March  2021, 41(3): 1359-1385. 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  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. $
Citation: João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321
References:
[1]

M. D. AmaralJ. V. da SilvaG. 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.

[2]

J. AnderssonE. 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.

[3]

D. J. AraújoG. 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.

[4]

D. J. AraújoE. 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.

[5]

A. AttouchiM. 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.

[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.

[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.

[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.

[9]

S.-S. ByunK.-A. LeeJ. 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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

G. DávilaP. 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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[27]

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

show all references

References:
[1]

M. D. AmaralJ. V. da SilvaG. 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.

[2]

J. AnderssonE. 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.

[3]

D. J. AraújoG. 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.

[4]

D. J. AraújoE. 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.

[5]

A. AttouchiM. 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.

[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.

[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.

[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.

[9]

S.-S. ByunK.-A. LeeJ. 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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

G. DávilaP. 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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[27]

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

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