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March  2020, 40(3): 1813-1846. doi: 10.3934/dcds.2020095

An epiperimetric inequality approach to the parabolic Signorini problem

9 Rainforest Walk, Level 4, Monash University, 3800 VIC, Australia

Received  July 2019 Revised  October 2019 Published  December 2019

In this note, we use an epiperimetric inequality approach to study the regularity of the free boundary for the parabolic Signorini problem. We show that if the "vanishing order" of a solution at a free boundary point is close to $ 3/2 $ or an even integer, then the solution is asymptotically homogeneous. Furthermore, one can derive a convergence rate estimate towards the asymptotic homogeneous solution. As a consequence, we obtain the regularity of the regular free boundary as well as the frequency gap.

Citation: Wenhui Shi. An epiperimetric inequality approach to the parabolic Signorini problem. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1813-1846. doi: 10.3934/dcds.2020095
References:
[1]

A. Arkhipova and N. Uraltseva, Sharp estimates for solutions of a parabolic Signorini problem, Math. Nachr., 177 (1996), 11-29.  doi: 10.1002/mana.19961770103.

[2]

I. AthanasopoulosL. Caffarelli and E. Milakis, On the regularity of the non-dynamic parabolic fractional obstacle problem, J. Differential Equations, 265 (2018), 2614-2647.  doi: 10.1016/j.jde.2018.04.043.

[3]

A. Audrito and S. Terracini, On the nodal set of solutions to a class of nonlocal parabolic reaction-diffusion equations, (2018), arXiv: 1807.10135.

[4]

A. Banerjee, D. Danielli, N. Garofalo and A. Petrosyan, The regular free boundary in the thin obstacle problem for degenerate parabolic equations, (2019), arXiv: 1906.06885.

[5]

A. Banerjee, D. Danielli, N. Garofalo and A. Petrosyan, The structure of the singular set in the thin obstacle problem for degenerate parabolic equations, (2019), arXiv: 1902.07457.

[6]

A. Banerjee and N. Garofalo, Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations, Adv. Math., 336 (2018), 149-241.  doi: 10.1016/j.aim.2018.07.021.

[7]

A. Banerjee, M. Smit Vega Garcia and A. K. Zeller, Higher regularity of the free boundary in the parabolic Signorini problem, Calc. Var. Partial Differential Equations, 56 (2017), Art. 7, 26 pp. doi: 10.1007/s00526-016-1103-7.

[8]

T. H. Colding and W. P. Minicozzi II, Sharp frequency bounds for eigenfunctions of the Ornstein-Uhlenbeck operator, Calc. Var. Partial Differential Equations, 57 (2018), Art. 138, 16 pp. doi: 10.1007/s00526-018-1405-z.

[9]

M. Colombo, L. Spolaor and B. Velichkov, Direct epiperimetric inequalities for the thin obstacle problem and applications, (2017), arXiv: 1709.03120.

[10]

M. ColomboL. Spolaor and B. Velichkov, A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal., 28 (2018), 1029-1061.  doi: 10.1007/s00039-018-0451-1.

[11]

M. Colombo, L. Spolaor and B. Velichkov, On the asymptotic behavior of the solutions to parabolic variational inequalities, (2018), arXiv: 1809.06075.

[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). doi: 10.1090/memo/1181.

[13]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976.

[14]

X. Fernández-Real and Y. Jhaveri, On the singular set in the thin obstacle problem: Higher order blow-ups and the very thin obstacle problem, (2019), arXiv: 1812.01515.

[15]

A. Figalli and J. Serra, On the fine structure of the free boundary for the classical obstacle problem, Invent. Math., 215 (2019), 311-366.  doi: 10.1007/s00222-018-0827-8.

[16]

M. Focardi and E. Spadaro, An epiperimetric inequality for the thin obstacle problem, Adv. Differential Equations, 21 (2016), 153-200. 

[17]

M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower dimensional obstacle problem, Arch. Ration. Mech. Anal., 230 (2018), 125-184.  doi: 10.1007/s00205-018-1242-4.

[18]

M. Focardi and E. Spadaro, The local structure of the free boundary in the fractional obstacle problem, (2019), arXiv: 1903.05909.

[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), 745–787. doi: 10.1016/j.matpur.2015.11.013.

[20]

A. Petrosyan and W. H. Shi, Parabolic boundary Harnack principles in domains with thin Lipschitz complement, Anal. PDE, 7 (2014), 1421-1463.  doi: 10.2140/apde.2014.7.1421.

[21]

A. Petrosyan and A. Zeller, Boundedness and continuity of the time derivative in the parabolic Signorini problem, Math. Res. Lett., 26 (2019), 281-292.  doi: 10.4310/MRL.2019.v26.n1.a13.

[22]

A. Rüland and W. H. Shi, Optimal regularity for the thin obstacle problem with $C^{0, \alpha}$ coefficients, Calc. Var. Partial Differential Equations, 56 (2017), Art. 129, 41 pp. doi: 10.1007/s00526-017-1230-9.

[23]

Y. Sire, G. Tortone and S. Terracini, On the nodal set of solutions to degenerate or singular elliptic equations with an application to s-harmonic functions, (2019), arXiv: 1808.01851.

[24]

G. S. Weiss, A homogeneity improvement approach to the obstacle problem, Invent. Math., 138 (1999), 23-50.  doi: 10.1007/s002220050340.

show all references

References:
[1]

A. Arkhipova and N. Uraltseva, Sharp estimates for solutions of a parabolic Signorini problem, Math. Nachr., 177 (1996), 11-29.  doi: 10.1002/mana.19961770103.

[2]

I. AthanasopoulosL. Caffarelli and E. Milakis, On the regularity of the non-dynamic parabolic fractional obstacle problem, J. Differential Equations, 265 (2018), 2614-2647.  doi: 10.1016/j.jde.2018.04.043.

[3]

A. Audrito and S. Terracini, On the nodal set of solutions to a class of nonlocal parabolic reaction-diffusion equations, (2018), arXiv: 1807.10135.

[4]

A. Banerjee, D. Danielli, N. Garofalo and A. Petrosyan, The regular free boundary in the thin obstacle problem for degenerate parabolic equations, (2019), arXiv: 1906.06885.

[5]

A. Banerjee, D. Danielli, N. Garofalo and A. Petrosyan, The structure of the singular set in the thin obstacle problem for degenerate parabolic equations, (2019), arXiv: 1902.07457.

[6]

A. Banerjee and N. Garofalo, Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations, Adv. Math., 336 (2018), 149-241.  doi: 10.1016/j.aim.2018.07.021.

[7]

A. Banerjee, M. Smit Vega Garcia and A. K. Zeller, Higher regularity of the free boundary in the parabolic Signorini problem, Calc. Var. Partial Differential Equations, 56 (2017), Art. 7, 26 pp. doi: 10.1007/s00526-016-1103-7.

[8]

T. H. Colding and W. P. Minicozzi II, Sharp frequency bounds for eigenfunctions of the Ornstein-Uhlenbeck operator, Calc. Var. Partial Differential Equations, 57 (2018), Art. 138, 16 pp. doi: 10.1007/s00526-018-1405-z.

[9]

M. Colombo, L. Spolaor and B. Velichkov, Direct epiperimetric inequalities for the thin obstacle problem and applications, (2017), arXiv: 1709.03120.

[10]

M. ColomboL. Spolaor and B. Velichkov, A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal., 28 (2018), 1029-1061.  doi: 10.1007/s00039-018-0451-1.

[11]

M. Colombo, L. Spolaor and B. Velichkov, On the asymptotic behavior of the solutions to parabolic variational inequalities, (2018), arXiv: 1809.06075.

[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). doi: 10.1090/memo/1181.

[13]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976.

[14]

X. Fernández-Real and Y. Jhaveri, On the singular set in the thin obstacle problem: Higher order blow-ups and the very thin obstacle problem, (2019), arXiv: 1812.01515.

[15]

A. Figalli and J. Serra, On the fine structure of the free boundary for the classical obstacle problem, Invent. Math., 215 (2019), 311-366.  doi: 10.1007/s00222-018-0827-8.

[16]

M. Focardi and E. Spadaro, An epiperimetric inequality for the thin obstacle problem, Adv. Differential Equations, 21 (2016), 153-200. 

[17]

M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower dimensional obstacle problem, Arch. Ration. Mech. Anal., 230 (2018), 125-184.  doi: 10.1007/s00205-018-1242-4.

[18]

M. Focardi and E. Spadaro, The local structure of the free boundary in the fractional obstacle problem, (2019), arXiv: 1903.05909.

[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), 745–787. doi: 10.1016/j.matpur.2015.11.013.

[20]

A. Petrosyan and W. H. Shi, Parabolic boundary Harnack principles in domains with thin Lipschitz complement, Anal. PDE, 7 (2014), 1421-1463.  doi: 10.2140/apde.2014.7.1421.

[21]

A. Petrosyan and A. Zeller, Boundedness and continuity of the time derivative in the parabolic Signorini problem, Math. Res. Lett., 26 (2019), 281-292.  doi: 10.4310/MRL.2019.v26.n1.a13.

[22]

A. Rüland and W. H. Shi, Optimal regularity for the thin obstacle problem with $C^{0, \alpha}$ coefficients, Calc. Var. Partial Differential Equations, 56 (2017), Art. 129, 41 pp. doi: 10.1007/s00526-017-1230-9.

[23]

Y. Sire, G. Tortone and S. Terracini, On the nodal set of solutions to degenerate or singular elliptic equations with an application to s-harmonic functions, (2019), arXiv: 1808.01851.

[24]

G. S. Weiss, A homogeneity improvement approach to the obstacle problem, Invent. Math., 138 (1999), 23-50.  doi: 10.1007/s002220050340.

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