. 10.1016/j.jde.2022.03.010Search in Google Scholar

[22] D. Kinderlehrer, The smoothness of the solution of the boundary obstacle problem, J. Math. Pures Appl. (9) 60 (1981), no. 2, 193–212. Search in Google Scholar

[23] H. Koch, A. Petrosyan and W. Shi, Higher regularity of the free boundary in the elliptic Signorini problem, Nonlinear Anal. 126 (2015), 3–44. 10.1016/j.na.2015.01.007Search in Google Scholar

[24] H. Koch, A. Rüland and W. Shi, The variable coefficient thin obstacle problem: optimal regularity and regularity of the regular free boundary, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 4, 845–897. 10.1016/j.anihpc.2016.08.001Search in Google Scholar

[25] H. Lewy, On the coincidence set in variational inequalities, J. Differential Geometry 6 (1972), 497–501. 10.4310/jdg/1214430639Search in Google Scholar

[26] E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math. 217 (2008), no. 3, 1301–1312. 10.1016/j.aim.2007.08.009Search in Google Scholar

[27] D. L. Richardson, Variational problems with thin obstacles, Ph.D. Thesis, University of British Columbia, 1978. Search in Google Scholar

[28] X. Ros-Oton and J. Serra, The structure of the free boundary in the fully nonlinear thin obstacle problem, Adv. Math. 316 (2017), 710–747. 10.1016/j.aim.2017.06.032Search in Google Scholar

[29] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), no. 1, 67–112. 10.1002/cpa.20153Search in Google Scholar

[30] N. N. Uraltseva, On the regularity of solutions of variational inequalities, Uspekhi Mat. Nauk 42 (1987), no. 6(258), 151–174. Search in Google Scholar

[31] L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math. 45 (1992), no. 1, 27–76. 10.1002/cpa.3160450103Search in Google Scholar

[32] L. Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math. 45 (1992), no. 2, 141–178. 10.1002/cpa.3160450202Search in Google Scholar

Received: 2023-11-14
Accepted: 2024-03-27
Published Online: 2024-04-24
Published in Print: 2025-04-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 2.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/acv-2023-0126/html
Scroll to top button