Weakly semibounded boundary problems and sesquilinear forms

Gerd Grubb

Displaying similar documents to “Weakly semibounded boundary problems and sesquilinear forms”

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)...

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...

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty} : L^{\infty} (X) \to L ¹ (X)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T \circ i_{\infty} : L^{\infty} (X) \to Y$ is a weakly compact operator. Moreover, we obtain that...

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.

Matrix triangulation of hypoelliptic boundary value problems

R. A. Artino, J. Barros-Neto (1992)

Annales de l'institut Fourier

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Given a hypoelliptic boundary value problem on $ω \times [0, T)$ with $ω$ an open set in $R^{n}$ , $(n > 1)$ , we show by matrix triangulation how to reduce it to two uncoupled first order systems, and how to estimate the eigenvalues of the corresponding matrices. Parametrices for the first order systems are constructed. We then characterize hypoellipticity up to the boundary in terms of the Calderon operator corresponding to the boundary value problem.

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

Serhii V. Gryshchuk, Sergiy A. Plaksa (2017)

Open Mathematics

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We consider a commutative algebra over the field of complex numbers with a basis e1, e2 satisfying the conditions [...] (e12+e22)2=0,e12+e22≠0. ${(e_{1}^{2} + e_{2}^{2})}^{2} = 0, e_{1}^{2} + e_{2}^{2} \neq 0 .$ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic -valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain Dζ = xe1 + ye2 : (x, y) ∈ D: to find a monogenic in Dζ function Φ, which is continuously extended to the boundary ∂Dζ, when...

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

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Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X, | | \cdot | |_{X})$ and $(Y, | | \cdot | |_{Y})$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $| | T (1_{A ₙ} f) {| |}_{Y} \to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

Regularity of domains of parameterized families of closed linear operators

Teresa Winiarska, Tadeusz Winiarski (2003)

Annales Polonici Mathematici

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The purpose of this paper is to provide a method of reduction of some problems concerning families $A_{t} = {(A (t))}_{t \in}$ of linear operators with domains ${(_{t})}_{t \in}$ to a problem in which all the operators have the same domain . To do it we propose to construct a family ${(Ψ_{t})}_{t \in}$ of automorphisms of a given Banach space X having two properties: (i) the mapping $t \mapsto Ψ_{t}$ is sufficiently regular and (ii) $Ψ_{t} () =_{t}$ for t ∈ . Three effective constructions are presented: for elliptic operators of second order with the Robin boundary condition...

Liftings of forms to Weil bundles and the exterior derivative

Jacek Dębecki (2009)

Annales Polonici Mathematici

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In a previous paper we have given a complete description of linear liftings of p-forms on n-dimensional manifolds M to q-forms on $T^{A} M$ , where $T^{A}$ is a Weil functor, for all non-negative integers n, p and q, except the case p = n and q = 0. We now establish formulas connecting such liftings and the exterior derivative of forms. These formulas contain a boundary operator, which enables us to define a homology of the Weil algebra A. We next study the case p = n and q = 0 under the condition...

Weak precompactness and property (V*) in spaces of compact operators

Ioana Ghenciu (2015)

Colloquium Mathematicae

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We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w *} (E *, F)$ (resp. $K_{w *} (E *, F)$ ) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w *} (E *, F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w *} (E *, F)$ has property (wV*). Suppose...

Representing non-weakly compact operators

Manuel González, Eero Saksman, Hans-Olav Tylli (1995)

Studia Mathematica

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For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E**/E) is defined by R(S)(x** + E) = S**x** + E(x** ∈ E**). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non-zero compact operators in Im(R) in the case of $L^{1}$ and C(0,1), but R(L(E)/W(E)) identifies isometrically...

Existence Theorems for a Fourth Order Boundary Value Problem

A. El-Haffaf (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper treats the question of the existence of solutions of a fourth order boundary value problem having the following form: $x^{(4)} (t) + f (t, x (t), x^{''} (t)) = 0$ , 0 < t < 1, x(0) = x’(0) = 0, x”(1) = 0, $x^{(3)} (1) = 0$ . Boundary value problems of very similar type are also considered. It is assumed that f is a function from the space C([0,1]×ℝ²,ℝ). The main tool used in the proof is the Leray-Schauder nonlinear alternative.

Asymptotic analysis of the initial boundary value problem for the thermoelastic system in a perforated domain

M. Sango (2003)

Colloquium Mathematicae

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We study the initial boundary value problem for the system of thermoelasticity in a sequence of perforated cylindrical domains $Q_{T}^{(s)}$ , s = 1,2,... We prove that as s → ∞, the solution of the problem converges in appropriate topologies to the solution of a limit initial boundary value problem of the same type but containing some additional terms which are expressed in terms of quantities related to the geometry of $Q_{T}^{(s)}$ . We give an explicit construction of that limit problem.

Weakly precompact operators on $C_{b} (X, E)$ with the strict topology

Juliusz Stochmal (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_{b} (X, E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study weakly precompact operators $T : C_{b} (X, E) \to F$ . In particular, we show that if X is a paracompact k-space and E contains no isomorphic copy of l¹, then every strongly bounded operator $T : C_{b} (X, E) \to F$ is weakly precompact.