A family of fundamental solutions of elliptic partial differential operators with quaternion constant coefficients

The purpose of this paper is to construct a family of fundamental solutions for elliptic partial differential operators with quaternion constant coefficients. The elements of such family are expressed by means of functions, which depend jointly real analytically on the coefficients of the operators and on the spatial variable. We show some regularity properties in the frame of Schauder spaces for the corresponding single layer potentials. Ultimately, we exploit our construction by showing a real analyticity result for perturbations of the layer potentials corresponding to complex elliptic partial differential operators of order two. Copyright © 2012 John Wiley & Sons, Ltd.     

Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients

Summary I give a sufficient condition in order that a Dirichlet problem is solvable in H ² (Ω) for a class of linear second order elliptic partial differential equations. Such a class includes some particular cases for which the result is known.

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Sunto Si prova una condizione sufficiente affinchè un problema di Dirichlet sia risolubile in H ² (Ω) per una classe di equazioni differenziali alle derivate parziali lineari ellittiche del secondo ordine. Tale classe comprende alcuni casi particolari per i quali il risultato è noto.

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The present work was written while the author was a member of the ? Centro di Matematica e Fisica Teorica del C.N.R. ? at the University of Genova, directed by professorJ. Cecconi.

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Entrata in Redazione il 25 febbraio 1971.     

On the supports of solutions of linear partial differential equations with analytic coefficients

Sunto In questo lavoro vengono dimostrati, con un metodo basato sui funzionali analitici, il teorema di Holmgren e varie sue estensioni. Il punto essenziale è che nelle dimostrazioni sono usate soltanto transformazioni lineari. Così è messo in luce che la variabile ? tempo ? ha un ruolo completamente diverso dal ruolo delle variabili ? spaziali ? per l'unicità del problema lineare di Cauchy. Viene anche studiato il problema globale di Cauchy in un semispazio.

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Entrato in Redazione il 1° aprile 1971.     

Reflection principles for linear elliptic second order partial differential equations with constant coefflcients

Summary Reflection principles, analogous to the classicalSchwarz reflection principle for harmonic functions, are obtained for solutions of linear elliptic second order partial differential equations with constant coefficients. The boundary conditions employed are supposed to be satisfied in a limiting sense only, and do not require (a priori) the existence of derivatives on the boundary. To Mauro Picone on his 70^th birth day. This research was supported in part by the United States Air Force under Contract N^o. AF(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command. The work of this author was sponsored by the Office of Ordnance Research, U.S. Army, under the Contract DA-36-034-ORD-1486.     

Surjectivity of partial differential operators with good fundamental solutions

We characterize the surjectivity of a linear partial differential operator with constant coefficients on E(Ω) as well as on D^′(Ω) in terms of the existence of “good” shifted fundamental solutions. This characterization complements results of Meise, Taylor, and Vogt as well as Frerick and the present author.     

Stability properties of solutions of linear second order differential equations with random coefficients

The equation

     

On the growth of solutions of second order linear differential equations with extremal coefficients

In this paper, we consider the differential equation f + A(z)f + B(z)f = 0, where A and B ≡ 0 are entire functions. Assume that A is extremal for Yang's inequality, then we will give some conditions on B which can guarantee that every non-trivial solution f of the equation is of infinite order.     

On meromorphic solutions of linear partial differential equations of second order

This paper is concerned with entire and meromorphic solutions of linear partial differential equations of second order with polynomial coefficients. We will characterize entire solutions for a class of partial differential equations associated with the Jacobi differential equations, and give a uniqueness theorem for their meromorphic solutions in the sense of the value distribution theory, which also applies to general linear partial differential equations of second order. The results are complemented by various examples for completeness.     

Asymptotic behavior of solutions of second order parabolic partial differential equations with unbounded coefficients

Classical solutions of a second order parabolic partial differential equation are considered in unbounded domains. The coefficients are allowed to have unbounded growth as the space variables tend to infinity. A Phragmèn-Lindelöf principle is proved for such equations. That principle is used together with comparison functions to derive sufficient conditions for the asymptotic decay in time of solutions.     

Markovian extensions of symmetric second order elliptic differential operators

Let be bounded with a smooth boundary Γ and let S be the symmetric operator in given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self‐adjoint extensions of S by providing an explicit one‐to‐one correspondence between such extensions and the class of Dirichlet forms in which are additively decomposable by the bilinear form of the Dirichlet‐to‐Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Kre?n's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell‐type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.     

First order linear partial differential equations with discontinuous coefficients

Summary Si dà un teorema di esistenza e di unicità di soluzione generalizzata per il problema di Cauchy relativo all'equazione . Le ipotesi fatte sui coefficienti a_i(t, x) permettono discontinuità di tipo molto generale, purchè sia soddisfatta una certa «condizione di trasversalità» tra le caratteristiche dell'equazione e le discontinuità del campo di vettori che le definisce.     

Estimates for the asymptotic behavior of solutions of the Helmholtz equation,with an application to second order elliptic differential operators with variable coefficients

The online version of the original article can be found at     

Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients

Consider a second-order elliptic partial differential operatorL in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 <p < ∞ and μ > 0. ThenL has a bounded H_∞ functional calculus in L^p(Ω), in the sense that ¦¦f (L +cI)u¦¦_p ≤C sup_¦arλ¦<μ ^¦f¦ ¦‖u¦‖_p for some constantsc andC, and all bounded holomorphic functionsf on the sector ¦ argλ¦ < μ that contains the spectrum ofL +cI. We prove this by showing that the operatorsf(L + cI) are Calderón-Zygmund singular integral operators.     

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.