-
Previous Article
Global large solutions and optimal time-decay estimates to the Korteweg system
- DCDS Home
- This Issue
-
Next Article
Gromov-Hausdorff stability for group actions
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 |
$ \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*} $ |
$ \gamma>0 $ |
$ \phi \in C^{1, \alpha}(B_1) $ |
$ \alpha\in(0,1] $ |
$ g $ |
$ f\in L^\infty(B_1)\cap C^0(B_1) $ |
$ C^{1,\beta}(B_{1/2}) $ |
$ \beta = \min\left\{\alpha, \frac{1}{\gamma+1}\right\} $ |
$ \partial\{u>\phi\} $ |
$ n $ |
$ |Du|^\gamma \Delta u = \chi_{\{u>\phi\}} \quad \text{with}\quad \gamma>0. $ |
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. |
[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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. |
[1] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 |
[2] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
[3] |
Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021028 |
[4] |
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 |
[5] |
Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286 |
[6] |
Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179 |
[7] |
Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 |
[8] |
Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083 |
[9] |
Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002 |
[10] |
Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084 |
[11] |
Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 |
[12] |
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 |
[13] |
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 |
[14] |
Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 |
[15] |
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, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318 |
[16] |
Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477 |
[17] |
Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 |
[18] |
Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387 |
[19] |
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 |
[20] |
Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]