# American Institute of Mathematical Sciences

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

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

 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 $\nabla u$.
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:
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##### 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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