On the principal eigenvalue of elliptic operators in $\mathbb {R}^N$ and applications

Henry Berestycki; Luca Rossi

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Displaying similar documents to “On the principal eigenvalue of elliptic operators in $ℝ^{N}$ and applications”

Sequences of differential operators: exponentials, hypercyclicity and equicontinuity

L. Bernal-González, J. A. Prado-Tendero (2001)

Annales Polonici Mathematici

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An eigenvalue criterion for hypercyclicity due to the first author is improved. As a consequence, some new sufficient conditions for a sequence of infinite order linear differential operators to be hypercyclic on the space of holomorphic functions on certain domains of $ℂ^{N}$ are shown. Moreover, several necessary conditions are furnished. The equicontinuity of a family of operators as above is also studied, and it is characterized if the domain is $ℂ^{N}$ . The results obtained extend or improve...

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...

A weak comparison principle for some quasilinear elliptic operators: it compares functions belonging to different spaces

Akihito Unai (2018)

Applications of Mathematics

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We shall prove a weak comparison principle for quasilinear elliptic operators $- div (a (x, \nabla u))$ that includes the negative $p$ -Laplace operator, where $a : Ω \times ℝ^{N} \to ℝ^{N}$ satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

$L^{p, Φ}$ spaces and their application to elliptic partial differential operators

Magdalena Jaroszewska (1983)

Annales Polonici Mathematici

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Tunnel effect and symmetries for non-selfadjoint operators

Michael Hitrik (2013)

Journées Équations aux dérivées partielles

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We study low lying eigenvalues for non-selfadjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and $𝒫𝒯$ -symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low temperature) limit. This analysis also applies sometimes to chains of oscillators coupled to two...

On extensions of families of operators

Oleg Lihvoinen (2023)

Commentationes Mathematicae Universitatis Carolinae

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The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of...

Multiple summing operators on $l_{p}$ spaces

Dumitru Popa (2014)

Studia Mathematica

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We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_{p}$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_{p}$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T : l_{4 / 3} \times l_{4 / 3} \to l ₂$ such that none of the associated linear operators...

On graphs whose second least eigenvalue is at least $- 1$

M. Petrović (1990)

Matematički Vesnik

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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.

On a proof of the boundedness and nuclearity of pseudodifferential operators in $R^{n}$

Jan A. Rempała (1990)

Annales Polonici Mathematici

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Besov spaces and 2-summing operators

M. A. Fugarolas (2004)

Colloquium Mathematicae

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Let Π₂ be the operator ideal of all absolutely 2-summing operators and let $I_{m}$ be the identity map of the m-dimensional linear space. We first establish upper estimates for some mixing norms of $I_{m}$ . Employing these estimates, we study the embedding operators between Besov function spaces as mixing operators. The result obtained is applied to give sufficient conditions under which certain kinds of integral operators, acting on a Besov function space, belong to Π₂; in this context, we also...

Semi-elliptic operators generated by vector fields

E. Shargorodsky

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Let M be a $C^{\infty}$ -smooth n-dimensional manifold and ν₁, ..., νₙ be $C^{\infty}$ -smooth vector fields on M which span the tangent space $T_{x} M$ at each point x ∈ M. The vector fields ν₁, ..., νₙ may have nonzero commutators. We construct a calculus of pseudodifferential operators (ψDOs) which act on sections of vector bundles over M and have symbols belonging to anisotropic analogues of the Hörmander classes $S_{ϱ, δ}^{r}$ , and apply it to semi-elliptic operators generated by ν₁,...,νₙ. The results obtained include the...

A general homogenization result of spectral problem for linearized elasticity in perforated domains

Mohamed Mourad Lhannafi Ait Yahia, Hamid Haddadou (2021)

Applications of Mathematics

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The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the $H^{0}$ -convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor $A^{0}$ , the $H^{0}$ -limit of $A^{ε}$ , is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using...

On the principal eigencurve of the p-Laplacian related to the Sobolev trace embedding

Abdelouahed El Khalil, Mohammed Ouanan (2005)

Applicationes Mathematicae

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We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ $Δ ₚ u = {| u |}^{p - 2} u$ in Ω, ⎨ ⎩ ${| \nabla u |}^{p - 2} \partial u / \partial ν = {λ ϱ (x) | u |}^{p - 2} u + μ {| u |}^{p - 2} u$ on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.

Isomorphic properties in spaces of compact operators

Ioana Ghenciu (2023)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the definition of $p$ -limited completely continuous operators, $1 \leq p < \infty$ . The question of whether a space of operators has the property that every $p$ -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using $p$ -limited completely continuous evaluation operators.

On a semilinear elliptic eigenvalue problem

Mario Michele Coclite (1997)

Annales Polonici Mathematici

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We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, ${u |}_{\partial Ω} = 0$ , where f(x) and h(u) satisfy minimal regularity assumptions.

Partially elliptic differential equations having distributions as their right members

H. Marcinkowska

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ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p, q; Y} (Ω, ℬ)$ ..........................................................73. Some properties of the spaces $H_{p, q; Y} (Ω, ℬ)$ ...................................................104. Some examples of the spaces $H_{p, q; Y} (Ω, ℬ)$ ....................................................155....

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

M. Sango (2001)

Colloquium Mathematicae

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We investigate the behaviour of a sequence $λ_{s}$ , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains $Ω_{s}$ , s = 1,2,..., obtained by removing from a given domain Ω a set $E_{s}$ whose diameter vanishes when s → ∞. We estimate the deviation of $λ_{s}$ from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

A new characterization of Anderson’s inequality in $C_{1}$ -classes

S. Mecheri (2007)

Czechoslovak Mathematical Journal

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Let $ℋ$ be a separable infinite dimensional complex Hilbert space, and let $ℒ (ℋ)$ denote the algebra of all bounded linear operators on $ℋ$ into itself. Let $A = (A_{1}, A_{2}, \dots, A_{n})$ , $B = (B_{1}, B_{2}, \dots, B_{n})$ be $n$ -tuples of operators in $ℒ (ℋ)$ ; we define the elementary operators $Δ_{A, B} ℒ (ℋ) \mapsto ℒ (ℋ)$ by $Δ_{A, B} (X) = \sum_{i = 1}^{n} A_{i} X B_{i} - X .$ In this paper, we characterize the class of pairs of operators $A, B \in ℒ (ℋ)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A, B \in ℒ (ℋ)$ such that $\sum_{i = 1}^{n} B_{i} T A_{i} = T$ implies $\sum_{i = 1}^{n} A_{i}^{*} T B_{i}^{*} = T$ for all $T \in 𝒞_{1} (ℋ)$ (trace class operators). The main result is the equivalence between this property and the...

Displaying similar documents to “On the principal eigenvalue of elliptic operators in ℝ N and applications”

Displaying similar documents to “On the principal eigenvalue of elliptic operators in $ℝ^{N}$ and applications”