We study the growth rate of a cell population that follows an age-structured PDE with time-periodic coefficients. Our motivation comes from the comparison between experimental tumor growth curves in mice endowed with intact or disrupted circadian clocks, known to exert their influence on the cell division cycle. We compare the growth rate of the model controlled by a time-periodic control on its coefficients with the growth rate of stationary models of the same nature, but with averaged coefficients....

The paper concerns an approximation of an eigenvalue problem for two forms on a Hilbert space $X$. We investigate some approximation methods generated by sequences of forms $a_n$ and $b_n$ defined on a dense subspace of $X$. The proof of convergence of the methods is based on the theory of the external approximation of eigenvalue problems. The general results are applied to Aronszajn’s method.

The least Steklov eigenvalue d1 for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the L2-norm of harmonic functions. These applications suggest to address the problem of minimizing d1 in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular, the existence of a convex domain...

Consider the family uₜ = Δu + G(u), t > 0, $x\in {\Omega }_{\epsilon }$, ${\partial }_{{\nu }_{\epsilon }}u=0$, t > 0, $x\in \partial {\Omega }_{\epsilon }$, $\left(E_{\epsilon }\right)$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set ${\Omega }_{\epsilon }$ is a thin domain in ${ℝ}^l$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ${ℝ}^l$. If G is dissipative, then equation $\left(E_{\epsilon }\right)$ has a global attractor ${}_{\epsilon }$. We identify a “limit” equation for the family $\left(E_{\epsilon }\right)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${}_{\epsilon }$ as ε → 0⁺.

The Li-Yau semiclassical lower bound for the sum of the first $N$ eigenvalues of the Dirichlet–Laplacian is extended to Dirichlet– Laplacians with constant magnetic fields. Our method involves a new diamagnetic inequality for constant magnetic fields.


The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a fixed number of random walkers evolving according to a stochastic differential equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove...

We give sufficient conditions for the discreteness of the spectrum of differential operators of the form $Au=-\mathrm{△}u+\left(\nabla F,\nabla u\right)$ in $L_{\mu }^2\left({\mathbb{R}}^n\right)$ where $d\mu \left(x\right)=e^{-F\left(x\right)}dx$ and for Schrödinger operators in $L^2\left({\mathbb{R}}^n\right)$. Our conditions are also necessary in the case of polynomial coefficients.

In [2] Kenig, Ruiz and Sogge proved$${\parallel u\parallel }_{L^{\frac{2n}{n-2}}\left({ℝ}^n\right)}\lesssim {\parallel Lu\parallel }_{L^{\frac{2n}{n+2}}\left({ℝ}^n\right)}$$provided $n\ge 3$, $u\in C_0^{\infty }\left({ℝ}^n\right)$ and $L$ is a second order operator with constant coefficients such that the second order coefficients are real and nonsingular. As a consequence of [3] we state local versions of this inequality for operators with $C^2$ coefficients. In this paper we show how to apply these local versions to the absence of embedded eigenvalues for potentials in $L^{\frac{n+1}{2}}$ and variants thereof.

The self-organization of porous nanostructures in anodic metal oxide is considered. A mathematical model which incorporates the chemical reactions at the metal-oxide and oxide-electrolyte interfaces and elastic stress caused by the electrostrictive effects is developed. It is shown through linear stability analysis, that a short-wave instability exists in certain parameter regimes which can lead to the formation of hexagonally ordered pores observed in anodized aluminum oxide.