    doi: 10.3934/dcds.2020321

## Sharp regularity for degenerate obstacle type problems: A geometric approach

 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
 $|Du|^\gamma \Delta u = \chi_{\{u>\phi\}} \quad \text{with}\quad \gamma>0.$
Citation: João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020321
##### References:
  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  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  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  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  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  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  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  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  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  L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS Colloquium Publications, Providence, Vol 43, 1995. doi: 10.1090/coll/043.  Google Scholar  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  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  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  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  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  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  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  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  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  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  N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108. Google Scholar  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  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  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  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  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  L. Zajíček, Porosity and $\sigma-$porosity, Real Anal. Exchange, 13 (1987/88), 314-350.  doi: 10.2307/44151885.  Google Scholar

show all references

##### References:
  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  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  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  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  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  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  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  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  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  L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, AMS Colloquium Publications, Providence, Vol 43, 1995. doi: 10.1090/coll/043.  Google Scholar  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  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  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  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  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  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  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  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  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  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  N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108. Google Scholar  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  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  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  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  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  L. Zajíček, Porosity and $\sigma-$porosity, Real Anal. Exchange, 13 (1987/88), 314-350.  doi: 10.2307/44151885.  Google Scholar
  Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272  Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267  Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380  Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $p$ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442  Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436  Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318  Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456  Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247  Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340  Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248  Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018  Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052  Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297  Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119  Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448  Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469  Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450  Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168  Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137  Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

2019 Impact Factor: 1.338