Hopf-Lax type formula for non monotonic autonomous Hamilton Jacobi equations

We give an explicit formula for a solution of Hamilton-Jacobi equation u_t+H(u,Du)=0 in with initial condition u(x,0)=u₀(x), where p H(u,p) is convex, positively homogeneous of degree one and the Hamiltonian H need not satisfy the monotonicity condition in u.     

Multiple equilibria,periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Consider the Dirichlet problem for the parabolic equation in , where $\Omega$ is a bounded domain in and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.     

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

On a theorem by Sohr for the Navier-Stokes equations

In this paper we study some criteria for the full (space-time) regularity of weak solutions to the Navier-Stokes equations. In particular, we generalize some classical and very recent criteria involving the velocity, or its derivatives. More specifically, we show with elementary tools that if a weak solution, or its vorticity, is small in appropriate Marcinkiewicz spaces, then it is regular.     

Projective Limits via Inner Premeasures and the True Wiener Measure

The paper continues the authors work in measure and integration, which is an attempt at unified systematization. It establishes projective limit theorems of the Prokhorov and Kolmogorov types in terms of inner premeasures. Then it specializes to obtain the (one-dimensional) Wiener measure on the space of real-valued functions on the positive halfline as a probability measure defined on an immense domain: In particular the subspace of continuous functions will be measurable of full measure - and not merely of full outer measure, as the usual projective limit theorems permit to conclude. Dedicated to Professor Gustave Choquet     

Diffusive Logistic Equations with Degenerate Boundary Conditions

The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights that model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment.Dedicated to Professor Mitsuru Ikawa on the occasion of his 60th birthday     

High-efficiency clean EUV plasma source at 10–30 nm, driven by a long-pulse-width excimer laser

A long-pulse-width high-output energy (120 ns FWHM, 7 J) XeCl laser has been focused on thin tape targets (Cu and Ta) to generate more than 100-ns-long (FWHM) EUV pulses in the 10–30 nm spectral region, suitable for projection microlithography. The conversion efficiency was more than 20% over a 2π solid angle. We observed debris emission using a gated CCD camera, and measured the debris speed for different irradiation conditions. We found irradiation conditions such that the measured velocities were low enough that simple mechanical devices combined with krypton at low-pressure could efficiently stop both ionic debris and cluster debris. Our results show that a suitable combination of driving-laser characteristics, target material and thickness, environment gas and mechanical choppers can make clean and increase the power of EUV solid-target laser-plasma sources. Received: 21 October 2002 / Published online: 26 February 2003 RID="*" ID="*"Corresponding author. Fax: +39-06/9400-5334, E-mail: bollanti@frascati.enea.it     

Analysis of the correction factor in the spreading-resistance technique for monitoring epitaxial silicon

The range of applicability of the mixed-boundary-value method for calculating spreading resistance for a homogeneous slab with an effective contact-radius source and backed by a substrate of arbitrary but finite resistivity is investigated. Solutions are presented in terms of the correction factor and the source current density distributions for a slab of varying thickness and with various high-resistivity substrates. The correction factors for different schemes of calculation are correctly obtained by means of introducing an additional current source of opposite sign, but with the same absolute value, located at the drain point contact. This combination of source and drain currents gives the true value for the potential distribution along the surface of the semiconductor. This, in turn, leads to new terms in the equations obtained for the correction factors, which have been omitted in previously published works. A comparison between several schemes of calculation is presented. Within the framework of “uniform” and “variable” current distributions underneath the contact probe, there are two limits for the correction factor. A model based on a combination of these approaches is discussed, and a comparison between the proposed method and the Schumann–Gardner formulation is made. Received: 30 January 2002 / Accepted: 30 January 2002 / Published online: 28 October 2002 RID="*" ID="*"Corresponding author. Fax: +7-095/531-83-54, E-mail: telkom@df.ru     

Malliavin calculus in abstract Wiener space using infinitesimals

Using infinitesimals, we develop Malliavin calculus on spaces which result from the classical Wiener space by replacing with any abstract Wiener space .We start from a Brownian motion b on a Loeb probability space Ω with values in the Banach space is the standard part of a ∗finite-dimensional Brownian motion B. Then we define iterated Itô integrals as standard parts of internal iterated Itô integrals. The integrator of the internal integrals is B and the values of the integrands are multilinear forms on , where is a ∗finite-dimensional linear space over between the Hilbert space and its ∗-extension .In the first part we prove a chaos decomposition theorem for L²-functionals on Ω that are measurable with respect to the σ-algebra generated by b. This result yields a chaos decomposition of L²-functionals with respect to the Wiener measure on the standard space of -valued continuous functions on [0,1]. In the second part we define the Malliavin derivative and the Skorohod integral as standard parts of internal operators defined on ∗finite-dimensional spaces. In an application we use the transformation rule for finite-dimensional Euclidean spaces to study time anticipating and non-anticipating shifts of Brownian motion by Bochner integrals (Girsanov transformations).