Search results for "Differential operator"

Spaces of weighted symbols and weighted sobolev spaces on manifolds

1987

This paper gives an approach to pseudodifferential operators on noncompact manifolds using a suitable class of weighted symbols and Sobolev spaces introduced by H.O. Cordes on ℙ. Here, these spaces are shown to be invariant under certain changes of coordinates. It is therefore possible to transfer them to manifolds with a compatible structure.

Statistics of nonlinear stochastic dynamical systems under Lévy noises by a convolution quadrature approach

2010

This paper describes a novel numerical approach to find the statistics of the non-stationary response of scalar non-linear systems excited by L\'evy white noises. The proposed numerical procedure relies on the introduction of an integral transform of Wiener-Hopf type into the equation governing the characteristic function. Once this equation is rewritten as partial integro-differential equation, it is then solved by applying the method of convolution quadrature originally proposed by Lubich, here extended to deal with this particular integral transform. The proposed approach is relevant for two reasons: 1) Statistics of systems with several different drift terms can be handled in an efficie…

Itô formula for an integro-differential operator without an associated stochastic process

2010

Nonlinear Nonhomogeneous Elliptic Problems

2019

We consider nonlinear elliptic equations driven by a nonhomogeneous differential operator plus an indefinite potential. The boundary condition is either Dirichlet or Robin (including as a special case the Neumann problem). First we present the corresponding regularity theory (up to the boundary). Then we develop the nonlinear maximum principle and present some important nonlinear strong comparison principles. Subsequently we see how these results together with variational methods, truncation and perturbation techniques, and Morse theory (critical groups) can be used to analyze different classes of elliptic equations. Special attention is given to (p, 2)-equations (these are equations driven…

Pseudodifferential operators of Beurling type and the wave front set

2008

AbstractWe investigate the action of pseudodifferential operators of Beurling type on the wave front sets. More precisely, we show that these operators are microlocal, that is, preserve or reduce wave front sets. Some consequences on micro-hypoellipticity are derived.

A Symplectic Kovacic's Algorithm in Dimension 4

2018

Let $L$ be a $4$th order differential operator with coefficients in $\mathbb{K}(z)$, with $\mathbb{K}$ a computable algebraically closed field. The operator $L$ is called symplectic when up to rational gauge transformation, the fundamental matrix of solutions $X$ satisfies $X^t J X=J$ where $J$ is the standard symplectic matrix. It is called projectively symplectic when it is projectively equivalent to a symplectic operator. We design an algorithm to test if $L$ is projectively symplectic. Furthermore, based on Kovacic's algorithm, we design an algorithm that computes Liouvillian solutions of projectively symplectic operators of order $4$. Moreover, using Klein's Theorem, algebraic solution…

Solutions with sign information for nonlinear Robin problems with no growth restriction on reaction

2019

We consider a parametric nonlinear Robin problem driven by a nonhomogeneous differential operator. The reaction is a Carathéodory function which is only locally defined (that is, the hypotheses concern only its behaviour near zero). The conditions on the reaction are minimal. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we show that for all small values of the parameter λ > 0, the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal.

Quantitative Approximation Properties for the Fractional Heat Equation

2017

In this note we analyse \emph{quantitative} approximation properties of a certain class of \emph{nonlocal} equations: Viewing the fractional heat equation as a model problem, which involves both \emph{local} and \emph{nonlocal} pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain \emph{qualitative} approximation results from \cite{DSV16}. Using propagation of smallness arguments, we then provide bounds on the \emph{cost} of approximate controllability and thus quantify the approximation properties of solutions to the fractional heat equation. Finally, we discuss genera…

Resolvent estimates for elliptic quadratic differential operators

2011

Sharp resolvent bounds for non-selfadjoint semiclassical elliptic quadratic differential operators are established, in the interior of the range of the associated quadratic symbol.

Distribution of Large Eigenvalues for Elliptic Operators

2019

In this chapter we consider elliptic differential operators on a compact manifold and rather than taking the semi-classical limit (h →), we let h = 1 and study the distribution of large eigenvalues. Bordeaux Montrieux (Loi de Weyl presque sure et resolvante pour des operateurs differentiels non-autoadjoints, these, CMLS, Ecole Polytechnique, 2008. https://pastel.archives-ouvertes.fr/pastel-00005367, Ann Henri Poincare 12:173–204, 2011) studied elliptic systems of differential operators on S1 with random perturbations of the coefficients, and under some additional assumptions, he showed that the large eigenvalues obey the Weyl law almost surely. His analysis was based on a reduction to the s…