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We provide a potential theoretic characterization of vanishing chord-arc domains under mild assumptions. In particular we show that, if a domain has Ahlfors regular boundary, the oscillation of the logarithm of the interior and exterior Poisson kernels yields a great deal of geometric information about the domain. We use techniques from classical calculus of variations, potential theory, and quantitative geometric measure theory to accomplish this. One feature of this work, compared to [KT06] and [BH16], is that a priori we only require that the domains in question are connected.

In Kenig and Toro’s two-phase free boundary problem, one studies how the regularity of the Radon–Nikodym derivative h = d ω - / d ω + of harmonic measures on complementary NTA domains controls the geometry of their common boundary. It is now known that log h ∈ C 0 , α ( ∂ Ω ) implies that pointwise the boundary has a unique blow-up, which is the zero set of a homogeneous harmonic polynomial. In this note, we give examples of domains with log h ∈ C ( ∂ Ω ) whose boundaries have points with non-unique blow-ups. Philosophically the examples arise from oscillating or rotating a blow-up limit by an infinite amount, but very slowly.

We discuss the notion of domain variation solution for some degenerate elliptic two-phase free boundary problems as well as the viscosity definition of the problem when the operator is degenerate.

In this paper we construct a viscosity solution of a two-phase free boundary problem for a class of fully nonlinear equation with distributed sources, via an adaptation of the Perron method. Our results extend those in [Ca arelli, 1988], [Wang, 2003] for the homogeneous case, and of [De Silva, Ferrari, Salsa, 2015] for divergence form operators with right hand side.

Consider the two-phase free boundary problem subject to surface tension and gravitational forces for a class of non-Newtonian fluids with stress tensors T n of the form T n = - q I + μ n ( | D ( v ) | 2 ) D ( v ) for n = 1 , 2 , respectively, where the viscosity functions μ n satisfy μ n ∈ C 3 ( [ 0 , ∞ ) ) and μ n ( 0 ) > 0 for n = 1 , 2 . It is shown that for given T > 0 this problem admits a unique strong solution on (0, T ) provided the initial data are sufficiently small in their natural norms.

In this paper we prove some results concerning inverse/free boundary type problems, below the continuous threshold, for the heat equation in the setting of parabolic regular graph domains.

A version of a famous and important result due to Alt-Caffarelli, relevant to the analysis of elliptic free boundary problems, states that there exists δn > 0 such that if Ω ⊂ ℝn is an unbounded δ-Reifenberg flat domain, δ ϵ (0, δn), and if ∂Ω satisfies an Ahlfors condition, then the following is true. Assume that there exist functions u (Green function with pole at infinity) and k (the Poisson kernel) such that Δu = 0 in Ω, u > 0 in Ω, u = 0 on ∂Ω and dω = kdHn–1 where ω is the harmonic measure at infinity. If furthermore supXϵΩ |∇u(X)| ≤ 1 and k(Q) ≥ 1 for Hn–1 a.e. point Q ϵ ∂Ω, then in suitable coordinates, Ω = {(x,xn) : xn > 0} and u(x,xn) = xn. This result is crucial in recent work on the analysis of elliptic free boundary problems beyond the continuous threshold by Kenig and Toro. In this paper we consider the corresponding parabolic problems in the setting of time-varying domains Ω = {(x0,x,t) ∈ R × ℝn–1 × ℝ : x0 > Ψ(x,t)} where Ψ is a Lip(1,½) function. Defining Ω1 = Ω and Ω2 = ℝn+1 \\ Ω̄, we let ωi(X̂i,t̂i,·), for i ∈ {1,2} and (X̂i,t̂i) ∈ Ωi be the caloric measure defined with respect to Ωi. Assuming that ωi(X̂i,t̂i,·) is absolutely continuous with respect to an appropriate surface measure σ for at least one i ∈ {1,2}, we study the implication of the condition log ki(X̂i,t̂i,·) ∈ VMO(dσ) on the 'free boundary' ∂Ω. We show that this information on the Poisson kernel(s) can be explored in a delicate blow-up argument and that results on the regularity of ∂Ω can be deduced from classification theorems for global solutions to parabolic free boundary problems appearing in the limit. In fact, we prove a number of such classification theorems and, in particular, we prove weaker parabolic analogues of the result of Alt-Caffarelli.

The proposed work is devoted to two-dimensional problems with free boundaries that arise in mathematical modeling of problems in medicine. A new formulation of two-dimensional, two-phase problems with free boundaries is considered, and a simplified two-dimensional stationary Stefan problem is obtained. A system of nonlinear equations is obtained by using the general solution of the differential equation in both frozen and unfrozen areas of the biological tissue, after satisfying the boundary conditions and the conjugation conditions of the problem. The original problem is also reduced to the system of nonlinear integral equations.

In this paper we revisit the proof of the Alt-Caffarelli-Friedman monotonicity formula. Then, in the framework of the Heisenberg group, we discuss the existence of an analogous monotonicity formula introducing a necessary condition for its existence, recently proved in [18].

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Navier–Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. The parametric finite element approximation of the evolving interface is then coupled to a standard finite element approximation of the two-phase Navier–Stokes equations in the bulk. Here enriching the pressure approximation space with the help of an XFEM function ensures good volume conservation properties for the two phase regions. In addition, the mesh quality of the parametric approximation of the interface in general does not deteriorate over time, and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, our finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results in two and three space dimensions.