Existence of an Absolute Minimizer via Perron's Method

Vesa Julin

doi:10.68381/jca18015

Abstract

The existence of an absolute minimizer for a functional

F(u,\Omega) = \underset{x \in \Omega}{ \text{ess sup}} \, f (x, u(x), Du(x))

is proved by using Perron's method. The function is assumed to be quasiconvex and uniformly coercive. This completes the result by T. Champion, L. De Pascale and F. Prinari [Gamma-convergence and absolute minimizers for supremal functionals, ESAIM Control Optim. Calc. Var. 10 (2004), No. 1, 14–27 (electronic)].

Author information

Contact details are reproduced from the original publication and may be historical.

Vesa Julin

Dept. of Mathematics and Statistics, P. O. Box 35, University of Jyväskylä, 40014 Jyväskylä, Finland

vesa.julin@jyu.fi

Keywords

Supremal functionals
absolute minimizer

Mathematics Subject Classification

49J45, 49J99

References

G. Aronsson: Minimization problems for the functional supx F (x, f (x), f ′ (x)). I, Ark. Mat. 6 (1965) 33–53.https://doi.org/10.1007/BF02591326 Google Scholar
G. Aronsson: Minimization problems for the functional supx F (x, f (x), f ′ (x)). II, Ark. Mat. 6 (1966) 409–431.https://doi.org/10.1007/BF02590964 Google Scholar
G. Aronsson: Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6(28) (1967) 551–561.https://doi.org/10.1007/BF02591928 Google Scholar
E. N. Barron, R. Jensen, C. Y. Wang: The Euler equation and absolute minimizers of L∞ functionals, Arch. Ration. Mech. Anal. 157(4) (2001) 255–283.https://doi.org/10.1007/PL00004239 Google Scholar
T. Bhattacharya, E. DiBenedetto, J. Manfredi: Limits as p → ∞ of △p up = f and related extremal problems, Rend. Sem. Mat. Fascicolo Speciale Non Linear PDE’s, Univ. di Torino (1989) 15–68.Google Scholar
T. Champion, L. De Pascale, F. Prinari: Γ-convergence and absolute minimizers for supremal functionals, ESAIM, Control Optim. Calc. Var. 10(1) (2004) 14–27.https://doi.org/10.1051/cocv:2003036 Google Scholar
P. Juutinen: Absolutely minimizing Lipschitz extensions on a metric space, Ann. Acad. Sci. Fenn., Math. 27(1) (2002) 57–67.Google Scholar
V. A. Milman: Absolutely minimal extensions of functions on metric spaces, Mat. Sb. 190(6) (1999) 83–110 (in Russian); Sb. Math. 190(6) (1999) 859–885 (in English).https://doi.org/10.4213/sm409 https://doi.org/10.1070/SM1999v190n06ABEH000409 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.

[1] G. Aronsson: Minimization problems for the functional supx F (x, f (x), f ′ (x)). I, Ark. Mat. 6 (1965) 33–53.https://doi.org/10.1007/BF02591326 Google Scholar

[2] G. Aronsson: Minimization problems for the functional supx F (x, f (x), f ′ (x)). II, Ark. Mat. 6 (1966) 409–431.https://doi.org/10.1007/BF02590964 Google Scholar

[3] G. Aronsson: Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6(28) (1967) 551–561.https://doi.org/10.1007/BF02591928 Google Scholar

[4] E. N. Barron, R. Jensen, C. Y. Wang: The Euler equation and absolute minimizers of L∞ functionals, Arch. Ration. Mech. Anal. 157(4) (2001) 255–283.https://doi.org/10.1007/PL00004239 Google Scholar

[5] T. Bhattacharya, E. DiBenedetto, J. Manfredi: Limits as p → ∞ of △p up = f and related extremal problems, Rend. Sem. Mat. Fascicolo Speciale Non Linear PDE’s, Univ. di Torino (1989) 15–68.Google Scholar

[6] T. Champion, L. De Pascale, F. Prinari: Γ-convergence and absolute minimizers for supremal functionals, ESAIM, Control Optim. Calc. Var. 10(1) (2004) 14–27.https://doi.org/10.1051/cocv:2003036 Google Scholar

[7] P. Juutinen: Absolutely minimizing Lipschitz extensions on a metric space, Ann. Acad. Sci. Fenn., Math. 27(1) (2002) 57–67.Google Scholar

[8] V. A. Milman: Absolutely minimal extensions of functions on metric spaces, Mat. Sb. 190(6) (1999) 83–110 (in Russian); Sb. Math. 190(6) (1999) 859–885 (in English).https://doi.org/10.4213/sm409 https://doi.org/10.1070/SM1999v190n06ABEH000409 Google Scholar