Yoshihiro Tonegawa

Convergence of Phase Interfaces in the van der Waals-Cahn-Hilliard theory

Abstract. We study the general asymptotic behavior of critical points, including those of non-minimal energy type, of the functional for the van der Waals-Cahn-Hilliard theory of phase transitions. We prove that the interface is close to a hypersurface with mean curvature zero when no Lagrange multiplier is present, and with locally constant mean curvature in general. The energy density of the limiting measure has integer multiplicity almost everywhere modulo division by a surface energy constant.

On the convergence of stable phase transitions

We consider the local behavior of critical points of the functional as e 0. Here, W is a double-well potential and U is a regular domain in ℝn, n ≥ 2. Assuming that {ue}e>0 is stable for n = 2 and locally energy-minimizing for n = 3, we show that the level sets of solutions converge in an average sense to a stationary (n − 1)-rectifiable varifold. Our study is based on estimates derived from the second variation formula and is entirely local.

A general regularity theory for weak mean curvature flow

We give a new proof of Brakke’s partial regularity theorem up to

Existence and regularity of mean curvature flow with transport term in higher dimensions

C^{1,\varsigma }

Stable phase interfaces in the van der Waals–Cahn–Hilliard theory

for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard’s regularity theorem in the sense that the special time-independent case reduces to Allard’s theorem.

Partial Regularity for a Selective Smoothing Functional for Image Restoration in BV Space

We prove the global-in-time existence of weak solution for a hypersurface evolution problem where the velocity is the sum of the mean curvature and arbitrarily given non-smooth vector field in a suitable Sobolev space. The approximate solution is obtained by the Allen–Cahn equation with transport term. By establishing the density ratio upper bound on the phase boundary measure it is shown that the limiting surface moves with the desired velocity in the sense of Brakke.

Two-phase flow problem coupled with mean curvature flow

Given an initial