Positive solutions of Schr\"odinger equations and fine regularity of boundary points

Given a Lipschitz domain $\Omega$ in ${\mathbb R} ^N$ and a nonnegative potential $V$ in $\Omega$ such that $V(x)\, d(x,\partial \Omega)^2$ is bounded in $\Omega$ we study the fine regularity of boundary points with respect to the Schr\"odinger operator $L_V:= \Delta -V$ in $\Omega$. Using potential theoretic methods, several conditions equivalent to the fine regularity of $z \in \partial \Omega$ are established. The main result is a simple (explicit if $\Omega$ is smooth) necessary and sufficient condition involving the size of $V$ for $z$ to be finely regular. An essential intermediate result consists in a majorization of $\int_A | {\frac {u} {d(.,\partial \Omega)}} | ^2\, dx$ for $u$ positive harmonic in $\Omega$ and $A \subset \Omega$. Conditions for almost everywhere regularity in a subset $A$ of $\partial \Omega$ are also given as well as an extension of the main results to a notion of fine ${\mathcal L}_1 | {\mathcal L}_0$-regularity, if ${\mathcal L}_j={\mathcal L}-V_j$, $V_0,\, V_1$ being two potentials, with $V_0 \leq V_1$ and ${\mathcal L}$ a second order elliptic operator.Comment: version 1. 23 pages version 3. 28 pages. Mainly a typo in Theorem 1.1 is correcte

Sur la théorie du potentiel dans les domaines de John

Using rather elementary and direct methods, we first recover and add on some results of Aikawa-Hirata-Lundh about the Martin boundary of a John domain. In particular we answer a question raised by these authors. Some applications are given and the case of more general second order elliptic operators is also investigated. In the last parts of the paper two potential theoretic results are shown in the framework of uniform domains or the framework of hyperbolic manifolds

A note on the Rellich formula in Lipschitz domains

Let $L$ be a symmetric second order uniformly elliptic operator in divergence form acting in a bounded Lipschitz domain $\Omega$ of $\Bbb R^N$ and having Lipschitz coefficients in $\Omega$. It is shown that the Rellich formula with respect to $\Omega$ and $L$ extends to all functions in the domain \Cal D=\{u \in H_0^1(\Omega);\, L(u) \in L^2(\Omega)\} of $L$. This answers a question of A. Chaïra and G. Lebeau

Coinductive subtyping for abstract compilation of object-oriented languages into Horn formulas

In recent work we have shown how it is possible to define very precise type systems for object-oriented languages by abstractly compiling a program into a Horn formula f. Then type inference amounts to resolving a certain goal w.r.t. the coinductive (that is, the greatest) Herbrand model of f. Type systems defined in this way are idealized, since in the most interesting instantiations both the terms of the coinductive Herbrand universe and goal derivations cannot be finitely represented. However, sound and quite expressive approximations can be implemented by considering only regular terms and derivations. In doing so, it is essential to introduce a proper subtyping relation formalizing the notion of approximation between types. In this paper we study a subtyping relation on coinductive terms built on union and object type constructors. We define an interpretation of types as set of values induced by a quite intuitive relation of membership of values to types, and prove that the definition of subtyping is sound w.r.t. subset inclusion between type interpretations. The proof of soundness has allowed us to simplify the notion of contractive derivation and to discover that the previously given definition of subtyping did not cover all possible representations of the empty type

Some Results on the Boundary Control of Systems of Conservation Laws

This note is concerned with the study of the initial boundary value problem for systems of conservation laws from the point of view of control theory, where the initial data is fixed and the boundary data are regarded as control functions. We first consider the problem of controllability at a fixed time for genuinely nonlinear Temple class systems, and present a description of the set of attainable configurations of the corresponding solutions in terms of suitable Oleinik-type estimates. We next present a result concerning the asymptotic stabilization near a constant state for general $n\times n$ systems. Finally we show with an example that in general one cannot achieve exact controllability to a constant state in finite time.Comment: 10 pages, 4 figures, conferenc

Laser fabrication of anti-icing surfaces: A review

In numerous fields such as aerospace, the environment, and energy supply, ice generation and accretion represent a severe issue. For this reason, numerous methods have been developed for ice formation to be delayed and/or to inhibit ice adhesion to the substrates. Among them, laser micro/nanostructuring of surfaces aiming to obtain superhydrophobic behavior has been taken as a starting point for engineering substrates with anti-icing properties. In this review article, the key concept of surface wettability and its relationship with anti-icing is discussed. Furthermore, a comprehensive overview of the laser strategies to obtain superhydrophobic surfaces with anti-icing behavior is provided, from direct laser writing (DLW) to laser-induced periodic surface structuring (LIPSS), and direct laser interference patterning (DLIP). Micro-/nano-texturing of several materials is reviewed, from aluminum alloys to polymeric substrates

Sorting of particles using inertial focusing and laminar vortex technology: A review

The capability of isolating and sorting specific types of cells is crucial in life science, particularly for the early diagnosis of lethal diseases and monitoring of medical treatments. Among all the micro-fluidics techniques for cell sorting, inertial focusing combined with the laminar vortex technology is a powerful method to isolate cells from flowing samples in an efficient manner. This label-free method does not require any external force to be applied, and allows high throughput and continuous sample separation, thus offering a high filtration efficiency over a wide range of particle sizes. Although rather recent, this technology and its applications are rapidly growing, thanks to the development of new chip designs, the employment of new materials and microfabrication technologies. In this review, a comprehensive overview is provided on the most relevant works which employ inertial focusing and laminar vortex technology to sort particles. After briefly summarizing the other cells sorting techniques, highlighting their limitations, the physical mechanisms involved in particle trapping and sorting are described. Then, the materials and microfabrication methods used to implement this technology on miniaturized devices are illustrated. The most relevant evolution steps in the chips design are discussed, and their performances critically analyzed to suggest future developments of this technology

A locally quadratic Glimm functional and sharp convergence rate of the Glimm scheme for nonlinear hyperbolic systems

Consider the Cauchy problem for a strictly hyperbolic, $N\times N$ quasilinear system in one space dimension u_t+A(u) u_x=0,\qquad u(0,x)=\bar u(x), \eqno (1) where $u \mapsto A(u)$ is a smooth matrix-valued map, and the initial data $\overline u$ is assumed to have small total variation. We investigate the rate of convergence of approximate solutions of (1) constructed by the Glimm scheme, under the assumption that, letting $\lambda_k(u)$, $r_k(u)$ denote the $k$-th eigenvalue and a corresponding eigenvector of $A(u)$, respectively, for each $k$-th characteristic family the linearly degenerate manifold $$\mathcal{M}_k \doteq \big\{u\in\Omega : \nabla\lambda_k(u)\cdot r_k(u)=0\big\}$$ is either the whole space, or it is empty, or it consists of a finite number of smooth, $N-1$-dimensional, connected, manifolds that are transversal to the characteristic vector field $r_k$. We introduce a Glimm type functional which is the sum of the cubic interaction potential defined in \cite{sie}, and of a quadratic term that takes into account interactions of waves of the same family with strength smaller than some fixed threshold parameter. Relying on an adapted wave tracing method, and on the decrease amount of such a functional, we obtain the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems satisfying the classical Lax assumptions of genuine nonlinearity or linear degeneracy of the characteristic families.Comment: To appear on Archive for Rational Mechanics and Analysi