Théorème d'existence pour des problèmes variationnels non convexes

1987 ◽  
Vol 107 (1-2) ◽  
pp. 43-64 ◽  
Author(s):  
J. P. Raymond
Keyword(s):  
Convex Subset ◽  
Existence Results ◽  
Open Convex ◽  
Image Position ◽  
Open Convex Subset

SynopsisOn donne dans cet article un théorème d'existence de solutions lipschitziennes pour des problèmes du type:où Ω est un ouvert convexe borné de ℝn, n ≧ 2, p ≧ 2, aucune hypothèse de convexité n'est faite sur g ou f. On étend de la sorte des ŕesultats d'existence obtenus en dimension 1.where Ω is a bounded open convex subset of ℝn, n ≧ 2, p ≧ 2; we suppose no assumption of convexity on g or f. In this way we extend existence results proved in dimension 1.)

Amenability and invariant subspaces

1974 ◽  
Vol 18 (2) ◽  
pp. 200-204 ◽  
Author(s):  
Anthony To-Ming Lau
Keyword(s):  
Convex Subset ◽  
Invariant Subspaces ◽  
Complex Field ◽  
Open Convex ◽  
Amenable Semigroup ◽  
Linear Action ◽  
Banach Theorem ◽  
Amenable Semigroups ◽  
Open Convex Subset ◽  
Invariant Element

Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

scholarly journals Quasianalytic functionals and projective descriptions

2004 ◽  
Vol 94 (2) ◽  
pp. 249 ◽  
Author(s):  
José Bonet ◽  
Reinhold Meise
Keyword(s):  
Laplace Transform ◽  
Convex Subset ◽  
Inductive Limit ◽  
Entire Functions ◽  
Fréchet Spaces ◽  
Open Convex ◽  
Open Convex Subset

The topology of the weighted inductive limit of Fréchet spaces of entire functions in $N$ variables which is obtained as the Fourier Laplace transform of the space of quasianalytic functionals on an open convex subset of $\mathrm{R}^N$ cannot be described by means of weighted sup-seminorms.

Asymptotic behaviour of positive solutions of semilinear elliptic problems with increasing powers

2021 ◽  
pp. 1-18
Author(s):  
Lucio Boccardo ◽  
Liliane Maia ◽  
Benedetta Pellacci
Keyword(s):  
Linear Operator ◽  
Asymptotic Behaviour ◽  
Positive Solutions ◽  
Elliptic Problems ◽  
Existence Results ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Image Position

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \] where $L(v)=-\textrm {div}(M(x)\nabla v)$ is a linear operator, $p\in (2,2^{*}]$ and $\lambda$ and $m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviours.

scholarly journals Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’

2020 ◽  
Vol 150 (5) ◽  
pp. 2682-2718 ◽  
Author(s):  
Boumediene Abdellaoui ◽  
Antonio J. Fernández
Keyword(s):  
Bounded Domain ◽  
Lebesgue Space ◽  
Fractional Laplacian ◽  
Existence Result ◽  
Existence Results ◽  
Real Parameter ◽  
Nonlocal Diffusion ◽  
Smooth Bounded Domain ◽  
Image Position ◽  
Regularity Results

AbstractLet$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form $$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$ where λ > 0 is a real parameter, f belongs to a suitable Lebesgue space, $\mu \in L^{\infty}$ and $\mathbb {D}_s^2$ is a nonlocal ‘gradient square’ term given by $$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$ Depending on the real parameter λ > 0, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón–Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

scholarly journals A generalised quasi-variational inequality without upper semicontinuity

1996 ◽  
Vol 54 (2) ◽  
pp. 247-254 ◽  
Author(s):  
Paolo Cubiotti ◽  
Xian-Zhi Yuan
Keyword(s):  
Variational Inequality ◽  
Convex Subset ◽  
Existence Result ◽  
Upper Semicontinuity ◽  
Nonempty Closed Convex Subset ◽  
Positive Answer ◽  
Upper Semicontinuous ◽  
Quasi Variational Inequality ◽  
General Existence ◽  
Image Position

In this note we deal with the following problem: given a nonempty closed convex subset X of Rn and two multifunctions Γ : X → 2X and , to find ( such thatWe prove a very general existence result where neither Γ nor Φ are assumed to be upper semicontinuous. In particular, our result give a positive answer to an open problem raised by the first author recently.

Ground state solutions of Hamiltonian elliptic systems in dimension two

2019 ◽  
Vol 150 (4) ◽  
pp. 1737-1768 ◽  
Author(s):  
Djairo G. de Figueiredo ◽  
João Marcos do Ó ◽  
Jianjun Zhang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Elliptic Systems ◽  
Nehari Manifold ◽  
Existence Results ◽  
Ground State Solutions ◽  
Variational Framework ◽  
Hamiltonian Elliptic System ◽  
Schrödinger Systems ◽  
Image Position

AbstractThe aim of this paper is to study Hamiltonian elliptic system of the form 0.1$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$ where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).

scholarly journals Some fixed-point theorems on locally convex linear topological spaces

1975 ◽  
Vol 13 (2) ◽  
pp. 241-254 ◽  
Author(s):  
E. Tarafdar
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Topological Spaces ◽  
The Family ◽  
Image Position ◽  
Commutative Family ◽  
Locally Convex

Let (E, τ) be a locally convex linear Hausdorff topological space. We have proved mainly the following results.(i) Let f be nonexpansive on a nonempty τ-sequentially complete, τ-bounded, and starshaped subset M of E and let (I-f) map τ-bounded and τ-sequentially closed subsets of M into τ-sequentially closed subsets of M. Then f has a fixed-point in M.(ii) Let f be nonexpansive on a nonempty, τ-sequentially compact, and starshaped subset M of E. Then f has a fixed-point in M.(iii) Let (E, τ) be τ-quasi-complete. Let X be a nonempty, τ-bounded, τ-closed, and convex subset of E and M be a τ-compact subset of X. Let F be a commutative family of nonexpansive mappings on X having the property that for some f1 ∈ F and for each x ∈ X, τ-closure of the setcontains a point of M. Then the family F has a common fixed-point in M.

scholarly journals Three proofs of Minkowski's second inequality in the geometry of numbers

1965 ◽  
Vol 5 (4) ◽  
pp. 453-462 ◽  
Author(s):  
R. P. Bambah ◽  
Alan Woods ◽  
Hans Zassenhaus
Keyword(s):  
Convex Set ◽  
Geometry Of Numbers ◽  
Real Numbers ◽  
Open Convex ◽  
Positive Real ◽  
Original Proof ◽  
Linearly Independent ◽  
Image Position

Let K be a bounded, open convex set in euclidean n-space Rn, symmetric in the origin 0. Further let L be a lattice in Rn containing 0 and put extended over all positive real numbers ui for which uiK contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowski's second inequality in the geometry of numbers states that Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].

scholarly journals A dual differentiation space without an equivalent locally uniformly rotund norm

2004 ◽  
Vol 77 (3) ◽  
pp. 357-364 ◽  
Author(s):  
Petar S. Kenderov ◽  
Warren B. Moors
Keyword(s):  
Dense Subset ◽  
Continuum Hypothesis ◽  
Open Convex ◽  
Residual Subset ◽  
Continuous Convex Function ◽  
Dual Differentiation ◽  
Fréchet Differentiable ◽  
Rotund Norm ◽  
Open Convex Subset ◽  
The Continuum

AbstractA Banach space (X, ∥ · ∥) is said to be a dual differentiation space if every continuous convex function defined on a non-empty open convex subset A of X* that possesses weak* continuous subgradients at the points of a residual subset of A is Fréchet differentiable on a dense subset of A. In this paper we show that if we assume the continuum hypothesis then there exists a dual differentiation space that does not admit an equivalent locally uniformly rotund norm.

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