# American Institute of Mathematical Sciences

January  2015, 14(1): 341-360. doi: 10.3934/cpaa.2015.14.341

## 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

Received  January 2014 Revised  March 2014 Published  September 2014

The variational problem of minimizing the functional $u \mapsto \int_\Omega |Du| + \frac{1}{p}\int_\Omega |Du|^p - \int_\Omega au$ on a domain $\Omega\subset R^2$ under zero boundary values, which among other things models the laminar flow of a Bingham fluid, shows an interesting phenomenon: its minimizer has a maximum set with positive measure (a "plateau"). In this work we show properties of the minimizer and its plateau, most notably, connectedness and a lower bound of its measure. In addition we look at the related boundary value problem where $a=0$, $\Omega$ is a convex ring, and two boundary values are given. For this problem we show various results, including quasiconcavity of the minimizer and regularity.
Citation: Florian Krügel. Some properties of minimizers of a variational problem involving the total variation functional. Communications on Pure & Applied Analysis, 2015, 14 (1) : 341-360. doi: 10.3934/cpaa.2015.14.341
##### References:
 [1] F. Alter and V. Caselles, Uniqueness of the Cheeger set of a convex body,, \emph{Nonlinear Analysis}, 70 (2009), 32. doi: 10.1016/j.na.2007.11.032. [2] X. Cabré and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains,, \emph{Selecta Math. (N.S.)}, 4 (1998), 1. doi: 10.1007/s000290050022. [3] Ph. Clément and R. Hagmeijer and G. Sweers, On the invertibility of mappings arising in 2D grid generation problems,, \emph{Numerische Mathematik}, 73 (1996), 37. doi: 10.1007/s002110050182. [4] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics,, Springer, (1976). [5] I. Fonseca and N. Fusco and P. Marcellini, An existence result for a nonconvex variational problem via regularity,, \emph{ESAIM Control, 7 (2002), 69. doi: 10.1051/cocv:2002004. [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 1977, (). [7] E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, 1984, (). doi: 10.1007/978-1-4684-9486-0. [8] R. Glowinski and J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities,, 1981, (). [9] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lecture Notes in Mathematics, (1150). [10] B. Kawohl and T. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane,, \emph{Pacific Journal of Mathematics}, 225 (2006), 103. doi: 10.2140/pjm.2006.225.103. [11] H. Kielhöfer, Bifurcation Theory,, Second, 156 (2012). doi: 10.1007/978-1-4614-0502-3. [12] N. J. Korevaar, Convexity of level sets for solutions to elliptic ring problems,, \emph{Comm. Partial Differential Equations}, 15 (1990), 541. doi: 10.1080/03605309908820698. [13] F. Krügel, A variational problem leading to a singular elliptic equation involving the 1-Laplacian,, Universit\, (2013). [14] X.-N. Ma and Q. Ou, The convexity of level sets for solutions to partial differential equations,, in \emph{Trends in Partial Differential Equations}, 10 (2010), 295. [15] P. Marcellini and K. Miller, Elliptic versus parabolic regularization for the equation of prescribed mean curvature,, \emph{Journal of Differential Equations}, 137 (1997), 1. doi: 10.1006/jdeq.1997.3247. [16] L. Robbiano, Sur les zéros de solutions d'inégalités différentielles elliptiques,, \emph{Comm. Partial Differential Equations}, 12 (1987), 903. doi: 10.1080/03605308708820513. [17] L. Robbiano, Dimension de zéros d'une solution faible d'un opérateur elliptique,, \emph{Journal de Math\'ematiques Pures et Appliqu\'ees}, 67 (1988), 339. [18] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory,, Cambridge University Press, 44 (1993). doi: 10.1017/CBO9780511526282. [19] F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampere Equations in Two Dimensions,, Springer, 1445 (1990). [20] L. Xu, A microscopic convexity theorem of level sets for solutions to elliptic equations,, \emph{Calculus of Variations}, 40 (2011), 51. doi: 10.1007/s00526-010-0333-3.

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