Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions

The main result of this paper concerns the behavior of a free boundary arising from a minimization problem, close to the fixed boundary in two dimensions

A new discrepancy principle

The aim of this note is to prove a new discrepancy principle. The advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem approximately, rather than exactly, and in the proof of a stability result

Maximally multipartite entangled states

We introduce the notion of maximally multipartite entangled states of n qubits as a generalization of the bipartite case. These pure states have a bipartite entanglement that does not depend on the bipartition and is maximal for all possible bipartitions. They are solutions of a minimization problem. Examples for small n are investigated, both analytically and numerically.Comment: 5 pages, 1 figure, final verso

A minimization problem for the lapse and the initial-boundary value problem for Einstein's field equations

We discuss the initial-boundary value problem of General Relativity. Previous considerations for a toy model problem in electrodynamics motivate the introduction of a variational principle for the lapse with several attractive properties. In particular, it is argued that the resulting elliptic gauge condition for the lapse together with a suitable condition for the shift and constraint-preserving boundary conditions controlling the Weyl scalar Psi_0 are expected to yield a well posed initial-boundary value problem for metric formulations of Einstein's field equations which are commonly used in numerical relativity. To present a simple and explicit example we consider the 3+1 decomposition introduced by York of the field equations on a cubic domain with two periodic directions and prove in the weak field limit that our gauge condition for the lapse and our boundary conditions lead to a well posed problem. The method discussed here is quite general and should also yield well posed problems for different ways of writing the evolution equations, including first order symmetric hyperbolic or mixed first-order second-order formulations. Well posed initial-boundary value formulations for the linearization about arbitrary stationary configurations will be presented elsewhere.Comment: 34 pages, no figure

Minimization of a fractional perimeter-Dirichlet integral functional

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely \int_\Om |\nabla u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big), with $\sigma\in(0,1)$. We obtain regularity results for the minimizers and for their free boundaries \p \{u>0\} using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems