Uniform Lipschitz regularity for classes of minimizers in two phase free boundary problems in Orlicz spaces with small density on the negative phase
Braga, J. Ederson M. ; Moreira, Diego R.
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 823-850 / Harvested from Numdam
Access to full text

In this paper we investigate Lipschitz regularity of minimizers for classes of functionals including ones of the type E G (u,Ω)=∫ Ω [G(|∇u|)+f 2 χ {u>0} +f 1 χ {u⩽0} ]dx. We prove that there exists a universal “tolerance” (depending only on the degenerate ellipticity and other intrinsic parameters) for the density of the negative phase along the free boundary under which uniform Lipschitz regularity holds. We also prove density estimates from below for the negative phase on points inside the contact set between the negative and positive free boundaries in the case where Lipschitz regularity fails to be the optimal one.

Publié le : 2014-01-01
DOI : https://doi.org/10.1016/j.anihpc.2013.07.006
Classification:  35J60,  35J70,  35J75,  35R35,  46E30,  49K20 
@article{AIHPC_2014__31_4_823_0,
     author = {Braga, J. Ederson M. and Moreira, Diego R.},
     title = {Uniform Lipschitz regularity for classes of minimizers in two phase free boundary problems in Orlicz spaces with small density on the negative phase},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {31},
     year = {2014},
     pages = {823-850},
     doi = {10.1016/j.anihpc.2013.07.006},
     zbl = {1301.49097},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_4_823_0}
}

Braga, J. Ederson M.; Moreira, Diego R. Uniform Lipschitz regularity for classes of minimizers in two phase free boundary problems in Orlicz spaces with small density on the negative phase. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 823-850. doi : 10.1016/j.anihpc.2013.07.006. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_4_823_0/

[1] R. Adams, J. Fournier, Sobolev Spaces, Academic Press (2003) | MR 2424078 | Zbl 1098.46001

[2] H.W. Alt, L.A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105 -144 | MR 618549 | Zbl 0449.35105

[3] H.W. Alt, L.A. Caffarelli, A. Friedman, A free boundary problem for quasi-linear elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 11 no. 1 (1984), 1 -44 | Numdam | MR 752578 | Zbl 0554.35129

[4] M. Allen, H.C. Lara, Free boundary on a cone, arXiv:1301.6047 [math.AP]

[5] H.W. Alt, L.A. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), 431 -461 | MR 732100 | Zbl 0844.35137

[6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext , Springer (2010) | MR 2759829

[7] L.A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on X , Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 15 no. 4 (1989), 583 -602 | Numdam | MR 1029856 | Zbl 0702.35249

[8] L.A. Caffarelli, D. Jerison, C. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, Noncompact Problems at the Intersection of Geometry, Analysis, and Topology, Contemp. Math. vol. 350 , Amer. Math. Soc., Providence, RI (2004), 83 -97 | Zbl 1330.35545

[9] D. De Silva, D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math. 635 (2009), 1 -21 | MR 2572253 | Zbl 1185.35050

[10] D. Danielli, A. Petrosyan, A minimum problem with free boundary for a degenerate quasilinear operator, Calc. Var. Partial Differential Equations 23 no. 1 (2005), 97 -124 | MR 2133664 | Zbl 1068.35187

[11] A. Kharakhanyan, On the Lipschitz regularity of solutions of a minimum problem with free boundary, Interfaces Free Bound. 10 (2008), 79 -86 | MR 2383537 | Zbl 1147.35116

[12] C. Lederman, N. Wolanski, A two phase elliptic singular perturbation problem with a forcing term, J. Math. Pures Appl. (9) 86 no. 6 (2006), 552 -589 | MR 2281453 | Zbl 1111.35136

[13] G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Differential Equations 16 no. 2–3 (1991), 311 -361 | MR 1104103 | Zbl 0742.35028

[14] Q. Han, F. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math. vol. 1 , Courant Institute of Mathematical Sciences/AMS, New York/Providence, RI (2011) | MR 2777537 | Zbl 1210.35031

[15] S. Martinez, N. Wolanski, A minimum problem with free boundary in Orlicz spaces, Adv. Math. 218 no. 6 (2008), 1914 -1971 | MR 2431665 | Zbl 1170.35030