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

Giuseppe Zampieri (1980)

Rendiconti del Seminario Matematico della Università di Padova

Analytic continuation of fundamental solutions to differential equations with constant coefficients

Christer O. Kiselman (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

If $P$ is a polynomial in $R^{n}$ such that $1 / P$ integrable, then the inverse Fourier transform of $1 / P$ is a fundamental solution $E_{P}$ to the differential operator $P (D)$ . The purpose of the article is to study the dependence of this fundamental solution on the polynomial $P$ . For $n = 1$ it is shown that $E_{P}$ can be analytically continued to a Riemann space over the set of all polynomials of the same degree as $P$ . The singularities of this extension are studied.

Caratterizzazione dei polinomi di convoluzione in una variabile a decrescenza rapida, a coefficienti costanti, che hanno soluzioni quasi periodiche per ogni termine noto quasi periodico

Giuliano Bratti (1972)

Rendiconti del Seminario Matematico della Università di Padova

Caratterizzazione dei sistemi ipoellittici a coefficienti costanti, sovradeterminati, con il metodo della paramatrice

Giuliano Bratti (1974)

Rendiconti del Seminario Matematico della Università di Padova

Characterization of surjective partial differential operators on spaces of real analytic functions

Michael Langenbruch (2004)

Studia Mathematica

Let A(Ω) denote the real analytic functions defined on an open set Ω ⊂ ℝⁿ. We show that a partial differential operator P(D) with constant coefficients is surjective on A(Ω) if and only if for any relatively compact open ω ⊂ Ω, P(D) admits (shifted) hyperfunction elementary solutions on Ω which are real analytic on ω (and if the equation P(D)f = g, g ∈ A(Ω), may be solved on ω). The latter condition is redundant if the elementary solutions are defined on conv(Ω). This extends and improves previous...

Continuous linear right inverses for partial differential operators of order 2 and fundamental solutions in half spaces.

Reinhold Meise, B.A. Taylor, D. Vogt (1996)

Manuscripta mathematica

Diffraction pour l'équation de la chaleur

Thierry Hargé (1993)

Journées équations aux dérivées partielles

Division Problem and Partial Differential Equations with Constant Coefficients in Colombeau's Space of New Generalized Functions.

M. Nedeljkov, S. Pilipovic, D. Scarpalezos (1996)

Monatshefte für Mathematik

Estimates for Green's matrices of elliptic systems by Lp theory.

G. Dolzmann, S. Müller (1995)

Manuscripta mathematica

Extension and lacunas of solutions of linear partial differential equations

Uwe Franken, Reinhold Meise (1996)

Annales de l'institut Fourier

Let $K \subset Q$ be compact, convex sets in $ℝ^{n}$ with $\overset{\circ}{K} \neq \emptyset$ and let $P (D)$ be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of $P (D)$ in the space $ℰ (K)$ of all $C^{\infty}$ -functions on $K$ extends to a zero solution in $ℰ (Q)$ resp. in $ℰ (ℝ^{n})$ . The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of $P$ in $ℂ^{n}$ and in terms of fundamental solutions for $P (D)$ with lacunas.

Fundamental solution $E_{n}$ of the operator $\partial^{2} / \partial t^{2} - Δ_{n}$ for n>3

Zofia Szmydt, Bogdan Ziemian (1979)

Annales Polonici Mathematici

Fundamental solution for operators preserving a quadratic form

Zofia Szmydt, Bogdan Ziemian (1983)

Annales Polonici Mathematici

Fundamental solutions of the differential operator ${(- 1)}^{n} D_{1}^{n} D_{2}^{n} + a {(i D_{1})}^{n} + b {(i D^{2})}_{n} + c$

Věra Radochová (1980)

Časopis pro pěstování matematiky

Fundamentallösungen von homogenen Differentialoperatoren [Book]

Martin Galler (1989)

Heat kernels for class 2 nilpotent groups

Jacek Cygan (1979)

Studia Mathematica

Konstruktion von Fundamentallösungen für Convolutoren.

Olaf von Grudzinski (1976)

Manuscripta mathematica

$L^{p}$ - $L^{q}$ -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity

Jerzy Gawinecki (1991)

Annales Polonici Mathematici

We prove the $L^{p}$ - $L^{q}$ -time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman’s approach [12], [5] to the $L^{p}$ - $L^{q}$ -time decay estimates.

Large time behaviour of solutions to nonhomogeneous diffusion equations

Jean Dolbeault, Grzegorz Karch (2006)

Banach Center Publications

This note is devoted to the study of the long time behaviour of solutions to the heat and the porous medium equations in the presence of an external source term, using entropy methods and self-similar variables. Intermediate asymptotics and convergence results are shown using interpolation inequalities, Gagliardo-Nirenberg-Sobolev inequalities and Csiszár-Kullback type estimates.

Localization and wave fronts

K. G. Andersson (1972/1973)

Séminaire Équations aux dérivées partielles (Polytechnique)

Localizations of partial differential operators and surjectivity on real analytic functions

Michael Langenbruch (2000)

Studia Mathematica

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $Ω \subset ℝ^{n}$ . Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization $P_{m, Θ}$ of the principal part $P_{m}$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for $P_{m, Θ}$ . Under additional assumptions $P_{m}$ must be locally hyperbolic.

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