Regularity of flat free boundaries in two-phase problems for the p-Laplace operator
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 1, pp. 83-108.

In this paper we continue the study in Lewis and Nyström (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.

DOI: 10.1016/j.anihpc.2011.09.002
Classification: 35J25, 35J70
Keywords: p-Harmonic function, p-Subharmonic, Free boundary, Two-phase, Boundary Harnack inequality, Hopf boundary principle, Lipschitz domain, ϵ-Monotone, Monotone, Regularity
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     title = {Regularity of flat free boundaries in two-phase problems for the {\protect\emph{p}-Laplace} operator},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {83--108},
     publisher = {Elsevier},
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Lewis, John L.; Nyström, Kaj. Regularity of flat free boundaries in two-phase problems for the p-Laplace operator. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 1, pp. 83-108. doi : 10.1016/j.anihpc.2011.09.002. http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.002/

[1] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I, Lipschitz free boundaries are C 1,α , Rev. Mat. Iberoam. 3 (1987), 139-162 | EuDML | Zbl

[2] L. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 no. 1 (1989), 55-78 | Zbl

[3] L. 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 (1988), 583-602 | EuDML | Numdam | Zbl

[4] X. Cabre, L. Caffarelli, Fully Non-linear Elliptic Equations, American Mathematical Society Colloquium Publications vol. 43, Amer. Math. Soc., Providence, RI (1995) | Zbl

[5] L. Caffarelli, E. Fabes, S. Mortola, S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 no. 4 (1981), 621-640 | Zbl

[6] M.C. Cerutti, F. Ferrari, S. Salsa, Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are C 1,γ , Arch. Ration. Mech. Anal. 171 (2004), 329-448 | Zbl

[7] M. Feldman, Regularity for two non-isotropic two phase problems with Lipschitz free boundaries, Differential Integral Equations 10 (1997), 227-251 | Zbl

[8] M. Feldman, Regularity of Lipschitz free boundaries in two-phase problems for fully non-linear elliptic equations, Indiana Univ. Math. J. 50 no. 3 (2001), 1171-1200 | Zbl

[9] F. Ferrari, Two-phase problems for a class of fully non-linear elliptic operators, Lipschitz free boundaries are C 1,γ , Amer. J. Math. 128 no. 3 (2006), 541-571 | Zbl

[10] F. Ferrari, S. Salsa, Regularity of the free boundary in two-phase problems for linear elliptic operators, Adv. Math. 214 (2007), 288-322 | Zbl

[11] F. Ferrari, S. Salsa, Subsolutions of elliptic operators in divergence form and applications to two-phase free boundary problems, Bound. Value Probl. (2007), Art. ID 57049, 21 pp. | EuDML

[12] I.N. Krolʼ, On the behavior of the solutions of a quasilinear equation near null salient points of the boundary, Proc. Steklov Inst. Math. 125 (1973), 130-136

[13] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 no. 11 (1988), 1203-1219 | Zbl

[14] J. Lewis, K. Nyström, Boundary behaviour for p-harmonic functions in Lipschitz and starlike Lipschitz ring domains, Ann. Sc. École Norm. Sup. (4) 40 no. 4 (2007), 765-813 | EuDML | Numdam | Zbl

[15] J. Lewis, K. Nyström, Boundary behaviour and the Martin boundary problem for p-harmonic functions in Lipschitz domains, Ann. of Math. 172 (2010), 1907-1948 | Zbl

[16] J. Lewis, K. Nyström, Regularity and free boundary regularity for the p-Laplacian in Lipschitz and C 1 -domains, Ann. Acad. Sci. Fenn. 33 (2008), 523-548 | EuDML | Zbl

[17] J. Lewis, K. Nyström, New results on p-harmonic functions, Pure Appl. Math. Q. 7 no. 7 (2011), 345-363 | Zbl

[18] J. Lewis, K. Nyström, Boundary behaviour of p-harmonic functions in domains beyond Lipschitz domains, Adv. Calculus Variations 1 (2008), 133-170 | Zbl

[19] J. Lewis, K. Nyström, Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator, Adv. Math. 225 (2010), 2565-2597 | Zbl

[20] J. Lewis, N. Lundström, K. Nyström, Boundary Harnack inequalities for operators of p Laplace type in Reifenberg flat domains, Dorina Mitrea, Marius Mitrea (ed.)Proceedings of Symposia in Pure Mathematics vol. 79 (2008), 229-266 | Zbl

[21] W. Littman, G. Stampacchia, H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa (3) 17 (1963), 43-77 | EuDML | Numdam | Zbl

[22] P. Wang, Regularity of free boundaries of two-phase problems for fully non-linear elliptic equations of second order. Part 1: Lipschitz free boundaries are C 1,α , Comm. Pure Appl. Math. 53 no. 7 (2000), 799-810 | Zbl

[23] P. Wang, Regularity of free boundaries of two-phase problems for fully non-linear elliptic equations of second order. II: Flat free boundaries are Lipschitz, Comm. Partial Differential Equations 27 no. 7–8 (2002), 1497-1514 | Zbl

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