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A splitting theory for the space of distributions

P. Domański, D. Vogt (2000)

Studia Mathematica

The splitting problem is studied for short exact sequences consisting of countable projective limits of DFN-spaces (*) 0 → F → X → G → 0, where F or G are isomorphic to the space of distributions D'. It is proved that every sequence (*) splits for F ≃ D' iff G is a subspace of D' and that, for ultrabornological F, every sequence (*) splits for G ≃ D' iff F is a quotient of D'

About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains

Laurent Bourgeois (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces...

An overdetermined elliptic problem in a domain with countably rectifiable boundary

Przemysław Górka (2007)

Colloquium Mathematicae

We examine an elliptic equation in a domain Ω whose boundary ∂Ω is countably (m-1)-rectifiable. We also assume that ∂Ω satisfies a geometrical condition. We are interested in an overdetermined boundary value problem (examined by Serrin [Arch. Ration. Mech. Anal. 43 (1971)] for classical solutions on domains with smooth boundary). We show that existence of a solution of this problem implies that Ω is an m-dimensional Euclidean ball.

Boundary regularity and compactness for overdetermined problems

Ivan Blank, Henrik Shahgholian (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let D be either the unit ball B 1 ( 0 ) or the half ball B 1 + ( 0 ) , let f be a strictly positive and continuous function, and let u and Ω ⊂ D solve the following overdetermined problem: Δ u ( x ) = χ Ω ( x ) f ( x ) in D , 0 ∈ ∂ Ω , u = | ∇ u | = 0 in Ω c , where χ Ω denotes the characteristic function of Ω , Ω c denotes the set D ∖ Ω , and the equation is satisfied in the...

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