We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative...


We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain...

Tetrahedral finite $C^0$-elements of the Hermite type satisfying the maximum angle condition are presented and the corresponding finite element interpolation theorems in the maximum norm are proved.

Let $\alpha >0$, $\lambda ={\left(2\alpha \right)}^{-1/2}$, $S^{n-1}$ be the $(n-1)$-dimensional unit sphere, $\sigma$ be the surface measure on $S^{n-1}$ and $h\left(x\right)={\int }_{S^{n-1}}{e}^{\lambda \langle x,y\rangle }d\sigma \left(y\right)$. We characterize all subsets $M$ of ${ℝ}^n$ such that $$\underset{x\in {ℝ}^n}{inf}\frac{u\left(x\right)}{h\left(x\right)}=\underset{x\in M}{inf}\frac{u\left(x\right)}{h\left(x\right)}$$ for every positive solution $u$ of the Helmholtz equation on ${ℝ}^n$. A closely related problem of representing functions of $L_1\left(S^{n-1}\right)$ as sums of blocks of the form $e^{\lambda \langle x_k,.\rangle }/h\left(x_k\right)$ corresponding to points of $M$ is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References.

Material and shape derivatives for solutions to the Dirichlet Laplacian in a half-space are derived by an application of the speed method. The proposed method is general and can be used for shape sensitivity analysis in unbounded domains for the Neumann Laplacian as well as for the elasticity boundary value problems.

Let $Y$ be a hyperbolic surface and let $\phi$ be a Laplacian eigenfunction having eigenvalue $-1/4-{\tau }^2$ with $\tau >0$. Let $N\left(\phi \right)$ be the set of nodal lines of $\phi$. For a fixed analytic curve $\gamma$ of finite length, we study the number of intersections between $N\left(\phi \right)$ and $\gamma$ in terms of $\tau$. When $Y$ is compact and $\gamma$ a geodesic circle, or when $Y$ has finite volume and $\gamma$ is a closed horocycle, we prove that $\gamma$ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N\left(\phi \right)$ and $\gamma$ is $O\left(\tau \right)$. This bound is sharp.

Si studia la perturbazione dello spettro deiroperatore $-\mathrm{\Delta }+{|x|}^2$ dovuta all'introduzione di un potenziale singolare non polinomiale e si prova che la serie perturbativa del primo autovalore di tale operatore è sommabile secondo Borel.