### A link between ${C}^{\infty}$ and analytic solvability for P.D.E. with constant coefficients

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The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term,...

Soit $\mathbf{A}$ un ensemble quelconque d’opérateurs différentiels en deux variables à coefficients complexes constants. Soit ${C}_{0}$ l’espace des fonctions continues complexes tendant vers zéro à l’infini dans le plan euclidien. Soit ${C}_{0}\left(\mathbf{A}\right)$ l’espace $\{f:f\in {C}_{0},\phantom{\rule{0.277778em}{0ex}}{A}_{f}{C}_{0}$, tout $a\in \mathbf{A}\}$. Classifier ces espaces équivaut à trouver des conditions nécessaires et suffisantes sur des opérateurs différentiels ${P}_{1},...,{P}_{n}$ pour que $\parallel {P}_{1}\phi {\parallel}_{\infty}\le K(\parallel {P}_{2}\phi {\parallel}_{\infty}+\cdots +\parallel {P}_{n}\phi {\parallel}_{\infty})$. Il paraît que ce problème général est bien difficile. Nous présentons ici la solution complète dans le cas spécial des ${C}_{0}\left(\mathbf{A}\right)$ stables...

This paper is an extended version of an invited talk presented during the Orlicz Centenary Conference (Poznań, 2003). It contains a brief survey of applications to classical problems of analysis of the theory of the so-called PLS-spaces (in particular, spaces of distributions and real analytic functions). Sequential representations of the spaces and the theory of the functor Proj¹ are applied to questions like solvability of linear partial differential equations, existence of a solution depending...

We provide a general series form solution for second-order linear PDE system with constant coefficients and prove a convergence theorem. The equations of three dimensional elastic equilibrium are solved as an example. Another convergence theorem is proved for this particular system. We also consider a possibility to represent solutions in a finite form as partial sums of the series with terms depending on several complex variables.

In this paper we consider first-order systems with constant coefficients for two real-valued functions of two real variables. This is both a problem in itself, as well as an alternative view of the classical linear partial differential equations of second order with constant coefficients. The classification of the systems is done using elementary methods of linear algebra. Each type presents its special canonical form in the associated characteristic coordinate system. Then you can formulate initial...

The surjectivity of the operator ${D}_{2}+i{x}_{2}^{2k}{D}_{1}$ from the Gevrey space ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^{2}\right)$, $s\ge 1$, onto itself and its non-surjectivity from ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^{3}\right)$ to ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^{3}\right)$ is proved.

We prove that any zero solution of a hypoelliptic partial differential operator can be expanded in a generalized Laurent series near a point singularity if and only if the operator is semielliptic. Moreover, the coefficients may be calculated by means of a Cauchy integral type formula. In particular, we obtain explicit expansions for the solutions of the heat equation near a point singularity. To prove the necessity of semiellipticity, we additionally assume that the index of hypoellipticity with...

We discuss existence of global solutions of moderate growth to a linear partial differential equation with constant coefficients whose total symbol P(ξ) has the origin as its only real zero. It is well known that for such equations, global solutions tempered in the sense of Schwartz reduce to polynomials. This is a generalization of the classical Liouville theorem in the theory of functions. In our former work we showed that for infra-exponential growth the corresponding assertion is true if and...