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L 2 -error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes

Thomas Apel, Dieter Sirch (2011)

Applications of Mathematics

An L 2 -estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.

La quasi-continuité dans l'étude du problème de Dirichlet. Effilement minimal abstrait et ensembles convexes compacts

Denis Feyel (1979)

Annales de l'institut Fourier

Les problèmes de Dirichlet sur la frontière de Martin, sur la frontière de Choquet d’un simplexe métrisable compact, et sur la frontière de Silov d’un simplexe de Bauer métrisable sont tous susceptibles d’une seule méthode de résolution qui utilise un espace de fonctions dites quasi-continues. Cela contient aussi le théorème des limites fines de Fatou-Naïm qui exprime une quasi-continuité jusqu’à la frontière.

Laplace type operators: Dirichlet problem

Wojciech Kozł (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We investigate Laplace type operators in the Euclidean space. We give a purely algebraic proof of the theorem on existence and uniqueness (in the space of polynomial forms) of the Dirichlet boundary problem for a Laplace type operator and give a method of determining the exact solution to that problem. Moreover, we give a decomposition of the kernel of a Laplace type operator into 𝖲𝖮 ( n ) -irreducible subspaces.

L'equazione Δ 2 ⁢ u + a 10 ⁢ ( x , y ) ⁢ ∂ ⁡ u ∂ ⁡ x + a 01 ⁢ ( x , y ) ⁢ ∂ ⁡ u ∂ ⁡ y + a 00 ⁢ ( x , y ) ⁢ u = F ⁢ ( x , y ) . Teorema di esistenza per un generale problema al contorno

Alberto Cialdea (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Necessary and sufficient conditions are given for the existence of smooth solutions of the differential equations (1) with the boundary conditions (2). Coefficients of (1) and (2) are only supposed Hölder-continuous.

Linear and nonlinear abstract differential equations of high order

Veli B. Shakhmurov (2015)

Open Mathematics

The nonlocal boundary value problems for linear and nonlinear degenerate abstract differential equations of arbitrary order are studied. The equations have the variable coefficients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding differential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established....

Linear elliptic equations with BMO coefficients

Menita Carozza, Gioconda Moscariello, Antonia Passarelli di Napoli (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove an existence and uniqueness theorem for the Dirichlet problem for the equation div ⁢ a ⁢ x ⁢ ∇ ⁡ u = div ⁢ f in an open cube Ω ⊂ R N , when f belongs to some L p ⁢ Ω , with p close to 2. Here we assume that the coefficient a belongs to the space BMO( Ω ) of functions of bounded mean oscillation and verifies the condition a ⁢ x ≥ λ 0 > 0 for a.e. x ∈ Ω .

Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations

Pierre Bousquet (2007)

ESAIM: Control, Optimisation and Calculus of Variations

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form div a ( ∇ u ) + F [ u ] ( x ) = 0 , over the functions u ∈ W 1 , 1 ( Ω ) that assume given boundary values ϕ on ∂Ω. The vector field a : ℝ n → ℝ n satisfies an ellipticity condition and for a fixed x, F[u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math.115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions...

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