American Institute of Mathematical Sciences

Advanced
August  2014, 7(4): 631-652. doi: 10.3934/dcdss.2014.7.631

An excess-decay result for a class of degenerate elliptic equations

Maria Colombo 1, and  Alessio Figalli 2,

1. 

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa
2. 

University of Texas at Austin, Department of Mathematics, 2515 Speedway Stop C1200, Austin, TX 78712-1202

Received  October 2013 Revised  December 2013 Published  February 2014

We consider a family of degenerate elliptic equations of the form div $(\nabla F(\nabla u)) = f$, where $F\in C^{1,1}$ is a convex function which is elliptic outside a ball. We prove an excess-decay estimate at points where $\nabla u$ is close to a nondegenerate value for $F$. This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of $\nabla u$ outside the degeneracy, and to anisotropic versions of the $p$-laplacian, where we get Hölder regularity of $\nabla u$.
Keywords: traffic congestion, excess decay, Degenerate elliptic PDEs, continuity of the gradient..
Mathematics Subject Classification: Primary: 35J70; Secondary: 35B65, 49K2.
Citation: Maria Colombo, Alessio Figalli. An excess-decay result for a class of degenerate elliptic equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 631-652. doi: 10.3934/dcdss.2014.7.631
References:
[1]

E. Acerbi and N. Fusco, A regularity theorem for minimizers of quasiconvex integrals,, Arch. Rational Mech. Anal., 99 (1987), 261.  doi: 10.1007/BF00284509.  Google Scholar

[2]

E. Acerbi and N. Fusco, Local regularity for minimizers of nonconvex integrals,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1989), 603.   Google Scholar

[3]

G. Anzellotti and M. Giaquinta, Convex functionals and partial regularity,, Arch. Rational Mech. Anal., 102 (1988), 243.  doi: 10.1007/BF00281349.  Google Scholar

[4]

L. Brasco, Global $L^\infty$ gradient estimates for solutions to a certain degenerate elliptic equation,, Nonlinear Anal., 74 (2011), 516.  doi: 10.1016/j.na.2010.09.006.  Google Scholar

[5]

L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations,, J. Math. Pures Appl., 93 (2010), 652.  doi: 10.1016/j.matpur.2010.03.010.  Google Scholar

[6]

M. Colombo and A. Figalli, Regularity results for very degenerate elliptic equations,, J. Math. Pures Appl., 101 (2014), 94.  doi: 10.1016/j.matpur.2013.05.005.  Google Scholar

[7]

D. De Silva and O. Savin, Minimizers of convex functionals arising in random surfaces,, Duke Math. J., 151 (2010), 487.  doi: 10.1215/00127094-2010-004.  Google Scholar

[8]

E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[9]

L. Esposito, G. Mingione and C. Trombetti, On the Lipschitz regularity for certain elliptic problems,, Forum Math., 18 (2006), 263.  doi: 10.1515/FORUM.2006.016.  Google Scholar

[10]

L. C. Evans, A new proof of local $C^{1,\alpha }$ regularity for solutions of certain degenerate elliptic p.d.e.,, J. Differential Equations, 45 (1982), 356.  doi: 10.1016/0022-0396(82)90033-X.  Google Scholar

[11]

I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity,, ESAIM Control Optim. Calc. Var., 7 (2002), 69.  doi: 10.1051/cocv:2002004.  Google Scholar

[12]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,, Princeton Univ. Press, (1983).   Google Scholar

[13]

M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals,, Ann. Inst. H. Poincaré, 3 (1986), 185.   Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, reprint of the 1998 edition, (1998).   Google Scholar

[15]

C. Imbert and L. Silvestre, Estimates on elliptic equations that hold only where the gradient is large,, preprint, (2013).   Google Scholar

[16]

J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations,, Indiana Univ. Math. J., 32 (1983), 849.  doi: 10.1512/iumj.1983.32.32058.  Google Scholar

[17]

F. Santambrogio and V. Vespri, Continuity in two dimensions for a very degenerate elliptic equation,, Nonlinear Anal., 73 (2010), 3832.  doi: 10.1016/j.na.2010.08.008.  Google Scholar

[18]

O. Savin, Small perturbation solutions for elliptic equations,, Comm. Partial Differential Equations, 32 (2007), 557.  doi: 10.1080/03605300500394405.  Google Scholar

[19]

P. Tolksdorff, Regularity for a more general class of quasi-linear elliptic equations,, J. Differential Equations, 51 (1984), 126.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[20]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems,, Acta Math., 138 (1977), 219.  doi: 10.1007/BF02392316.  Google Scholar

[21]

N. N. Uraltseva, Degenerate quasilinear elliptic systems,, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184.   Google Scholar

[22]

L. Wang, Compactness methods for certain degenerate elliptic equations,, J. Differential Equations, 107 (1994), 341.  doi: 10.1006/jdeq.1994.1016.  Google Scholar

show all references

References:
[1]

E. Acerbi and N. Fusco, A regularity theorem for minimizers of quasiconvex integrals,, Arch. Rational Mech. Anal., 99 (1987), 261.  doi: 10.1007/BF00284509.  Google Scholar

[2]

E. Acerbi and N. Fusco, Local regularity for minimizers of nonconvex integrals,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1989), 603.   Google Scholar

[3]

G. Anzellotti and M. Giaquinta, Convex functionals and partial regularity,, Arch. Rational Mech. Anal., 102 (1988), 243.  doi: 10.1007/BF00281349.  Google Scholar

[4]

L. Brasco, Global $L^\infty$ gradient estimates for solutions to a certain degenerate elliptic equation,, Nonlinear Anal., 74 (2011), 516.  doi: 10.1016/j.na.2010.09.006.  Google Scholar

[5]

L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations,, J. Math. Pures Appl., 93 (2010), 652.  doi: 10.1016/j.matpur.2010.03.010.  Google Scholar

[6]

M. Colombo and A. Figalli, Regularity results for very degenerate elliptic equations,, J. Math. Pures Appl., 101 (2014), 94.  doi: 10.1016/j.matpur.2013.05.005.  Google Scholar

[7]

D. De Silva and O. Savin, Minimizers of convex functionals arising in random surfaces,, Duke Math. J., 151 (2010), 487.  doi: 10.1215/00127094-2010-004.  Google Scholar

[8]

E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[9]

L. Esposito, G. Mingione and C. Trombetti, On the Lipschitz regularity for certain elliptic problems,, Forum Math., 18 (2006), 263.  doi: 10.1515/FORUM.2006.016.  Google Scholar

[10]

L. C. Evans, A new proof of local $C^{1,\alpha }$ regularity for solutions of certain degenerate elliptic p.d.e.,, J. Differential Equations, 45 (1982), 356.  doi: 10.1016/0022-0396(82)90033-X.  Google Scholar

[11]

I. Fonseca, N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity,, ESAIM Control Optim. Calc. Var., 7 (2002), 69.  doi: 10.1051/cocv:2002004.  Google Scholar

[12]

M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems,, Princeton Univ. Press, (1983).   Google Scholar

[13]

M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals,, Ann. Inst. H. Poincaré, 3 (1986), 185.   Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, reprint of the 1998 edition, (1998).   Google Scholar

[15]

C. Imbert and L. Silvestre, Estimates on elliptic equations that hold only where the gradient is large,, preprint, (2013).   Google Scholar

[16]

J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations,, Indiana Univ. Math. J., 32 (1983), 849.  doi: 10.1512/iumj.1983.32.32058.  Google Scholar

[17]

F. Santambrogio and V. Vespri, Continuity in two dimensions for a very degenerate elliptic equation,, Nonlinear Anal., 73 (2010), 3832.  doi: 10.1016/j.na.2010.08.008.  Google Scholar

[18]

O. Savin, Small perturbation solutions for elliptic equations,, Comm. Partial Differential Equations, 32 (2007), 557.  doi: 10.1080/03605300500394405.  Google Scholar

[19]

P. Tolksdorff, Regularity for a more general class of quasi-linear elliptic equations,, J. Differential Equations, 51 (1984), 126.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar

[20]

K. Uhlenbeck, Regularity for a class of non-linear elliptic systems,, Acta Math., 138 (1977), 219.  doi: 10.1007/BF02392316.  Google Scholar

[21]

N. N. Uraltseva, Degenerate quasilinear elliptic systems,, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184.   Google Scholar

[22]

L. Wang, Compactness methods for certain degenerate elliptic equations,, J. Differential Equations, 107 (1994), 341.  doi: 10.1006/jdeq.1994.1016.  Google Scholar

[1]

Fethallah Benmansour, Guillaume Carlier, Gabriel Peyré, Filippo Santambrogio. Numerical approximation of continuous traffic congestion equilibria. Networks & Heterogeneous Media, 2009, 4 (3) : 605-623. doi: 10.3934/nhm.2009.4.605
[2]

Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507
[3]

Jingmei Zhou, Xiangmo Zhao, Xin Cheng, Zhigang Xu. Visualization analysis of traffic congestion based on floating car data. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1423-1433. doi: 10.3934/dcdss.2015.8.1423
[4]

Sachiko Ishida. $L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 335-344. doi: 10.3934/proc.2013.2013.335
[5]

Messoud Efendiev, Anna Zhigun. On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 651-673. doi: 10.3934/dcds.2018028
[6]

Antonio Vitolo. On the growth of positive entire solutions of elliptic PDEs and their gradients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1335-1346. doi: 10.3934/dcdss.2014.7.1335
[7]

Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081
[8]

Tariel Sanikidze, A.F. Tedeev. On the temporal decay estimates for the degenerate parabolic system. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1755-1768. doi: 10.3934/cpaa.2013.12.1755
[9]

Lucio Boccardo, Maria Michaela Porzio. Some degenerate parabolic problems: Existence and decay properties. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 617-629. doi: 10.3934/dcdss.2014.7.617
[10]

Massimiliano Berti, M. Matzeu, Enrico Valdinoci. On periodic elliptic equations with gradient dependence. Communications on Pure & Applied Analysis, 2008, 7 (3) : 601-615. doi: 10.3934/cpaa.2008.7.601
[11]

Agnese Di Castro, Mayte Pérez-Llanos, José Miguel Urbano. Limits of anisotropic and degenerate elliptic problems. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1217-1229. doi: 10.3934/cpaa.2012.11.1217
[12]

Tomasz Komorowski, Adam Bobrowski. A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020248
[13]

Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069
[14]

V. Lakshmikantham, S. Leela. Generalized quasilinearization and semilinear degenerate elliptic problems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 801-808. doi: 10.3934/dcds.2001.7.801
[15]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184
[16]

Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549
[17]

Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363
[18]

Yuxia Guo, Jianjun Nie. Classification for positive solutions of degenerate elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1457-1475. doi: 10.3934/dcds.2018130
[19]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[20]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences