This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all...

This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all...

We show that phase space bounds on the eigenvalues of Schr¨odinger operators can be derived from universal bounds recently obtained by E. M. Harrell and the author via a monotonicity property with respect to coupling constants. In particular, we provide a new proof of sharp Lieb– Thirring inequalities.

The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately to obstructions to the existence of solutions. Moreover, we will deduce that the solution space of such equations is always finite dimensional.

In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the nonlinear heat equation $u_t-\Delta u+{\left|u\right|}^{\alpha }u=0$ on ${ℝ}^N$ (where $\alpha \>0$) and the asymptotic behavior as $\left|x\right|\to \infty$ of its initial value $u_0$. In particular, we show that if the sequence of dilations ${\lambda }_n^{2/\alpha }{u}_0({\lambda }_n·)$ converges weakly to $z(·)$ as ${\lambda }_n\to \infty$, then the rescaled solution $t^{1/\alpha }u(t,·\sqrt{t})$ converges uniformly on ${ℝ}^N$ to $𝒰\left(1\right)z$ along the subsequence $t_n={\lambda }_n^2$, where $𝒰\left(t\right)$ is an appropriate flow. Moreover, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable in this...

A generalized approach to several universality results is given by replacing holomorphic or harmonic functions by zero solutions of arbitrary linear partial differential operators. Instead of the approximation theorems of Runge and others, we use an approximation theorem of Hörmander.

Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let $W$ be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of...

Following our previous paper in the radial case, we consider type II blow-up solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of $W$ concentrating at the origin.

L'autore dà una condizione necessaria e sufficiente per la risolubilità formale degli operatori differenziali a coefficienti costanti, lineari, $p\left(D\right)$, in termini di prolungamento, come distribuzioni, delle $u\in D^{\prime }\left({\mathbb{R}}^n-\left\{0\right\}\right)$, tali che $p\left(-D\right)u=0$.

In this paper we investigate the existence of solutions for the initial value problems (IVP for short), for a class of implicit impulsive hyperbolic differential equations by using the lower and upper solutions method combined with Schauder’s fixed point theorem.