Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations

Serhii V. Gryshchuk; Sergiy A. Plaksa

Displaying similar documents to “Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations”

On solutions of a fourth-order Lidstone boundary value problem at resonance

Mariusz Jurkiewicz (2009)

Annales Polonici Mathematici

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We consider a Lidstone boundary value problem in $ℝ^{k}$ at resonance. We prove the existence of a solution under the assumption that the nonlinear part is a Carathéodory map and conditions similar to those of Landesman-Lazer are satisfied.

Martin boundary and positive solutions of some boundary value problems

Evgeny B. Dynkin (1965)

Annales de l'institut Fourier

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Nous étudions le problème de Neumann avec dérivée oblique dans un domaine 2-dimensionnel borné par un contour régulier $C$ . Le champ vectoriel donné sur $C$ peut être tangent à $C$ en un nombre fini de points. En utilisant une extension de la méthode de Martin nous trouvons toutes les solutions positives de ce problème aux valeurs limites.

Boundary value problems with compatible boundary conditions

George L. Karakostas, P. K. Palamides (2005)

Czechoslovak Mathematical Journal

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If $Y$ is a subset of the space $ℝ^{n} \times ℝ^{n}$ , we call a pair of continuous functions $U$ , $V$ $Y$ -compatible, if they map the space $ℝ^{n}$ into itself and satisfy $U x \cdot V y \geq 0$ , for all $(x, y) \in Y$ with $x \cdot y \geq 0$ . (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential $n$ -dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its...

Boundary regularity and compactness for overdetermined problems

Ivan Blank, Henrik Shahgholian (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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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 $Ω \subset D$ solve the following overdetermined problem: $Δ u (x) = χ_{_{Ω}} (x) f (x) in D, 0 \in \partial Ω, u = | \nabla u | = 0 in Ω^{c},$ where $χ_{_{Ω}}$ denotes the characteristic function of $Ω,$ $Ω^{c}$ denotes the set $D ∖ Ω,$ and the equation is satisfied in the sense of distributions. When $D = B_{1}^{+} (0),$ then we impose in addition that $u (x) \equiv 0 on {(x^{'}, x_{n}) | x_{n} = 0} .$ We show that a fairly mild thickness assumption on $Ω^{c}$ will ensure enough compactness on...

Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications

Roberto Triggiani (2008)

Applicationes Mathematicae

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This paper is divided into two parts and focuses on the linear independence of boundary traces of eigenfunctions of boundary value problems. Part I deals with second-order elliptic operators, and Part II with Stokes (and Oseen) operators. Part I: Let $λ_{i}$ be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain Ω in ℝⁿ, with Neumann homogeneous boundary conditions on Γ = tial Ω. Let ${φ_{i j}}_{j = 1}^{ℓ_{i}}$ be the corresponding linearly independent (normalized)...

Weakly semibounded boundary problems and sesquilinear forms

Gerd Grubb (1973)

Annales de l'institut Fourier

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Let $A$ be a $2 m$ order differential operator in a hermitian vector bundle $E$ over a compact riemannian manifold $\overline{Ω}$ with boundary $Γ$ ; and denote by $A_{B}$ the realization defined by a normal differential boundary condition $B ρ u = 0$ ( $u \in H^{2 m} (E)$ , $ρ u =$ Cauchy data). We characterize, by an explicit condition on $A$ and $B$ near $Γ$ , the realizations $A_{B}$ for which there exists an integro-differential sesquilinear form $a_{B} (u, ν)$ on $H^{m} (E)$ such that $(A u, ν) = a_{B} (u, ν)$ on $D (A_{B})$ ; moreover we show that these are exactly the realizations satisfying a weak semiboundedness...

Remarks on a three-point boundary value problem in a differential equation of the third order

M. Greguš (1974)

Annales Polonici Mathematici

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Nonuniqueness for some linear oblique derivative problems for elliptic equations

Gary M. Lieberman (1999)

Commentationes Mathematicae Universitatis Carolinae

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It is well-known that the “standard” oblique derivative problem, $Δ u = 0$ in $Ω$ , $\partial u / \partial ν - u = 0$ on $\partial Ω$ ( $ν$ is the unit inner normal) has a unique solution even when the boundary condition is not assumed to hold on the entire boundary. When the boundary condition is modified to satisfy an obliqueness condition, the behavior at a single boundary point can change the uniqueness result. We give two simple examples to demonstrate what can happen.