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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  March 2021 Early access  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

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