• Previous Article
    Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition
  • DCDS Home
  • This Issue
  • Next Article
    Topological cubic polynomials with one periodic ramification point
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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

M. Colombo, L. Spolaor and B. Velichkov, On the asymptotic behavior of the solutions to parabolic variational inequalities, (2018), arXiv: 1809.06075. 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). doi: 10.1090/memo/1181.  Google Scholar

[13]

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

[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. Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

M. Focardi and E. Spadaro, The local structure of the free boundary in the fractional obstacle problem, (2019), arXiv: 1903.05909. 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), 745–787. doi: 10.1016/j.matpur.2015.11.013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[24]

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

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

M. Colombo, L. Spolaor and B. Velichkov, On the asymptotic behavior of the solutions to parabolic variational inequalities, (2018), arXiv: 1809.06075. 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). doi: 10.1090/memo/1181.  Google Scholar

[13]

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

[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. Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

M. Focardi and E. Spadaro, The local structure of the free boundary in the fractional obstacle problem, (2019), arXiv: 1903.05909. 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), 745–787. doi: 10.1016/j.matpur.2015.11.013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[24]

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

[1]

Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44

[2]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

[3]

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure & Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957

[4]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667

[5]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122

[6]

Daniela De Silva, Ovidiu Savin. A note on higher regularity boundary Harnack inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6155-6163. doi: 10.3934/dcds.2015.35.6155

[7]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605

[8]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737

[9]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176

[10]

Xinfu Chen, Huibin Cheng. Regularity of the free boundary for the American put option. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1751-1759. doi: 10.3934/dcdsb.2012.17.1751

[11]

Carlos E. Kenig, Tatiana Toro. On the free boundary regularity theorem of Alt and Caffarelli. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 397-422. doi: 10.3934/dcds.2004.10.397

[12]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[13]

Brahim Bougherara, Jacques Giacomoni, Jesus Hernández. Some regularity results for a singular elliptic problem. Conference Publications, 2015, 2015 (special) : 142-150. doi: 10.3934/proc.2015.0142

[14]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431

[15]

G. P. Trachanas, Nikolaos B. Zographopoulos. A strongly singular parabolic problem on an unbounded domain. Communications on Pure & Applied Analysis, 2014, 13 (2) : 789-809. doi: 10.3934/cpaa.2014.13.789

[16]

Huiqiang Jiang. Regularity of a vector valued two phase free boundary problems. Conference Publications, 2013, 2013 (special) : 365-374. doi: 10.3934/proc.2013.2013.365

[17]

Panagiota Daskalopoulos, Eunjai Rhee. Free-boundary regularity for generalized porous medium equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 481-494. doi: 10.3934/cpaa.2003.2.481

[18]

María Teresa Cao-Rial, Peregrina Quintela, Carlos Moreno. Numerical solution of a time-dependent Signorini contact problem. Conference Publications, 2007, 2007 (Special) : 201-211. doi: 10.3934/proc.2007.2007.201

[19]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

[20]

Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks & Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (36)
  • HTML views (48)
  • Cited by (0)

Other articles
by authors

[Back to Top]