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Some degenerate parabolic problems: Existence and decay properties
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 |
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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