Regularity for degenerate two-phase free boundary problems

  • Raimundo Leitão

    CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal
  • Olivaine S. de Queiroz

    Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Rua Sérgio Buarque de Holanda, 651, Campinas, SP, CEP 13083-859, Brazil
  • Eduardo V. Teixeira

    Universidade Federal do Ceará, Campus of Pici, Bloco 914, Fortaleza, Ceará, 60455-760, Brazil


We provide a rather complete description of the sharp regularity theory to a family of heterogeneous, two-phase free boundary problems, , 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 becomes singular along the free interface . The degree of singularity is, in turn, dimmed by the parameter . For we show that local minima are locally of class for a sharp α that depends on dimension, p and γ. For 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.

Cite this article

Raimundo Leitão, Olivaine S. de Queiroz, Eduardo V. Teixeira, Regularity for degenerate two-phase free boundary problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 4, pp. 741–762

DOI 10.1016/J.ANIHPC.2014.03.004