The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics 2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons,...

The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons,...

This paper deals with the use of wavelets in the framework of the Mortar method. We first review in an abstract framework the theory of the mortar method for non conforming domain decomposition, and point out some basic assumptions under which stability and convergence of such method can be proven. We study the application of the mortar method in the biorthogonal wavelet framework. In particular we define suitable multiplier spaces for imposing weak continuity. Unlike in the classical mortar method,...

This paper deals with the use of wavelets in the framework of the Mortar method. We first review in an abstract framework the theory of the mortar method for non conforming domain decomposition, and point out some basic assumptions under which stability and convergence of such method can be proven. We study the application of the mortar method in the biorthogonal wavelet framework. In particular we define suitable multiplier spaces for imposing weak continuity. Unlike in the classical mortar method,...

The definition of multiple layer potential for the biharmonic equation in ${\mathbb{R}}^n$ is given. In order to represent the solution of Dirichlet problem by means of such a potential, a singular integral system, whose symbol determinant identically vanishes, is considered. The concept of bilateral reduction is introduced and employed for investigating such a system.

Let Ω be a bounded domain in ${ℝ}^n$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from $L^2\left(\Omega \right)$ into itself with compact inverse, with eigenvalues $-{\lambda }_i$, each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem $Lu+b{u}^{+}-a{u}^{-}=f\left(x\right)$ in Ω, u=0 on ∂ Ω. We assume that $a<{\lambda }_1$, ${\lambda }_2<b<{\lambda }_3$ and f is generated by ${\varphi }_1$ and ${\varphi }_2$. We show a relation between the multiplicity of solutions and source terms in the equation....

In this note we prove the existence of extremal solutions of the quasilinear Neumann problem $-\left(\right|x^{{}^{\text{'}}}\left(t\right){{|}^{p-2}{x}^{{}^{\text{'}}}\left(t\right))}^{{}^{\text{'}}}=f(t,x\left(t\right),x^{{}^{\text{'}}}\left(t\right))$, a.e. on $T$, $x^{{}^{\text{'}}}\left(0\right)=x^{{}^{\text{'}}}\left(b\right)=0$, $2\le p<\infty$ in the order interval $[\psi ,\varphi ]$, where $\psi$ and $\varphi$ are respectively a lower and an upper solution of the Neumann problem.

In the paper we study the equation $Lu=f$, where $L$ is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set $\Omega$. We prove existence and uniqueness of solutions in the space $H\left(\Omega \right)$ for the Neumann problem.

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set $G$ in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on $\partial G$. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on $G$ a necessary and sufficient condition for the solvability...

We prove sharp inequalities in weighted Sobolev spaces. Our approach is based on the blow-up technique applied to some nonlinear Neumann problems.