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, Nonlinear Analysis, 70 (2009), 32-44. doi: 10.1016/j.na.2007.11.032.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976.  Google Scholar

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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.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 1977, ().   Google Scholar

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E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, 1984, ().  doi: 10.1007/978-1-4684-9486-0.  Google Scholar

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R. Glowinski and J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities,, 1981, ().   Google Scholar

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B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, 1150, Springer, Berlin, 1985.  Google Scholar

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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.  Google Scholar

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H. Kielhöfer, Bifurcation Theory, Second, Applied Mathematical Sciences, 156, Springer, Berlin, 2012. doi: 10.1007/978-1-4614-0502-3.  Google Scholar

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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.  Google Scholar

[13]

F. Krügel, A variational problem leading to a singular elliptic equation involving the 1-Laplacian, Universität zu Köln, 2013. Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[16]

L. Robbiano, Sur les zéros de solutions d'inégalités différentielles elliptiques, Comm. Partial Differential Equations, 12 (1987), 903-919. doi: 10.1080/03605308708820513.  Google Scholar

[17]

L. Robbiano, Dimension de zéros d'une solution faible d'un opérateur elliptique, Journal de Mathématiques Pures et Appliquées, 67 (1988), 339-357.  Google Scholar

[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.  Google Scholar

[19]

F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampere Equations in Two Dimensions, Springer, Berlin, Lecture Notes in Mathematics, 1445, 1990.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin, 1976.  Google Scholar

[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.  Google Scholar

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 1977, ().   Google Scholar

[7]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, 1984, ().  doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[8]

R. Glowinski and J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities,, 1981, ().   Google Scholar

[9]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics, 1150, Springer, Berlin, 1985.  Google Scholar

[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.  Google Scholar

[11]

H. Kielhöfer, Bifurcation Theory, Second, Applied Mathematical Sciences, 156, Springer, Berlin, 2012. doi: 10.1007/978-1-4614-0502-3.  Google Scholar

[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.  Google Scholar

[13]

F. Krügel, A variational problem leading to a singular elliptic equation involving the 1-Laplacian, Universität zu Köln, 2013. Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[16]

L. Robbiano, Sur les zéros de solutions d'inégalités différentielles elliptiques, Comm. Partial Differential Equations, 12 (1987), 903-919. doi: 10.1080/03605308708820513.  Google Scholar

[17]

L. Robbiano, Dimension de zéros d'une solution faible d'un opérateur elliptique, Journal de Mathématiques Pures et Appliquées, 67 (1988), 339-357.  Google Scholar

[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.  Google Scholar

[19]

F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampere Equations in Two Dimensions, Springer, Berlin, Lecture Notes in Mathematics, 1445, 1990.  Google Scholar

[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.  Google Scholar

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