We study a 2-phase free boundary problem for harmonic measure first considered by Kenig and Toro [KT06] and prove a sharp Hölder regularity result. The central difficulty is that there is no a priori non-degeneracy in the free boundary condition. Thus we must establish non-degeneracy by means of monotonicity formulae.

This paper studies adaptive regularized methods for nonlinear least-squares problems where the model of the objective function used at each iteration is either the Euclidean residual regularized by a quadratic term or the Gauss-Newton model regularized by a cubic term. For suitable choices of the regularization parameter the role of the regularization term is to provide global convergence. In t...

Many different methods can be used to treat open boundary conditions in lattice Boltzmann method. Zou-He method, finite difference velocity gradient method, and regularized method are reviewed and compared for velocity Dirichlet condition for Poiseuille flow with different Reynolds numbers. Using same convergence criterion, all the numerical procedures are carried on till steady-states are reac...

Abstract An approach to find the field equation solution of the Randall-Sundrum model with the S/Z2 extra axis is presented. We closely examine the infrared singularity. The vacuum is set by the 5 dimensional Higgs field. Both the domain-wall and the anti-domain-wall naturally appear, at the ends of the extra compact axis, by taking a new infrared regularization. The stability is guaranteed fro...

We consider the problem of determining a pair of functions (u, f) satisfying the two-dimensional backward heat equation ut −∆u = φ(t)f(x, y), t ∈ (0, T ), (x, y) ∈ (0, 1)× (0, 1), u(x, y, T ) = g(x, y) with a homogeneous Cauchy boundary condition, where φ and g are given approximately. The problem is severely ill-posed. Using an interpolation method and the truncated Fourier series, we construc...

We consider the problem of determining a pair of functions (u, f) satisfying the heat equation ut−∆u = φ(t)f(x, y), where (x, y) ∈ Ω = (0, 1)× (0, 1) and the function φ is given. The problem is ill-posed. Under a slight condition on φ, we show that the solution is determined uniquely from some boundary data and the initial temperature. Using the interpolation method and the truncated Fourier se...

The regularized total Casimir energy in spacetimes with boundaries is not in general equal to the integral of the regularized energy density. This paradoxical phenomenon is most transparently analyzed in the simple example of a massless scalar field in 1+1 dimensions confined to a line element of length L and obeying Dirichlet boundary conditions. PACS: 03.70.+k; 11.10.Kk; 32.80.-t