Minimizing L -norm functional on divergence-free fields
Yan, Baisheng
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011), p. 325-355 / Harvested from Numdam

In this paper, we study the minimization problem on the L -norm functional over the divergence-free fields with given boundary normal component. We focus on the computation of the minimum value and the classification of certain special minimizers including the so-called absolute minimizers. In particular, several alternative approaches for computing the minimum value are given using L q -approximations and the sets of finite perimeter. For problems in two dimensions, we establish the existence of absolute minimizers using a similar technique for the absolute minimizers of L -functionals of gradient fields. In some special cases, precise characterizations of all minimizers and the absolute minimizers are also given based on equivalent descriptions of the absolutely minimizing Lipschitz extensions of boundary functions.

Publié le : 2011-01-01
DOI : https://doi.org/10.1016/j.anihpc.2011.02.004
Classification:  49J45,  49K30,  26B30,  35J92
@article{AIHPC_2011__28_3_325_0,
     author = {Yan, Baisheng},
     title = {Minimizing ${L}^{\infty }$-norm functional on divergence-free fields},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {28},
     year = {2011},
     pages = {325-355},
     doi = {10.1016/j.anihpc.2011.02.004},
     mrnumber = {2795710},
     zbl = {1233.49010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2011__28_3_325_0}
}
Yan, Baisheng. Minimizing ${L}^{\infty }$-norm functional on divergence-free fields. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) pp. 325-355. doi : 10.1016/j.anihpc.2011.02.004. http://gdmltest.u-ga.fr/item/AIHPC_2011__28_3_325_0/

[1] R.A. Adams, J. Fournier, Sobolev Spaces, Academic Press, New York (2003) | MR 2424078 | Zbl 0201.16301

[2] F. Alter, V. Caselles, Uniqueness of the Cheeger set of a convex body, Nonlinear Anal. 70 (2009), 32-44 | MR 2468216 | Zbl 1167.52005

[3] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon Press, Oxford University Press, New York (2000) | MR 1857292 | Zbl 0957.49001

[4] F. Andreu, J.M. Mazón, J.D. Rossi, The best constant for the Sobolev trace embedding from W 1,1 (Ω) into L 1 (Ω), Nonlinear Anal. 59 (2004), 1125-1145 | MR 2098509 | Zbl 1108.35067

[5] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. 135 no. 4 (1983), 293-318 | MR 750538 | Zbl 0572.46023

[6] G. Aronsson, M. Crandall, P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. 41 no. 4 (2004), 439-505 | MR 2083637 | Zbl 1150.35047

[7] E. Barron, R. Jensen, C. Wang, The Euler equation and absolute minimizers of L functionals, Arch. Rational Mech. Anal. 157 (2001), 255-283 | MR 1831173 | Zbl 0979.49003

[8] G. Bellettini, V. Caselles, M. Novaga, Explicit solutions of the eigenvalue problem - div (Du |Du|)=u in 𝐑 2 , SIAM J. Math. Anal. 36 no. 4 (2005), 1095-1129 | MR 2139202 | Zbl 1162.35379

[9] M. Bocea, V. Nesi, Γ-convergence of power-law functionals, variational principles in L , and applications, SIAM J. Math. Anal. 39 no. 5 (2008), 1550-1576 | MR 2377289 | Zbl 1166.35300

[10] J. Bourgain, H. Brezis, On the equation div Y=f and application to control of phases, J. Amer. Math. Soc. 16 no. 2 (2002), 393-426 | MR 1949165 | Zbl 1075.35006

[11] G. Chen, M. Torres, W. Ziemer, Gauss–Green Theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math. LXII (2009), 0242-0304 | MR 2468610

[12] F. Demengel, Compactness theorems for spaces of functions with bounded derivatives and applications to limit analysis problems in plasticity, Arch. Rational Mech. Anal. 105 no. 2 (1989), 123-161 | MR 968458 | Zbl 0669.73030

[13] F. Demengel, One some nonlinear equation involving the 1-Laplacian and trace map inequalities, Nonlinear Anal. 48 (2002), 1151-1163 | MR 1880578 | Zbl 1016.49001

[14] L.C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence (1998) | MR 1625845

[15] I. Fonseca, S. Müller, 𝒜-quasiconvexity, lower semicontinuity and Young measures, SIAM J. Math. Anal. 30 (1999), 1355-1390 | MR 1718306 | Zbl 0940.49014

[16] M. Giaquinta, G. Modica, J. Soucek, Cartesian Currents in the Calculus of Variations I, Springer, Berlin (1998) | MR 1645082 | Zbl 0914.49001

[17] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston (1984) | MR 775682 | Zbl 0545.49018

[18] I. Ionescu, T. Lachand-Robert, Generalized Cheeger sets related to landslides, Calc. Var. 23 (2005), 227-249 | MR 2138084 | Zbl 1062.49036

[19] R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup-norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), 51-74 | MR 1218686 | Zbl 0789.35008

[20] B. Kawohl, V. Fridman, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolin. 44 no. 4 (2003), 659-667 | MR 2062882 | Zbl 1105.35029

[21] J. Malý, D. Swanson, W. Ziemer, The co-area formula for Sobolev mappings, Trans. Amer. Math. Soc. 355 no. 2 (2002), 477-492 | MR 1932709 | Zbl 1034.46032

[22] P. Pedregal, B. Yan, A duality method for micromagnetics, SIAM J. Math. Anal. 41 no. 6 (2010), 2431-2452 | MR 2607317 | Zbl 1203.49052

[23] Y. Peres, O. Schramm, S. Sheffields, D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 no. 1 (2009), 167-210 | MR 2449057 | Zbl 1206.91002

[24] N. Saintier, Estimates of the best Sobolev constant of the embedding of 𝐵𝑉(Ω) into L 1 (Ω) and related shape optimization problems, Nonlinear Anal. 69 (2008), 2479-2491 | MR 2446345 | Zbl 1151.49036