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

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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

Abstract
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Under a Creative Commons license

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 and 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), 261-281. doi: 10.1007/BF00284509.
[2]	E. Acerbi and N. Fusco, Local regularity for minimizers of nonconvex integrals, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1989), 603-636.
[3]	G. Anzellotti and M. Giaquinta, Convex functionals and partial regularity, Arch. Rational Mech. Anal., 102 (1988), 243-272. doi: 10.1007/BF00281349.
[4]	L. Brasco, Global $L^\infty$ gradient estimates for solutions to a certain degenerate elliptic equation, Nonlinear Anal., 74 (2011), 516-531. doi: 10.1016/j.na.2010.09.006.
[5]	L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl., 93 (2010), 652-671. doi: 10.1016/j.matpur.2010.03.010.
[6]	M. Colombo and A. Figalli, Regularity results for very degenerate elliptic equations, J. Math. Pures Appl., 101 (2014), 94-117. doi: 10.1016/j.matpur.2013.05.005.
[7]	D. De Silva and O. Savin, Minimizers of convex functionals arising in random surfaces, Duke Math. J., 151 (2010), 487-532. doi: 10.1215/00127094-2010-004.
[8]	E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.
[9]	L. Esposito, G. Mingione and C. Trombetti, On the Lipschitz regularity for certain elliptic problems, Forum Math., 18 (2006), 263-292. doi: 10.1515/FORUM.2006.016.
[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), 356-373. doi: 10.1016/0022-0396(82)90033-X.
[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), 69-95. doi: 10.1051/cocv:2002004.
[12]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton, 1983.
[13]	M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse non linéaire, 3 (1986), 185-208.
[14]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
[15]	C. Imbert and L. Silvestre, Estimates on elliptic equations that hold only where the gradient is large, preprint, (2013).
[16]	J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32 (1983), 849-858. doi: 10.1512/iumj.1983.32.32058.
[17]	F. Santambrogio and V. Vespri, Continuity in two dimensions for a very degenerate elliptic equation, Nonlinear Anal., 73 (2010), 3832-3841. doi: 10.1016/j.na.2010.08.008.
[18]	O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578. doi: 10.1080/03605300500394405.
[19]	P. Tolksdorff, Regularity for a more general class of quasi-linear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.
[20]	K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.
[21]	N. N. Uraltseva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184-222.
[22]	L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations, 107 (1994), 341-350. doi: 10.1006/jdeq.1994.1016.

show all references

References:

[1]	E. Acerbi and N. Fusco, A regularity theorem for minimizers of quasiconvex integrals, Arch. Rational Mech. Anal., 99 (1987), 261-281. doi: 10.1007/BF00284509.
[2]	E. Acerbi and N. Fusco, Local regularity for minimizers of nonconvex integrals, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 16 (1989), 603-636.
[3]	G. Anzellotti and M. Giaquinta, Convex functionals and partial regularity, Arch. Rational Mech. Anal., 102 (1988), 243-272. doi: 10.1007/BF00281349.
[4]	L. Brasco, Global $L^\infty$ gradient estimates for solutions to a certain degenerate elliptic equation, Nonlinear Anal., 74 (2011), 516-531. doi: 10.1016/j.na.2010.09.006.
[5]	L. Brasco, G. Carlier and F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations, J. Math. Pures Appl., 93 (2010), 652-671. doi: 10.1016/j.matpur.2010.03.010.
[6]	M. Colombo and A. Figalli, Regularity results for very degenerate elliptic equations, J. Math. Pures Appl., 101 (2014), 94-117. doi: 10.1016/j.matpur.2013.05.005.
[7]	D. De Silva and O. Savin, Minimizers of convex functionals arising in random surfaces, Duke Math. J., 151 (2010), 487-532. doi: 10.1215/00127094-2010-004.
[8]	E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5.
[9]	L. Esposito, G. Mingione and C. Trombetti, On the Lipschitz regularity for certain elliptic problems, Forum Math., 18 (2006), 263-292. doi: 10.1515/FORUM.2006.016.
[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), 356-373. doi: 10.1016/0022-0396(82)90033-X.
[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), 69-95. doi: 10.1051/cocv:2002004.
[12]	M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton, 1983.
[13]	M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse non linéaire, 3 (1986), 185-208.
[14]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
[15]	C. Imbert and L. Silvestre, Estimates on elliptic equations that hold only where the gradient is large, preprint, (2013).
[16]	J. L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J., 32 (1983), 849-858. doi: 10.1512/iumj.1983.32.32058.
[17]	F. Santambrogio and V. Vespri, Continuity in two dimensions for a very degenerate elliptic equation, Nonlinear Anal., 73 (2010), 3832-3841. doi: 10.1016/j.na.2010.08.008.
[18]	O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations, 32 (2007), 557-578. doi: 10.1080/03605300500394405.
[19]	P. Tolksdorff, Regularity for a more general class of quasi-linear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.
[20]	K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. doi: 10.1007/BF02392316.
[21]	N. N. Uraltseva, Degenerate quasilinear elliptic systems, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968), 184-222.
[22]	L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations, 107 (1994), 341-350. doi: 10.1006/jdeq.1994.1016.

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