Epsilon-regularity for the solutions of a free boundary system

• Francesco Paolo Maiale ^[2] ; Giorgio Tortone ^[1] ; Bozhidar Velichkov ^[1]
  1. [1] University of Pisa

     University of Pisa

     Pisa, Italia

     
  2. [2] Scuola Normale Superiore, Pisa
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 5, 2023, págs. 1947-1972
• Idioma: inglés
• DOI: 10.4171/RMI/1430
• Enlaces
  • Texto completo
• Resumen

  • This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions u and v, and a domain ; with u and v being both positive in , vanishing simultaneously on @, and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on @. Precisely, we consider solutions u; v 2 C.B1/ of u D f and v D g in D ¹u > 0º D ¹v > 0º; @u @n @v @n D Q on @ \ B1: Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions p uv and 1 2 .u C v/. Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies C 1;˛ regularity.