Some properties of minimizers of a variational problem involving the total variation functional

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First eigenfunctions of the 1-Laplacian are viscosity solutions

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

Florian Krügel ^1,

1.	Mathematisches Institut der Universität zu Köln, Weyertal 86 -- 90, 50931 Köln, Germany

Received January 2014 Revised March 2014 Published September 2014

Abstract
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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.

Keywords: 1-Laplacian, Bingham fluid, convex ring., p-Laplacian, quasiconcavity.

Mathematics Subject Classification: Primary: 35J93, 35B38, 35B65; Secondary: 35J9.

Citation: Florian Krügel. Some properties of minimizers of a variational problem involving the total variation functional. Communications on Pure and 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, 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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