Singular behavior of the Dirichlet problem in Hölder spaces of the solutions to the Dirichlet problem in a cone

In the present study we consider the solution of the Dirichlet problem in conical domain. For general elliptic problems in non Hilbertian Sobolev spaces built on $L^{p},1<p<\infty$ the theory of sums of operators developed by Dore-Venni $\left[8\right]$ provides an optimal result. Holder spaces, as opposed to LP spaces, are not UMD. Using the results of Da Prato-Grisvard $\left[6\right]$ and Labbas $\left[14\right]$ we cope with the singular behaviour of the solution in the framework of H$\ddot{\textrm{o}}$lder and little H$\ddot{\textrm{o}}$lder spaces

Generalized diffusion problems in a conical domain, part I

The purpose of this article (composed of two parts) is the study of the generalized dispersal operator of a reaction-diffusion equation in $L^p$-spaces set in the finite conical domain $S_{\omega,\rho}$ of angle $\omega>0$ and radius $\rho>0$ in $\mathbb{R}^2$. This first part is devoted to the behaviour of the solution near the top of the cone which is completely described in the weighted Sobolev space $W^{4,p}_{3-\frac{1}{p}}(S_{\omega,\rho})$, see Theorem 2.2

Generalized diffusion problems in a conical domain, part II

After different variables and functions changes, the generalized dispersal problem, recalled in (1) below and considered in part I, see [14], leads us to invert a sum of linear operators in a suitable Banach space, see (2) below. The essential result of this second part lies in the complete study of this sum using the two well-known strategies: the one of Da Prato-Grisvard [4] and the one of Dore-Venni [6]

Étude unifiée d'équations aux dérivées partielles de type elliptique régies par des équations différentielles à coefficients opérateurs dans un cadre non commutatif (applications concrètes dans les espaces de Hölder et les espaces Lp)

L'objectif de ce travail est l'étude des équations différentielles complètes du second ordre de type elliptique à coefficients opérateurs dans un espace de Banach X quelconque. Une application concrète de ces équations est détaillée, il s'agit d'un problème de transmission du potentiel électrique dans une cellule biologique où la membrane constitue une couche mince. L'originalité de ce travail réside particulièrement dans le fait que les opérateurs non bornés considérés ne commutent pas nécessairement. Une nouvelle hypothèse dite de non commutativité est alors introduite. L'analyse est faite dans deux cadres fonctionnels distincts: les espaces de Hölder et les espaces Lp (avec X un espace UMD). L'équation est d'abord étudiée sur la droite réelle puis sur un intervalle borné avec conditions aux limites de Dirichlet. On donne des résultats d'existence, d'unicité et de régularité maximale de la solution classique sous des conditions sur les données dans des espaces d'interpolation. Les techniques utilisées sont basées sur la théorie des semi-groupes, le calcul fonctionnel de Dunford et la théorie de l'interpolation. Ces résultats sont tous appliqués à des équations aux dérivées partielles concrètes de type elliptique ou quasi-elliptique.The aim of this work is the study of complete elliptic differential equations of second order with operator coefficients in a Banach space X. A concrete application of these equations is detailed, it concerns a transmission problem of electric potential in a biological cell where the membrane is considered as a thin layer. The originality of this work is the fact that unbounded operators which are considered do not commute necessarily. A new noncommutativity hypothesis is then introduced. The analysis is performed in two distinct functional frameworks: the Hölder spaces and the Lp spaces (X being a UMD space). First, the equation is studied on the whole real line and secondly in a bounded interval with Dirichlet boundary conditions. Existence, uniqueness and maximal regularity of the classical solution are proved under some conditions on the data in interpolation spaces. The techniques used are based on semigroup theory, Dunford functional calculus and interpolation theory. All the results are applied to concrete partial differential equations of elliptic or quasi-elliptic type.LE HAVRE-BU Centrale (763512101) / SudocSudocFranceF

Study of a Complete Abstract Differential Equation of Elliptic Type with Variable Operator Coefficients, I

Differential Equation of Elliptic Type with Variable Operator Coefficients, I Angelo FAVINI, Rabah LABBAS, Keddour LEMRABET, and Boubaker-Khaled SADALLAH Universit` degli Studi di Bologna a Laboratoire de Math´ ematiques Appliqu´ ees Dipartimento di Matematica Universit´ du Havre e Piazza di Porta S. U.F.R Sciences et Techniques, B.P 540 Donato 5, 40126 Bologna -- Italy 76058 Le Havre -- France [email protected] [email protected] Laboratoire EDP et Hist. Maths Lab oraoire AMNEDP Ecole Normale Sup´ erieure Facult´ des Maths, USTHB e 16050 Kouba, Alger -- Algeria BP 32, El Alia, Bab Ezzouar 16111 Alger -- Algeria [email protected] [email protected] Received: May 16, 2007 Accepted: August 30, 2007The aim of this first work is the resolution of an abstract complete second order differential equation of elliptic type with variable operator coefficients set in a small length interval. We obtain existence, uniqueness and maximal regularity results under some appropriate differentiability assumptions combining those of Yagi [13] and Da Prato-Grisvard [6]. An example for the Laplacian in a regular domain of R³ will illustrate the theory. A forthcoming work (Part II) will complete the present one by the study of the Steklov-Poincaré operator related to this equation when the length &#948; of the interval tends to zero

NECESSARY AND SUFFICIENT CONDITIONS FOR MAXIMAL REGULARITY IN THE STUDY OF ELLIPTIC DIFFERENTIAL EQUATIONS IN HÖLDER SPACES

International audienceIn this paper we give new results on complete abstract second order differential equations of elliptic type in the framework of Hölder spaces. More precisely we study u′′+2Bu′+Au=f in the case when f is Hölder continuous and under some natural assumptions on the operators A and B. We give necessary and sufficient conditions of compatibility to obtain a strict solution u and also to ensure that the strict solution has the maximal regularity property

Sur l opérateur de Steklov-Poincaré dans une couche mince (analyse, résolution et résultats optimaux de deux classes de problèmes de transmission régis par des EDP et applications concrètes)

LE HAVRE-BU Centrale (763512101) / SudocSudocFranceF

New results on abstract elliptic problems with general Robin boundary conditions in Hölder spaces: non commutative cases

International audienceIn this paper, we prove some new results on operational second order differential equations of elliptic type with general Robin boundary conditions in a non-commutative framework. The study is developed in Hölder spaces under some natural assumptions generalizing those in [4]. We give necessary and sufficient conditions on the data to obtain a unique strict solution satisfying the maximal regularity property, see Theorems 1 and 2. This work completes the one given in [4] and [12]