Full and partial regularity for a class of nonlinear free boundary problems
Karakhanyan, Aram1
1 School of Mathematics, The University of Edinburgh, Peter Tait Guthrie Road, EH9 3FD Edinburgh, UK
Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 981-999.
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In this paper we classify the nonnegative global minimizers of the functional

JF(u)=∫ΩF(|∇u|2)+λ2χ{u>0},
where F satisfies some structural conditions and χD is the characteristic function of a set D⊂Rn. We compute the second variation of the energy and study the properties of the stability operator. The free boundary ∂{u>0} can be seen as a rectifiable n−1 varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if n=3 and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.

Reçu le :
Révisé le :
Accepté le :
DOI : 10.1016/j.anihpc.2020.09.008
Classification : 35R35
Mots-clés : Free boundary, Regularity theory, Monotonicity formula, Free boundary conditions
Karakhanyan, Aram 1

1 School of Mathematics, The University of Edinburgh, Peter Tait Guthrie Road, EH9 3FD Edinburgh, UK 
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Karakhanyan, Aram. Full and partial regularity for a class of nonlinear free boundary problems. Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 981-999. doi : 10.1016/j.anihpc.2020.09.008. http://archive.numdam.org/articles/10.1016/j.anihpc.2020.09.008/

[1] Allard, William K. On the first variation of a varifold, Ann. Math. (2), Volume 95 (1972), pp. 417-491 (MR0307015) | DOI | MR | Zbl

[2] Almgren, Fred; Taylor, Jean E.; Wang, Lihe Curvature-driven flows: a variational approach, SIAM J. Control Optim., Volume 31 (1993) no. 2, pp. 387-438 (MR1205983) | DOI | MR | Zbl

[3] Alt, H.W.; Caffarelli, L.A. Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., Volume 325 (1981), pp. 105-144 (MR618549) | MR | Zbl

[4] Alt, Hans Wilhelm; Caffarelli, Luis A.; Friedman, Avner A free boundary problem for quasilinear elliptic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 11 (1984) no. 1, pp. 1-44 (MR752578) | Numdam | MR | Zbl

[5] Barozzi, Elisabetta; Gonzalez, Eduardo; Massari, Umberto The mean curvature of a Lipschitz continuous manifold, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., Volume 14 (2003) no. 4, pp. 257-277 (2004) (in English, with English and Italian summaries), MR2104215 | MR | Zbl

[6] Caffarelli, Luis A.; Jerison, David; Kenig, Carlos E. 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, pp. 83-97 (MR2082392) | DOI | MR | Zbl

[7] Dai, Qiuyi; Trudinger, Neil S.; Wang, Xu-Jia The mean curvature measure, J. Eur. Math. Soc., Volume 14 (2012) no. 3, pp. 779-800 (MR2911884) | DOI | MR | Zbl

[8] Danielli, Donatella; Petrosyan, Arshak Full regularity of the free boundary in a Bernoulli-type problem in two dimensions, Math. Res. Lett., Volume 13 (2006) no. 4, pp. 667-681 (MR2250499) | DOI | MR | Zbl

[9] De Silva, Daniela; Jerison, David A singular energy minimizing free boundary, J. Reine Angew. Math., Volume 635 (2009), pp. 1-21 (MR2572253) | DOI | MR | Zbl

[10] Dipierro, Serena; Karakhanyan, Aram L. Stratification of free boundary points for a two-phase variational problem, Adv. Math., Volume 328 (2018), pp. 40-81 (MR3771123) | DOI | MR | Zbl

[11] Feldman, Mikhail Regularity of Lipschitz free boundaries in two-phase problems for fully nonlinear elliptic equations, Indiana Univ. Math. J., Volume 50 (2001) no. 3, pp. 1171-1200 (MR1871352) | DOI | MR | Zbl

[12] Gilbarg, David; Trudinger, Neil S. Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001 (Reprint of the 1998 edition, MR1814364) | DOI | MR | Zbl

[13] Jerison, David; Savin, Ovidiu Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal., Volume 25 (2015) no. 4, pp. 1240-1257 (MR3385632) | DOI | MR | Zbl

[14] Kellogg, Oliver Dimon Foundations of Potential Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967 (MR0222317) | MR | Zbl

[15] O'Neill, Barrett Elementary Differential Geometry, Academic Press, New York-London, 1966 (MR0203595) | MR | Zbl

[16] Simon, Leon Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983 (MR756417) | MR | Zbl

[17] Weiss, Georg S. Partial regularity for weak solutions of an elliptic free boundary problem, Commun. Partial Differ. Equ., Volume 23 (1998) no. 3–4, pp. 439-455 (MR1620644) | DOI | MR | Zbl

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