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

Similarity:

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

Similarity:

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

Similarity:

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...

The Existence of a Generalized Solution of an $m$ -Point Nonlocal Boundary Value Problem

David Devadze (2017)

Communications in Mathematics

Similarity:

An $m$ -point nonlocal boundary value problem is posed for quasilinear differential equations of first order on the plane. Nonlocal boundary value problems are investigated using the algorithm of reducing nonlocal boundary value problems to a sequence of Riemann-Hilbert problems for a generalized analytic function. The conditions for the existence and uniqueness of a generalized solution in the space are considered.

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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...

On weak supercyclicity II

Carlos S. Kubrusly, Bhagwati P. Duggal (2018)

Czechoslovak Mathematical Journal

Similarity:

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly $l$ -sequentially supercyclic, and (iii) weak $l$ -sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space...

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

Ioana Ghenciu (2015)

Colloquium Mathematicae

Similarity:

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...