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First eigenfunctions of the 1-Laplacian are viscosity solutions
Some properties of minimizers of a variational problem involving the total variation functional
1. | Mathematisches Institut der Universität zu Köln, Weyertal 86 -- 90, 50931 Köln, Germany |
References:
[1] |
F. Alter and V. Caselles, Uniqueness of the Cheeger set of a convex body, Nonlinear Analysis, 70 (2009), 32-44.
doi: 10.1016/j.na.2007.11.032. |
[2] |
X. Cabré and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.), 4 (1998), 1-10.
doi: 10.1007/s000290050022. |
[3] |
Ph. Clément and R. Hagmeijer and G. Sweers, On the invertibility of mappings arising in 2D grid generation problems, Numerische Mathematik, 73 (1996), 37-51.
doi: 10.1007/s002110050182. |
[4] |
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976. |
[5] |
I. Fonseca and N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity, ESAIM Control, Optimisation and Calculus of Variations, 7 (2002), 69-95.
doi: 10.1051/cocv:2002004. |
[6] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 1977, Springer, Berlin. |
[7] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, 1984, Birkhäuser, Basel.
doi: 10.1007/978-1-4684-9486-0. |
[8] |
R. Glowinski and J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, 1981, North Holland Publishing Co., Amsterdam. |
[9] |
B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, 1150, Springer, Berlin, 1985. |
[10] |
B. Kawohl and T. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane, Pacific Journal of Mathematics, 225 (2006), 103-118.
doi: 10.2140/pjm.2006.225.103. |
[11] |
H. Kielhöfer, Bifurcation Theory, Second, Applied Mathematical Sciences, 156, Springer, Berlin, 2012.
doi: 10.1007/978-1-4614-0502-3. |
[12] |
N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differential Equations, 15 (1990), 541-556.
doi: 10.1080/03605309908820698. |
[13] |
F. Krügel, A variational problem leading to a singular elliptic equation involving the 1-Laplacian, Universität zu Köln, 2013. |
[14] |
X.-N. Ma and Q. Ou, The convexity of level sets for solutions to partial differential equations, in Trends in Partial Differential Equations, Adv. Lect. Math., 10, 295-322, International Press of Boston, (2010). |
[15] |
P. Marcellini and K. Miller, Elliptic versus parabolic regularization for the equation of prescribed mean curvature, Journal of Differential Equations, 137 (1997), 1-53.
doi: 10.1006/jdeq.1997.3247. |
[16] |
L. Robbiano, Sur les zéros de solutions d'inégalités différentielles elliptiques, Comm. Partial Differential Equations, 12 (1987), 903-919.
doi: 10.1080/03605308708820513. |
[17] |
L. Robbiano, Dimension de zéros d'une solution faible d'un opérateur elliptique, Journal de Mathématiques Pures et Appliquées, 67 (1988), 339-357. |
[18] |
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, Encyclopedia of mathematics and its applications, 44, 1993.
doi: 10.1017/CBO9780511526282. |
[19] |
F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampere Equations in Two Dimensions, Springer, Berlin, Lecture Notes in Mathematics, 1445, 1990. |
[20] |
L. Xu, A microscopic convexity theorem of level sets for solutions to elliptic equations, Calculus of Variations, 40 (2011), 51-63.
doi: 10.1007/s00526-010-0333-3. |
show all references
References:
[1] |
F. Alter and V. Caselles, Uniqueness of the Cheeger set of a convex body, Nonlinear Analysis, 70 (2009), 32-44.
doi: 10.1016/j.na.2007.11.032. |
[2] |
X. Cabré and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N.S.), 4 (1998), 1-10.
doi: 10.1007/s000290050022. |
[3] |
Ph. Clément and R. Hagmeijer and G. Sweers, On the invertibility of mappings arising in 2D grid generation problems, Numerische Mathematik, 73 (1996), 37-51.
doi: 10.1007/s002110050182. |
[4] |
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976. |
[5] |
I. Fonseca and N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity, ESAIM Control, Optimisation and Calculus of Variations, 7 (2002), 69-95.
doi: 10.1051/cocv:2002004. |
[6] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 1977, Springer, Berlin. |
[7] |
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, 1984, Birkhäuser, Basel.
doi: 10.1007/978-1-4684-9486-0. |
[8] |
R. Glowinski and J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, 1981, North Holland Publishing Co., Amsterdam. |
[9] |
B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, 1150, Springer, Berlin, 1985. |
[10] |
B. Kawohl and T. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane, Pacific Journal of Mathematics, 225 (2006), 103-118.
doi: 10.2140/pjm.2006.225.103. |
[11] |
H. Kielhöfer, Bifurcation Theory, Second, Applied Mathematical Sciences, 156, Springer, Berlin, 2012.
doi: 10.1007/978-1-4614-0502-3. |
[12] |
N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differential Equations, 15 (1990), 541-556.
doi: 10.1080/03605309908820698. |
[13] |
F. Krügel, A variational problem leading to a singular elliptic equation involving the 1-Laplacian, Universität zu Köln, 2013. |
[14] |
X.-N. Ma and Q. Ou, The convexity of level sets for solutions to partial differential equations, in Trends in Partial Differential Equations, Adv. Lect. Math., 10, 295-322, International Press of Boston, (2010). |
[15] |
P. Marcellini and K. Miller, Elliptic versus parabolic regularization for the equation of prescribed mean curvature, Journal of Differential Equations, 137 (1997), 1-53.
doi: 10.1006/jdeq.1997.3247. |
[16] |
L. Robbiano, Sur les zéros de solutions d'inégalités différentielles elliptiques, Comm. Partial Differential Equations, 12 (1987), 903-919.
doi: 10.1080/03605308708820513. |
[17] |
L. Robbiano, Dimension de zéros d'une solution faible d'un opérateur elliptique, Journal de Mathématiques Pures et Appliquées, 67 (1988), 339-357. |
[18] |
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, Encyclopedia of mathematics and its applications, 44, 1993.
doi: 10.1017/CBO9780511526282. |
[19] |
F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampere Equations in Two Dimensions, Springer, Berlin, Lecture Notes in Mathematics, 1445, 1990. |
[20] |
L. Xu, A microscopic convexity theorem of level sets for solutions to elliptic equations, Calculus of Variations, 40 (2011), 51-63.
doi: 10.1007/s00526-010-0333-3. |
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