Advances in Differential Equations Volume 8, Number 10, October 2003, Pages 1259–1280 EXACT ESTIMATES FOR THE CLASSICAL SOLUTIONS TO THE FREE-BOUNDARY PROBLEM IN THE HELE-SHAW CELL Stanislav Antontsev Departamento de Matem´atica, Universidade da Beira Interior Rua Marquˆ es ´ Avila e Bolama, 6201-001 Covilh˜ a, Portugal C´ esar Gonc ¸alves Departamento de Matem´atica, ESTG - Instituto Polit´ ecnico da Guarda Av. Dr. Francisco S´ a Carneiro, n o 50, 6301-559 Guarda, Portugal Anvarbek Meirmanov Departamento de Matem´atica, Universidade da Beira Interior Rua Marquˆ es d’ ´ Avila e Bolama, 6201-001 Covilh˜ a, Portugal (Submitted by: Michel Chipot) Abstract. We consider the classical solutions to the multi-dimensional free-boundary problem in the Hele-Shaw cell with general boundary con- ditions on a given boundary. Exact regularity estimates in the H¨older space are established using the explicit form of the solution to the model linear problem in the half-space and a method for evaluating the con- volution integrals. This method was suggested by V. Solonnikov and is based on the use of Golovkin’s theorem. We prove that if the free boundary Γ(t) is initially C l -regular (l> 2 is noninteger), then it pre- serves the same regularity (Γ(t) ∈ C l ) till some instant T∗ depending on the C 2 -norm of the free boundary Γ(t) and on the topology of Γ(t). At this instant T∗, either the C 2 -norm of the free boundary Γ(t) tends to infinity or Γ(t) changes its topology. 1. Introduction The well-posed Hele-Shaw problem is studied in this paper. In this prob- lem the bounded region Ω(t) ⊂ R n occupied by some liquid is unknown. The liquid pressure p(t, x) satisfies the equation −△ p = f (t, x) ≡ div F, x ∈ Ω(t). (1.1) Accepted for publication: February 2003. AMS Subject Classifications: 35R35, 35J25, 35J60, 76D27. 1259