Regularity for degenerate two-phase free boundary problems

Leitão, Raimundo; de Queiroz, Olivaine S.; Teixeira, Eduardo V.

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 741-762.

We provide a rather complete description of the sharp regularity theory to a family of heterogeneous, two-phase free boundary problems, ${𝒥}_{\gamma }\to \min$, ruled by nonlinear, p-degenerate elliptic operators. Included in such family are heterogeneous cavitation problems of Prandtl–Batchelor type, singular degenerate elliptic equations; and obstacle type systems. The Euler–Lagrange equation associated to ${𝒥}_{\gamma }$ becomes singular along the free interface $\{u=0\}$. The degree of singularity is, in turn, dimmed by the parameter $\gamma \in [0,1]$. For $0<\gamma <1$ we show that local minima are locally of class $C^{1,\alpha }$ for a sharp α that depends on dimension, p and γ. For $\gamma =0$ we obtain a quantitative, asymptotically optimal result, which assures that local minima are Log-Lipschitz continuous. The results proven in this article are new even in the classical context of linear, nondegenerate equations.

Classification : 35R35, 35J70, 35J75, 35J20
Mots clés : Free boundary problems, Degenerate elliptic operators, Regularity theory

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

[2] H.W. Alt, L. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries, Trans. Am. Math. Soc. 282 no. 2 (1984), 431 -461 | MR | Zbl

[3] H.M. Alt, D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. 368 (1986), 63 -107 | EuDML | MR | Zbl

[4] L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 no. 6 (1992), 581 -597 | MR | Zbl

[5] L.A. Caffarelli, D. Jerison, C.E. Kenig, Some new monotonicity theorems with applications to free boundary problems, Ann. Math. 155 (2002), 369 -404 | MR | Zbl

[6] E. Dibenedetto, $C^{1,\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 no. 8 (1983), 827 -850 | MR | Zbl

[7] L.C. Evans, A new proof of local $C^{1,\alpha }$ regularity for solutions of certain degenerate elliptic p.d.e, J. Differ. Equ. 45 no. 3 (1982), 356 -373 | Zbl

[8] C. Fefferman, E.M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137 -193 | MR | Zbl

[9] M. Giaquinta, E. Giusti, Differentiability of minima of non-differentiable functionals, Invent. Math. 72 (1983), 285 -298 | EuDML | MR | Zbl

[10] L. Karp, T. Kilpeläinen, A. Petrosyan, H. Shahgholian, On the porosity of free boundaries in degenerate variational inequalities, J. Differ. Equ. 164 no. 1 (2000), 110 -117 | MR | Zbl

[11] A. Kiselev, F. Nazarov, A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 no. 3 (2007), 445 -453 | MR | Zbl

[12] D. Moreira, E. Teixeira, On the behavior of weak convergence under nonlinearities and applications, Proc. Am. Math. Soc. 133 no. 6 (2005), 1647 -1656 | MR | Zbl

[13] T. Iwaniec, J.J. Manfredi, Regularity of p-harmonic functions on the plane, Rev. Mat. Iberoam. 5 no. 1–2 (1989), 1 -19 | EuDML | MR | Zbl

[14] J.L. Lewis, Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 no. 6 (1983), 849 -858 | MR | Zbl

[15] G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Commun. Partial Differ. Equ. 16 no. 2–3 (1991), 311 -361 | MR | Zbl

[16] J. Malý, W.P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Math. Surv. Monogr. vol. 51 , American Mathematical Society, Providence, RI (1997) | MR | Zbl

[17] D. Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J. 32 (1983), 1 -17 | MR | Zbl

[18] D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem, Commun. Partial Differ. Equ. 8 (1983), 1409 -1454 | MR | Zbl

[19] H. Shahgholian, $C^{1,1}$ regularity in semilinear elliptic problems, Commun. Pure Appl. Math. 56 no. 2 (2003), 278 -281 | MR | Zbl

[20] J. Serrin, A Harnack inequality for nonlinear equations, Bull. Am. Math. Soc. 69 no. 4 (1963), 481 -486 | MR | Zbl

[21] E.V. Teixeira, Universal moduli of continuity for solutions to fully nonlinear elliptic equations, http://arxiv.org/abs/1111.2728 | MR

[22] E.V. Teixeira, Sharp regularity for general Poisson equations with borderline sources, J. Math. Pures Appl. (9) 99 no. 2 (2013), 150 -164 | MR | Zbl

[23] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ. 51 no. 1 (1984), 126 -150 | MR | Zbl

[24] O.A. Ladyzhenskaya, N.N. Ural'Tseva, Linear and Quasilinear Elliptic Equations, Math. Sci. Eng. vol. 46 , Academic Press, New York (1968) | MR | Zbl

[25] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 no. 3–4 (1977), 219 -240 | MR | Zbl

[26] N.N. Ural'Ceva, Degenerate quasilinear elliptic systems, Zap. Nauč. Semin. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 184 -222 | MR

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