Two-phase problems with distributed sources: regularity of the free boundary

Abstract

Abstract. We investigate the regularity of the free boundary for a general class of two-phase free boundary problems with non-zero right hand side. We prove that Lipschitz or flat free boundaries are C 1,γ . In particular, viscosity solutions are indeed classical.

Recent progresses on elliptic two-phase free boundary problems

“…Based on a Harnack type theorem and linearization, this technique avoids the use of supconvolutions, that in presence of distributed sources produces several complicacies. The method can be very well adapted to nonhomogeneous two-phase problems to prove that flat (see below) or Lipschitz free boundaries of (1) are C 1,γ , when the governing equation is the same in both phases (see [16], [17], [18]). Throughout this section, L 1 = L 2 and this common operator will be denoted by L. Also, f 1 = f 2 = f.…”

A two-phase free boundary problem for harmonic measure

“…Theorem 7.14. [Theorem 3.3 and Theorem 3.4 in [DFS14]] Let W be a viscosity solution to (7.10) in B 1 such that W L ∞ ≤ 1. Then, in B 1/2 , W is actually a classical solution to (7.10).…”

“…Therefore, we present them in detail here. Section 7 adapts the iterative argument of De Silva, Ferrari and Salsa [DFS14] to get C 1,s regularity for the free boundary. In Section 8 we first describe how to establish optimal C 1,α regularity and then C 2,α regularity (in analogy to the aforementioned work of Jerison [J90]).…”

“…Here we will borrow heavily from the arguments of De Silva et al [DFS14], who prove C 1,γ regularity for a wide class of non-homogenous free boundary problems. We cannot immediately apply their results, as they assume a non-degeneracy in the free boundary condition that our problem does not have (see condition (H2) in Section 7 of [DFS14]). It should also be noted that our main result in this section is not immediately implied by the remark at the end of Caffarelli's paper, [C87].…”

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

“…2 C k;˛, then locally @ is the graph of a C kC1;f unction. See also [1,6,13,14], as well as the references therein, for example.…”