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On the asymptotic behavior of the solutions to parabolic variational inequalities. (English) Zbl 1452.35033

The asymptotic behavior of the solutions to a parabolic variational inequality is studied in connection with a Łojasjiewicz-type inequality. The results are applied to parabolic obstacle and thin-obstacle problems.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K86 Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators
49J40 Variational inequalities
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[1] H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9) 51 (1972), 1-168. · Zbl 0237.35001
[2] L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383-402. · Zbl 0928.49030
[3] L. A Caffarelli, A. Petrosyan and H. Shahgholian, Regularity of a free boundary in parabolic potential theory, J. Amer. Math. Soc. 17 (2004), no. 4, 827-869. · Zbl 1054.35142
[4] L. A. Caffarelli and N. M. Rivière, Smoothness and analyticity of free boundaries in variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 3 (1976), no. 2, 289-310. · Zbl 0363.35009
[5] T. H. Colding and W. P. Minicozzi, II, Uniqueness of blowups and Łojasiewicz inequalities, Ann. of Math. (2) 182 (2015), no. 1, 221-285. · Zbl 1337.53082
[6] M. Colombo, L. Spolaor and B. Velichkov, A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal. 28 (2018), no. 4, 1029-1061. · Zbl 1428.49042
[7] M. Colombo, L. Spolaor and B. Velichkov, Direct epiperimetric inequalities for the thin obstacle problem and applications, Comm. Pure Appl. Math., to appear. · Zbl 1433.49010
[8] 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. · Zbl 1381.35249
[9] C. De Lellis, E. Spadaro and L. Spolaor, Uniqueness of tangent cones for two-dimensional almost-minimizing currents, Comm. Pure Appl. Math. 70 (2017), no. 7, 1402-1421. · Zbl 1369.49062
[10] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Grundlehren Math. Wiss. 219, Springer, Berlin 1976. · Zbl 0331.35002
[11] M. Engelstein, L. Spolaor and B. Velichkov, Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional, preprint (2018), https://arxiv.org/abs/1801.09276.
[12] M. Engelstein, L. Spolaor and B. Velichkov, (Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currents, Geom. Topol. 23 (2019), no. 1, 513-540. · Zbl 1409.53013
[13] A. Figalli and J. Serra, On the fine structure of the free boundary for the classical obstacle problem, Invent. Math. 215 (2019), no. 1, 311-366. · Zbl 1408.35228
[14] M. Focardi and E. Spadaro, An epiperimetric inequality for the thin obstacle problem, Adv. Differential Equations 21 (2016), no. 1-2, 153-200. · Zbl 1336.35370
[15] A. Friedman, Variational principles and free-boundary problems, 2nd ed., Robert E. Krieger Publishing, Malabar 1988. · Zbl 0671.49001
[16] 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. · Zbl 1175.35154
[17] 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. · Zbl 1341.35036
[18] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les équations aux dérivées partielles (Paris 1962), Éditions du Centre National de la Recherche Scientifique, Paris (1963), 87-89. · Zbl 0234.57007
[19] J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493-519. · Zbl 0152.34601
[20] R. Monneau, A brief overview on the obstacle problem, European congress of mathematics, Vol. II (Barcelona 2000), Progr. Math. 202, Birkhäuser, Basel (2001), 303-312. · Zbl 1027.35164
[21] E. R. Reifenberg, An epiperimetric inequality related to the analyticity of minimal surfaces, Ann. of Math. (2) 80 (1964), 1-14. · Zbl 0151.16701
[22] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), no. 3, 525-571. · Zbl 0549.35071
[23] L. Spolaor and B. Velichkov, An epiperimetric inequality for the regularity of some free boundary problems: The 2-dimensional case, Comm. Pure Appl. Math. 72 (2019), no. 2, 375-421. · Zbl 1429.35214
[24] J. E. Taylor, Regularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3 in \mathbb{R}^3, Invent. Math. 22 (1973), 119-159. · Zbl 0278.49046
[25] J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489-539. · Zbl 0335.49032
[26] G. S. Weiss, A homogeneity improvement approach to the obstacle problem, Invent. Math. 138 (1999), no. 1, 23-50. · Zbl 0940.35102
[27] B. White, Tangent cones to two-dimensional area-minimizing integral currents are unique, Duke Math. J. 50 (1983), no. 1, 143-160. · Zbl 0538.49030
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