Domain separation by means of sign changing eigenfunctions of p-laplacians

We are interested in algorithms for constructing surfaces Γ of possibly small measure that separate a given domain ω into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the p-Laplacians,p→1 under humogeneous Neumann houndary conditions. These eigenfunctions are proven to be limtes of a steepest descent method applied to suitable norm quotients. Finally we use these ideas for the construction of separators on simplex grids.     

On vector fields with simply connected trajectories and one invariant line

We study polynomial vector fields X on ${\mathbb{C}}^2$ which have simply connected trajectories and satisfy $dP(X)=a?P$, for a constant $a\in {\mathbb{C}}^{?}$ and a primitive polynomial $P\in \mathbb{C}[x,y]$. We determine X, up to an algebraic change of coordinates. In particular, we obtain that X is complete.     

Decay of eigenfunctions of elliptic PDE's,I

We study exponential decay of eigenfunctions of self-adjoint higher order elliptic operators on R^d${\mathbb{R}}^d$. We show that the possible (global) critical decay rates are determined algebraically. In addition we show absence of super-exponentially decaying eigenfunctions and a refined exponential upper bound.     

Parameter convexity and concavity of generalized trigonometric functions

We study the convexity properties of the generalized trigonometric functions viewed as functions of the parameter. We show that p→_sinp?(y)$p\to {\sin }_p?(y)$ and p→_cosp?(y)$p\to {\cos }_p?(y)$ are log-concave on the appropriate intervals while p→_tanp?(y)$p\to {\tan }_p?(y)$ is log-convex. We also prove similar facts about the generalized hyperbolic functions. In particular, our results settle a major part of the conjecture recently put forward in [4].     

On the Spectrum of Non-Selfadjoint Schrödinger Operators with Compact Resolvent

We determine the Schatten class for the compact resolvent of Dirichlet realizations, in unbounded domains, of a class of non-selfadjoint differential operators. This class consists of operators that can be obtained via analytic dilation from a Schrödinger operator with magnetic field and a complex electric potential. As an application, we prove, in a variety of examples motivated by physics, that the system of generalized eigenfunctions associated with the operator is complete, or at least the existence of an infinite discrete spectrum.     

Non-localization of eigenfunctions for Sturm–Liouville operators and applications

In this article, we investigate a non-localization property of the eigenfunctions of Sturm–Liouville operators $A_a=?{?}_{xx}+a(?)\mathrm{Id}$ with Dirichlet boundary conditions, where $a(?)$ runs over the bounded nonnegative potential functions on the interval $(0,L)$ with $L>0$. More precisely, we address the extremal spectral problem of minimizing the $L^2$-norm of a function $e(?)$ on a measurable subset ω of $(0,L)$, where $e(?)$ runs over all eigenfunctions of $A_a$, at the same time with respect to all subsets ω having a prescribed measure and all $L^{\infty }$ potential functions $a(?)$ having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted.     

Eigenvalues and eigenfunctions of one-dimensional fractal Laplacians defined by iterated function systems with overlaps

Under the assumption that a self-similar measure defined by a one-dimensional iterated function system with overlaps satisfies a family of second-order self-similar identities introduced by Strichartz et al., we obtain a method to discretize the equation defining the eigenvalues and eigenfunctions of the corresponding fractal Laplacian. This allows us to obtain numerical solutions by using the finite element method. We also prove that the numerical eigenvalues and eigenfunctions converge to the true ones, and obtain estimates for the rates of convergence. We apply this scheme to the fractal Laplacians defined by the well-known infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the open set condition or the post-critically finite condition; we use second-order self-similar identities to analyze the measures.     

Spectral theory of Volterra-composition operators

Given a Lebesgue measurable self-map of the interval [0, 1], the Volterra- composition operator is defined as We develop the spectral theory of these operators. In particular, for a class of natural symbols , finiteness of the spectrum is characterized and formulae for the trace and the convergence exponent of eigenvalues are provided. The positivity of the spectrum as well as the analyticity of the eigenfunctions are also treated. The theory of entire functions as well as solving some Cauchy Problems will play a fundamental role in this theory. We also supply some examples of symbols to which the theory can be applied and, in particular, eigenvalues and eigenfunctions are computed explicitly. Partially supported by Plan Nacional I+D+I grant no. MTM2006-09060, Junta de Andalucía FQM-260 and P06-FQM-02225.     

Solving eigenvalue problems on curved surfaces using the Closest Point Method

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace–Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach.