Phase diagram of softly repulsive systems: The Gaussian and inverse-power-law potentials

We redraw, using state-of-the-art methods for free-energy calculations, the phase diagrams of two reference models for the liquid state: the Gaussian and inverse-power-law repulsive potentials. Notwithstanding the different behavior of the two potentials for vanishing interparticle distances, their thermodynamic properties are similar in a range of densities and temperatures, being ruled by the competition between the body-centered-cubic (BCC) and face-centered-cubic (FCC) crystalline structures and the fluid phase. We confirm the existence of a reentrant BCC phase in the phase diagram of the Gaussian-core model, just above the triple point. We also trace the BCC-FCC coexistence line of the inverse-power-law model as a function of the power exponent $n$ and relate the common features in the phase diagrams of such systems to the softness degree of the interaction.Comment: 22 pages, 8 figure

Geometric approach to nonvariational singular elliptic equations

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter $(\gamma -1)$, for $0 < \gamma < 1$, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases. We establish existence as well sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and obtain fine geometric-measure properties of the free boundary $\mathfrak{F} = \partial \{u > 0 \}$. In particular we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region $\{u > 0 \}$ and $\mathcal{H}^{n-1}$ a.e. weak differentiability property of $\mathfrak{F}$.Comment: Paper from D. Araujo's Ph.D. thesis, distinguished at the 2013 Carlos Gutierrez prize for best thesis, Archive for Rational Mechanics and Analysis 201

Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.Comment: 24 pages, to be submitte

Hamiltonians for curves

We examine the equilibrium conditions of a curve in space when a local energy penalty is associated with its extrinsic geometrical state characterized by its curvature and torsion. To do this we tailor the theory of deformations to the Frenet-Serret frame of the curve. The Euler-Lagrange equations describing equilibrium are obtained; Noether's theorem is exploited to identify the constants of integration of these equations as the Casimirs of the euclidean group in three dimensions. While this system appears not to be integrable in general, it {\it is} in various limits of interest. Let the energy density be given as some function of the curvature and torsion, $f(\kappa,\tau)$. If $f$ is a linear function of either of its arguments but otherwise arbitrary, we claim that the first integral associated with rotational invariance permits the torsion $\tau$ to be expressed as the solution of an algebraic equation in terms of the bending curvature, $\kappa$. The first integral associated with translational invariance can then be cast as a quadrature for $\kappa$ or for $\tau$.Comment: 17 page

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

Boundary regularity for the Poisson equation in reifenberg-flat domains

This paper is devoted to the investigation of the boundary regularity for the Poisson equation {{cc} -\Delta u = f & \text{in} \Omega u= 0 & \text{on} \partial \Omega where $f$ belongs to some $L^p(\Omega)$ and $\Omega$ is a Reifenberg-flat domain of $\mathbb R^n.$ More precisely, we prove that given an exponent $\alpha\in (0,1)$, there exists an $\varepsilon>0$ such that the solution $u$ to the previous system is locally H\"older continuous provided that $\Omega$ is $(\varepsilon,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem

The mixed problem for the Laplacian in Lipschitz domains

We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that the boundary between the sets where we specify Dirichlet and Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p and the Dirichlet data is in the Sobolev space of functions having one derivative in L^p for some p near 1. Under these conditions, there is a unique solution to the mixed problem with the non-tangential maximal function of the gradient of the solution in L^p of the boundary. We also obtain results with data from Hardy spaces when p=1.Comment: Version 5 includes a correction to one step of the main proof. Since the paper appeared long ago, this submission includes the complete paper, followed by a short section that gives the correction to one step in the proo

Re-entrant spin susceptibility of a superconducting grain

We study the spin susceptibility chi of a small, isolated superconducting grain. Due to the interplay between parity effects and pairing correlations, the dependence of chi on temperature T is qualitatively different from the standard BCS result valid in the bulk limit. If the number of electrons on the grain is odd, chi shows a re-entrant behavior as a function of temperature. This behavior persists even in the case of ultrasmall grains where the mean level spacing is much larger than the BCS gap. If the number of electrons is even, chi(T) is exponentially small at low temperatures.Comment: 9 pages, 3 figures. To be published in PR