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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

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