American Institute of Mathematical Sciences

Advanced
March  2021, 41(3): 1359-1385. doi: 10.3934/dcds.2020321

Sharp regularity for degenerate obstacle type problems: A geometric approach

João Vitor da Silva 1,2, and  Hernán Vivas 2,3,,

1. 

Departamento de Matemática - Instituto de Ciências Exatas, Universidade de Brasília – UnB, Campus Universitário Darcy Ribeiro, 70910-900, Brasília - Distrito Federal - Brazil
2. 

Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS), UBA/CONICET, Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n - Buenos Aires, Argentina
3. 

Centro Marplatense de Investigaciones matemáticas, UNMdP/CIC, Dean Funes 3350, 7600 Mar del Plata, Argentina

* Corresponding author

Received  December 2019 Revised  July 2020 Published  September 2020

We prove sharp regularity estimates for solutions of obstacle type problems driven by a class of degenerate fully nonlinear operators. More specifically, we consider viscosity solutions of
$ \begin{equation*} \left\{ \begin{array}{rcll} |D u|^\gamma F(x, D^2u)& = & f(x)\chi_{\{u>\phi\}} & \ \rm{ in } \ B_1 \\ u(x) & \geq & \phi(x) & \ \rm{ in } \ B_1 \\ u(x) & = & g(x) & \ \rm{on } \ \partial B_1, \end{array} \right. \end{equation*} $
with
$ \gamma>0 $
,
$ \phi \in C^{1, \alpha}(B_1) $
 for some
$ \alpha\in(0,1] $
, a continuous boundary datum
$ g $
 and
$ f\in L^\infty(B_1)\cap C^0(B_1) $
 and prove that they are
$ C^{1,\beta}(B_{1/2}) $
 (and in particular at free boundary points) where
$ \beta = \min\left\{\alpha, \frac{1}{\gamma+1}\right\} $
. Moreover, we achieve such a feature by using a recently developed geometric approach which is a novelty for these types of free boundary problems. Furthermore, under a natural non-degeneracy assumption on the obstacle, we prove that the free boundary
$ \partial\{u>\phi\} $
 has Hausdorff dimension less than
$ n $
 (and in particular zero Lebesgue measure). Our results are new even for degenerate problems such as
$ |Du|^\gamma \Delta u = \chi_{\{u>\phi\}} \quad \text{with}\quad \gamma>0. $
Keywords: Free boundary problems, nonlinear elliptic equations, regularity of solutions.
Mathematics Subject Classification: Pirmary: 35R35, 35J60, 35B65.
Citation: João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321
References:
[1]

M. D. Amaral, J. V. da Silva, G. C. Ricarte and R. Teymurazyan, Sharp regularity estimates for quasilinear evolution equations, Israel J. Math., 231 (2019), 25-45.  doi: 10.1007/s11856-019-1842-1.  Google Scholar

[2]

J. Andersson, E. Lindgren and H. Shahgholian, Optimal regularity for the obstacle problem for the $p-$Laplacian, J. Differential Equations, 259 (2015), 2167-2179.  doi: 10.1016/j.jde.2015.03.019.  Google Scholar

[3]

D. J. Araújo, G. Ricarte and E. V. Teixeira, Geometric gradient estimates for solutions to degenerate elliptic equations, Calc. Var. Partial Differential Equations, 53 (2015), 605-625.  doi: 10.1007/s00526-014-0760-7.  Google Scholar

[4]

D. J. Araújo, E. V. Teixeira and J. M. Urbano, Towards the $C^{p^{\prime}}$ regularity conjecture in higher dimensions, Int. Math. Res. Not. IMRN, 2018 (2018), 6481-6495.  doi: 10.1093/imrn/rnx068.  Google Scholar

[5]

A. Attouchi, M. Parviainen and E. Ruosteenoja, $C^{1, \alpha}$ regularity for the normalized p-Poisson problem, J. Math. Pures Appl., 108 (2017), 553-591.  doi: 10.1016/j.matpur.2017.05.003.  Google Scholar

[6]

I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math., 13 (2004), 261-287.  doi: 10.5802/afst.1070.  Google Scholar

[7]

I. Birindelli and F. Demengel, $C^{1, \beta}$ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM Control Optim. Calc. Var., 20 (2014), 1009-1024.  doi: 10.1051/cocv/2014005.  Google Scholar

[8]

I. Blank and K. Teka, The Caffarelli alternative in measure for the nondivergence form elliptic obstacle problem with principal coefficients in VMO, Comm. Partial Differential Equations, 39 (2014), 321-353.  doi: 10.1080/03605302.2013.823988.  Google Scholar

[9]

S.-S. Byun, K.-A. Lee, J. Oh and J. Park, Nondivergence elliptic and parabolic problems with irregular obstacles, Math. Z., 290 (2018), 973-990.  doi: 10.1007/s00209-018-2048-7.  Google Scholar

[10]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS Colloquium Publications, Providence, Vol 43, 1995. doi: 10.1090/coll/043.  Google Scholar

[11]

J. V. da Silva, Sharp and Improved Regularity Estimates to Fully Nonlinear Equations and Free Boundary Problems, PhD. Thesis, Universidade Federal do Ceará - UFC, Brazil, 2015. http://www.repositorio.ufc.br/handle/riufc/41839. Google Scholar

[12]

J. V. da Silva and D. dos Prazeres, Schauder type estimates for "flat" viscosity solutions to non-convex fully nonlinear parabolic equations and applications, Potential Anal., 50 (2019), 149-170.  doi: 10.1007/s11118-017-9677-z.  Google Scholar

[13]

J. V. da Silva, R. A. Leitão and G. C. Ricarte, Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries, to appear in Mathematische Nachrichten, arXiv: 2008.04832. Google Scholar

[14]

J. V. da Silva and E. V. Teixeira, Sharp regularity estimates for second order fully nonlinear parabolic equations, Math. Ann., 369 (2017), 1623-1648.  doi: 10.1007/s00208-016-1506-y.  Google Scholar

[15]

J. V. da Silva and H. Vivas, The obstacle problem for a class of degenerate fully nonlinear operators, to appear in Revista Matemática Iberoamericana, arXiv: 1905.06146. Google Scholar

[16]

G. Dávila, P. Felmer and A. Quaas, Alexandroff-Bakelman-Pucci estimate for singular or degenerate fully nonlinear elliptic equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 1165-1168.  doi: 10.1016/j.crma.2009.09.009.  Google Scholar

[17]

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.  Google Scholar

[18]

A. Figalli and H. Shahgholian, A general class of free boundary problems for fully nonlinear elliptic equations, Archive for Rational Mechanics and Analysis, 213 (2014), 269-286.  doi: 10.1007/s00205-014-0734-0.  Google Scholar

[19]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully non-linear elliptic equations, Adv. Math., 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.  Google Scholar

[20]

E. Indrei and A. Minne, Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 33 (2016), 1259-1277.  doi: 10.1016/j.anihpc.2015.03.009.  Google Scholar

[21]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.   Google Scholar

[22]

K.-A. Lee, Obstacle Problems for the Fully Nonlinear Elliptic Operators, Thesis (Ph.D.)-New York University. 1998. 53 pp. ISBN: 978-0599-04972-7.  Google Scholar

[23]

K.-A. Lee and H. Shahgholian, Regularity of a free boundary for viscosity solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 54 (2001), 43-56.  doi: 10.1002/1097-0312(200101)54:1<43::AID-CPA2>3.0.CO;2-T.  Google Scholar

[24]

N. Nadirashvili and S. Vlăduţ, Nonclassical solutions of fully nonlinear elliptic equations, Geometric and Functional Analysis, 17 (2007), 1283-1296.  doi: 10.1007/s00039-007-0626-7.  Google Scholar

[25]

A. Petrosyan, H. Shahgholian and N. Uralt'seva, Regularity of Free Boundaries in Obstacle-Type Problems, Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3. doi: 10.1090/gsm/136.  Google Scholar

[26]

L. Silvestre and E. V. Teixeira, Regularity estimates for fully non linear elliptic equations which are asymptotically convex, in Contributions to nonlinear elliptic equations and systems, 425–438, Progr. Nonlinear Differential Equations Appl., 86, Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-19902-3_25.  Google Scholar

[27]

L. Zajíček, Porosity and $\sigma-$porosity, Real Anal. Exchange, 13 (1987/88), 314-350.  doi: 10.2307/44151885.  Google Scholar

show all references

References:
[1]

M. D. Amaral, J. V. da Silva, G. C. Ricarte and R. Teymurazyan, Sharp regularity estimates for quasilinear evolution equations, Israel J. Math., 231 (2019), 25-45.  doi: 10.1007/s11856-019-1842-1.  Google Scholar

[2]

J. Andersson, E. Lindgren and H. Shahgholian, Optimal regularity for the obstacle problem for the $p-$Laplacian, J. Differential Equations, 259 (2015), 2167-2179.  doi: 10.1016/j.jde.2015.03.019.  Google Scholar

[3]

D. J. Araújo, G. Ricarte and E. V. Teixeira, Geometric gradient estimates for solutions to degenerate elliptic equations, Calc. Var. Partial Differential Equations, 53 (2015), 605-625.  doi: 10.1007/s00526-014-0760-7.  Google Scholar

[4]

D. J. Araújo, E. V. Teixeira and J. M. Urbano, Towards the $C^{p^{\prime}}$ regularity conjecture in higher dimensions, Int. Math. Res. Not. IMRN, 2018 (2018), 6481-6495.  doi: 10.1093/imrn/rnx068.  Google Scholar

[5]

A. Attouchi, M. Parviainen and E. Ruosteenoja, $C^{1, \alpha}$ regularity for the normalized p-Poisson problem, J. Math. Pures Appl., 108 (2017), 553-591.  doi: 10.1016/j.matpur.2017.05.003.  Google Scholar

[6]

I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse Math., 13 (2004), 261-287.  doi: 10.5802/afst.1070.  Google Scholar

[7]

I. Birindelli and F. Demengel, $C^{1, \beta}$ regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM Control Optim. Calc. Var., 20 (2014), 1009-1024.  doi: 10.1051/cocv/2014005.  Google Scholar

[8]

I. Blank and K. Teka, The Caffarelli alternative in measure for the nondivergence form elliptic obstacle problem with principal coefficients in VMO, Comm. Partial Differential Equations, 39 (2014), 321-353.  doi: 10.1080/03605302.2013.823988.  Google Scholar

[9]

S.-S. Byun, K.-A. Lee, J. Oh and J. Park, Nondivergence elliptic and parabolic problems with irregular obstacles, Math. Z., 290 (2018), 973-990.  doi: 10.1007/s00209-018-2048-7.  Google Scholar

[10]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS Colloquium Publications, Providence, Vol 43, 1995. doi: 10.1090/coll/043.  Google Scholar

[11]

J. V. da Silva, Sharp and Improved Regularity Estimates to Fully Nonlinear Equations and Free Boundary Problems, PhD. Thesis, Universidade Federal do Ceará - UFC, Brazil, 2015. http://www.repositorio.ufc.br/handle/riufc/41839. Google Scholar

[12]

J. V. da Silva and D. dos Prazeres, Schauder type estimates for "flat" viscosity solutions to non-convex fully nonlinear parabolic equations and applications, Potential Anal., 50 (2019), 149-170.  doi: 10.1007/s11118-017-9677-z.  Google Scholar

[13]

J. V. da Silva, R. A. Leitão and G. C. Ricarte, Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries, to appear in Mathematische Nachrichten, arXiv: 2008.04832. Google Scholar

[14]

J. V. da Silva and E. V. Teixeira, Sharp regularity estimates for second order fully nonlinear parabolic equations, Math. Ann., 369 (2017), 1623-1648.  doi: 10.1007/s00208-016-1506-y.  Google Scholar

[15]

J. V. da Silva and H. Vivas, The obstacle problem for a class of degenerate fully nonlinear operators, to appear in Revista Matemática Iberoamericana, arXiv: 1905.06146. Google Scholar

[16]

G. Dávila, P. Felmer and A. Quaas, Alexandroff-Bakelman-Pucci estimate for singular or degenerate fully nonlinear elliptic equations, C. R. Math. Acad. Sci. Paris, 347 (2009), 1165-1168.  doi: 10.1016/j.crma.2009.09.009.  Google Scholar

[17]

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.  Google Scholar

[18]

A. Figalli and H. Shahgholian, A general class of free boundary problems for fully nonlinear elliptic equations, Archive for Rational Mechanics and Analysis, 213 (2014), 269-286.  doi: 10.1007/s00205-014-0734-0.  Google Scholar

[19]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully non-linear elliptic equations, Adv. Math., 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.  Google Scholar

[20]

E. Indrei and A. Minne, Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 33 (2016), 1259-1277.  doi: 10.1016/j.anihpc.2015.03.009.  Google Scholar

[21]

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.   Google Scholar

[22]

K.-A. Lee, Obstacle Problems for the Fully Nonlinear Elliptic Operators, Thesis (Ph.D.)-New York University. 1998. 53 pp. ISBN: 978-0599-04972-7.  Google Scholar

[23]

K.-A. Lee and H. Shahgholian, Regularity of a free boundary for viscosity solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 54 (2001), 43-56.  doi: 10.1002/1097-0312(200101)54:1<43::AID-CPA2>3.0.CO;2-T.  Google Scholar

[24]

N. Nadirashvili and S. Vlăduţ, Nonclassical solutions of fully nonlinear elliptic equations, Geometric and Functional Analysis, 17 (2007), 1283-1296.  doi: 10.1007/s00039-007-0626-7.  Google Scholar

[25]

A. Petrosyan, H. Shahgholian and N. Uralt'seva, Regularity of Free Boundaries in Obstacle-Type Problems, Graduate Studies in Mathematics, 136. American Mathematical Society, Providence, RI, 2012. x+221 pp. ISBN: 978-0-8218-8794-3. doi: 10.1090/gsm/136.  Google Scholar

[26]

L. Silvestre and E. V. Teixeira, Regularity estimates for fully non linear elliptic equations which are asymptotically convex, in Contributions to nonlinear elliptic equations and systems, 425–438, Progr. Nonlinear Differential Equations Appl., 86, Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-19902-3_25.  Google Scholar

[27]

L. Zajíček, Porosity and $\sigma-$porosity, Real Anal. Exchange, 13 (1987/88), 314-350.  doi: 10.2307/44151885.  Google Scholar

[1]

Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920
[2]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673
[3]

Xavier Cabré. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 331-359. doi: 10.3934/dcds.2002.8.331
[4]

Luigi C. Berselli, Carlo R. Grisanti. On the regularity up to the boundary for certain nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 53-71. doi: 10.3934/dcdss.2016.9.53
[5]

Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713
[6]

Junichi Harada, Mitsuharu Ôtani. $H^2$-solutions for some elliptic equations with nonlinear boundary conditions. Conference Publications, 2009, 2009 (Special) : 333-339. doi: 10.3934/proc.2009.2009.333
[7]

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
[8]

Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386
[9]

Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6945-6959. doi: 10.3934/dcds.2019238
[10]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006
[11]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184
[12]

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
[13]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. Recent progresses on elliptic two-phase free boundary problems. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6961-6978. doi: 10.3934/dcds.2019239
[14]

Geng Chen, Yannan Shen. Existence and regularity of solutions in nonlinear wave equations. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3327-3342. doi: 10.3934/dcds.2015.35.3327
[15]

Mingxin Wang. Existence and uniqueness of solutions of free boundary problems in heterogeneous environments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 415-421. doi: 10.3934/dcdsb.2018179
[16]

Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1307-1323. doi: 10.3934/dcds.2011.31.1307
[17]

Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685
[18]

Maria Francesca Betta, Olivier Guibé, Anna Mercaldo. Uniqueness for Neumann problems for nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1023-1048. doi: 10.3934/cpaa.2019050
[19]

Ryuji Kajikiya, Daisuke Naimen. Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1593-1612. doi: 10.3934/cpaa.2014.13.1593
[20]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (78)
  • HTML views (190)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences