Regularity and free boundary regularity for the 𝑝-Laplace operator in Reifenberg flat and Ahlfors regular domains

Lewis, John¹ ; Nyström, Kaj²

¹ Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
² Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden

Journal of the American Mathematical Society, Tome 25 (2012) no. 3, pp. 827-862

Résumé

In this paper we solve several problems concerning regularity and free boundary regularity, below the continuous threshold, for positive solutions to the $p$-Laplace equation, $1 p \infty$, vanishing on a portion of the boundary of an Ahlfors regular NTA-domain. In Theorem 1 of our paper we show that if $\Omega \subset \mathbf {R}^{n}, n \geq 2,$ is an Ahlfors regular NTA-domain and $u$ is a positive $p$-harmonic function in $\Omega \cap B (w, 4r)$, with continuous boundary value 0 on $\partial \Omega \cap B (w, 4r)$, then $\nabla u (x) \to \nabla u (y)$ nontangentially as $x \rightarrow y \in \partial \Omega \cap B (w, 4r),$ almost everywhere with respect to surface area, $\sigma ,$ on $\partial \Omega \cap B (w, 4 r).$ Moreover, $\log | \nabla u |$ is of bounded mean oscillation on $\partial \Omega \cap B (w, r)$ with $\| \log | \nabla u |\|_{\mathrm {BMO} (\partial \Omega \cap B(w, r))} \leq c$. If, in addition, $\Omega$ is Reifenberg flat with vanishing constant and $n\in \mathrm {VMO}(\partial \Omega \cap B(w, 4r))$, where $n$ denotes the unit inner normal to $\partial \Omega$ in the measure-theoretic sense, then in Theorem 2 we prove that $\log | \nabla u | \in \mathrm {VMO}(\partial \Omega \cap B(w, r))$. In Theorem 3 we prove the following converse to Theorem 2. Suppose $u$ is as in Theorem 1, $\log | \nabla u | \in \mathrm {VMO}(\partial \Omega \cap B(w, r))$, and that $\partial \Omega \cap B (w, r)$ is $(\delta , r_0)$-Reifenberg flat. Then there exists $\bar \delta = \bar \delta (p, n)$ such that if $0 \delta \leq \bar \delta ,$ then $\partial \Omega \cap B(w, r/2)$ is Reifenberg flat with vanishing constant and $n\in \mathrm {VMO}(\partial \Omega \cap B(w, r/2))$. Finally, in Theorem 4 we establish a two-phase version of Theorem 3 without the smallness assumption on $\delta .$

Détail

DOI : 10.1090/S0894-0347-2011-00726-1

Bibliographie

[1] Alt, Hans Wilhelm, Caffarelli, Luis A., Friedman, Avner A free boundary problem for quasilinear elliptic equations Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1984 1 44

[2] Bennewitz, Bjã¶Rn, Lewis, John L. On the dimension of 𝑝-harmonic measure Ann. Acad. Sci. Fenn. Math. 2005 459 505

[3] Caffarelli, Luis A. A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are 𝐶^{1,𝛼} Rev. Mat. Iberoamericana 1987 139 162

[4] Caffarelli, Luis A. A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz Comm. Pure Appl. Math. 1989 55 78

[5] Caffarelli, L., Fabes, E., Mortola, S., Salsa, S. Boundary behavior of nonnegative solutions of elliptic operators in divergence form Indiana Univ. Math. J. 1981 621 640

[6] Coifman, R. R., Fefferman, C. Weighted norm inequalities for maximal functions and singular integrals Studia Math. 1974 241 250

[7] Capogna, Luca, Kenig, Carlos E., Lanzani, Loredana Harmonic measure 2005

[8] Dahlberg, Bjã¶Rn E. J. Estimates of harmonic measure Arch. Rational Mech. Anal. 1977 275 288

[9] David, G., Jerison, D. Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals Indiana Univ. Math. J. 1990 831 845

[10] Danielli, Donatella, Petrosyan, Arshak A minimum problem with free boundary for a degenerate quasilinear operator Calc. Var. Partial Differential Equations 2005 97 124

[11] Evans, Lawrence C., Gariepy, Ronald F. Measure theory and fine properties of functions 1992

[12] Jerison, David Regularity of the Poisson kernel and free boundary problems Colloq. Math. 1990 547 568

[13] Jerison, David S., Kenig, Carlos E. The logarithm of the Poisson kernel of a 𝐶¹ domain has vanishing mean oscillation Trans. Amer. Math. Soc. 1982 781 794

[14] Jerison, David S., Kenig, Carlos E. Boundary behavior of harmonic functions in nontangentially accessible domains Adv. in Math. 1982 80 147

[15] Kenig, Carlos E., Toro, Tatiana Harmonic measure on locally flat domains Duke Math. J. 1997 509 551

[16] Kenig, Carlos E., Toro, Tatiana Free boundary regularity for harmonic measures and Poisson kernels Ann. of Math. (2) 1999 369 454

[17] Kenig, Carlos E., Toro, Tatiana Poisson kernel characterization of Reifenberg flat chord arc domains Ann. Sci. École Norm. Sup. (4) 2003 323 401

[18] Kenig, Carlos, Toro, Tatiana Free boundary regularity below the continuous threshold: 2-phase problems J. Reine Angew. Math. 2006 1 44

[19] Kenig, C., Preiss, D., Toro, T. Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions J. Amer. Math. Soc. 2009 771 796

[20] Lavrent′Ev, M. Boundary problems in the theory of univalent functions Amer. Math. Soc. Transl. (2) 1963 1 35

[21] Lewis, John L. Uniformly fat sets Trans. Amer. Math. Soc. 1988 177 196

[22] Lewis, John L., Nystrã¶M, Kaj Boundary behaviour for 𝑝 harmonic functions in Lipschitz and starlike Lipschitz ring domains Ann. Sci. École Norm. Sup. (4) 2007 765 813

[23] Lewis, John, Nystrã¶M, Kaj Boundary behavior and the Martin boundary problem for 𝑝 harmonic functions in Lipschitz domains Ann. of Math. (2) 2010 1907 1948

[24] Lewis, John L., Nystrã¶M, Kaj Regularity and free boundary regularity for the 𝑝 Laplacian in Lipschitz and 𝐶¹ domains Ann. Acad. Sci. Fenn. Math. 2008 523 548

[25] Lewis, John L., Nystrã¶M, Kaj New results for 𝑝 harmonic functions Pure Appl. Math. Q. 2011 345 363

[26] Lewis, John L., Nystrã¶M, Kaj Boundary behaviour of 𝑝-harmonic functions in domains beyond Lipschitz domains Adv. Calc. Var. 2008 133 170

[27] Lewis, John L., Nystrã¶M, Kaj Regularity of Lipschitz free boundaries in two-phase problems for the 𝑝-Laplace operator Adv. Math. 2010 2565 2597

[28] Lundstrã¶M, Niklas L. P., Nystrã¶M, Kaj On a two-phase free boundary condition for 𝑝-harmonic measures Manuscripta Math. 2009 231 249

[29] Lewis, John L., Vogel, Andrew L. Uniqueness in a free boundary problem Comm. Partial Differential Equations 2006 1591 1614

[30] Milakis, Emmanouil, Toro, Tatiana Divergence form operators in Reifenberg flat domains Math. Z. 2010 15 41

[31] Pommerenke, Ch. On univalent functions, Bloch functions and VMOA Math. Ann. 1978 199 208

[32] Semmes, Stephen Analysis vs. geometry on a class of rectifiable hypersurfaces in 𝑅ⁿ Indiana Univ. Math. J. 1990 1005 1035

[33] Semmes, Stephen Chord-arc surfaces with small constant. I Adv. Math. 1991 198 223

[34] Semmes, Stephen Chord-arc surfaces with small constant. II. Good parameterizations Adv. Math. 1991 170 199

[35] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals 1993