A limit involving functions in $W^{1,p}_0(Ω)$

Biagio Ricceri

Displaying similar documents to “A limit involving functions in $W_{0}^{1, p} (Ω)$ ”

Existence and multiplicity of solutions for a $p (x)$ -Kirchhoff type problem via variational techniques

A. Mokhtari, Toufik Moussaoui, D. O’Regan (2015)

Archivum Mathematicum

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This paper discusses the existence and multiplicity of solutions for a class of $p (x)$ -Kirchhoff type problems with Dirichlet boundary data of the following form

$\}\begin{matrix} - (a + b \int_{Ω} \frac{1}{p (x)} {| \nabla u |}^{p (x)} d x) {div (| \nabla u |}^{p (x) - 2} \nabla u) = f (x, u), & i n Ω \\ u = 0 & o n \partial Ω, \end{matrix}$ where $Ω$ is a smooth open subset of $ℝ^{N}$ and $p \in C (\bar{Ω})$ with $N < p^{-} = {inf}_{x \in Ω} p (x) \leq p^{+} = {sup}_{x \in Ω} p (x) < + \infty$ , $a$ , $b$ are positive constants and $f : \bar{Ω} \times ℝ \to ℝ$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

On D-connected sets in the space $V^{q}$

S. Topa (1976)

Annales Polonici Mathematici

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Domination numbers in graphs with removed edge or set of edges

Magdalena Lemańska (2005)

Discussiones Mathematicae Graph Theory

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It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number $γ_{w}$ and the connected domination number $γ_{c}$ , i.e., we show that $γ_{w} (G) \leq γ_{w} (G - e) \leq γ_{w} (G) + 1$ and $γ_{c} (G) \leq γ_{c} (G - e) \leq γ_{c} (G) + 2$ if G and G-e are connected. Additionally we show that $γ_{w} (G) \leq γ_{w} (G - E ₚ) \leq γ_{w} (G) + p - 1$ and $γ_{c} (G) \leq γ_{c} (G - E ₚ) \leq γ_{c} (G) + 2 p - 2$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order...

Does the endomorphism poset $P^{P}$ determine whether a finite poset $P$ is connected? An issue Duffus raised in 1978

Jonathan David Farley (2023)

Mathematica Bohemica

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Duffus wrote in his 1978 Ph.D. thesis, “It is not obvious that $P$ is connected and $P^{P} ≅ Q^{Q}$ imply that $Q$ is connected”, where $P$ and $Q$ are finite nonempty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P ≅ Q$ .

Connected components of sets of finite perimeter and applications to image processing

Luigi Ambrosio, Vicent Caselles, Simon Masnou, Jean-Michel Morel (2001)

Journal of the European Mathematical Society

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This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in $ℝ^{N}$ , introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the so-called $M$ -connected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set...

$L_{p}$ inequalities for the growth of polynomials with restricted zeros

Nisar A. Rather, Suhail Gulzar, Aijaz A. Bhat (2022)

Archivum Mathematicum

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Let $P (z) = \sum_{ν = 0}^{n} a_{ν} z^{ν}$ be a polynomial of degree at most $n$ which does not vanish in the disk $| z | < 1$ , then for $1 \leq p < \infty$ and $R > 1$ , Boas and Rahman proved

${∥P (R z)∥}_{p} \leq ({∥R^{n} + z∥}_{p} / {∥1 + z∥}_{p}) {∥P∥}_{p} .$ In this paper, we improve the above inequality for $0 \leq p < \infty$ by involving some of the coefficients of the polynomial $P (z)$ . Analogous result for the class of polynomials $P (z)$ having no zero in $| z | > 1$ is also given.

Geometric rigidity of $\times m$ invariant measures

Michael Hochman (2012)

Journal of the European Mathematical Society

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Let $μ$ be a probability measure on $[0, 1]$ which is invariant and ergodic for $T_{a} (x) = a x 𝚖𝚘𝚍 1$ , and $0 < 𝚍𝚒𝚖 μ < 1$ . Let $f$ be a local diffeomorphism on some open set. We show that if $E \subseteq ℝ$ and $(f μ) {|_{E} \sim μ|}_{E}$ , then $f^{'} (x) \in \{\pm a^{r} : r \in ℚ\}$ at $μ$ -a.e. point $x \in f^{- 1} E$ . In particular, if $g$ is a piecewise-analytic map preserving $μ$ then there is an open $g$ -invariant set $U$ containing supp $μ$ such that $g |_{U}$ is piecewise-linear with slopes which are rational powers of $a$ . In a similar vein, for $μ$ as above, if $b$ is another integer and $a, b$ are not powers of a common integer, and if $ν$ is...

On the topology of polynomials with bounded integer coefficients

De-Jun Feng (2016)

Journal of the European Mathematical Society

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For a real number $q > 1$ and a positive integer $m$ , let $Y_{m} (q) : = \{\sum_{i = 0}^{n} ϵ_{i} q^{i} : ϵ_{i} \in \{0, \pm 1, ..., \pm m\}, n = 0, 1, ...\}$ . In this paper, we show that $Y_{m} (q)$ is dense in $ℝ$ if and only if $q < m + 1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].

Essential norms of the Neumann operator of the arithmetical mean

Josef Král, Dagmar Medková (2001)

Mathematica Bohemica

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Let $K \subset ℝ^{m}$ ( $m \geq 2$ ) be a compact set; assume that each ball centered on the boundary $B$ of $K$ meets $K$ in a set of positive Lebesgue measure. Let $C_{0}^{(1)}$ be the class of all continuously differentiable real-valued functions with compact support in $ℝ^{m}$ and denote by $σ_{m}$ the area of the unit sphere in $ℝ^{m}$ . With each $ϕ \in C_{0}^{(1)}$ we associate the function

$W_{K} ϕ (z) = \frac{1}{σ_{m}} \underset{ℝ^{m} ∖ K}{\to} \int g r a d ϕ (x) \cdot \frac{z - x}{{| z - x |}^{m}} x$ of the variable $z \in K$ (which is continuous in $K$ and harmonic in $K ∖ B$ ). $W_{K} ϕ$ depends only on the restriction ${ϕ |}_{B}$ of $ϕ$ to the boundary $B$ of $K$ . This gives rise to a linear operator $W_{K}$ ...

A Hardy type inequality for $W_{0}^{m, 1} (Ω)$ functions

Hernán Castro, Juan Dávila, Hui Wang (2013)

Journal of the European Mathematical Society

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We consider functions $u \in W_{0}^{m, 1} (Ω)$ , where $Ω \subset ℝ^{N}$ is a smooth bounded domain, and $m \geq 2$ is an integer. For all $j \geq 0, 1 \leq k \leq m - 1$ , such that $1 \leq j + k \leq m$ , we prove that $\frac{\partial^{i} u (x)}{d {(x)}^{m - j - k}} \in W_{0}^{k, 1} (Ω)$ with $∥\partial^{k} {(\frac{\partial^{i} u (x)}{d {(x)}^{m - j - k}})}_{L^{1} (Ω)} \leq C∥ u ∥_{W^{m, 1} (Ω)}$ , where $d$ is a smooth positive function which coincides with dist $(x, \partial Ω)$ near $\partial Ω$ , and $\partial^{l}$ denotes any partial differential operator of order $l$ .

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X, d, μ)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B (x, r)$ converge to $f (x)$ when $r$ converges to $0$ . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

Calculus of variations with differential forms

Saugata Bandyopadhyay, Bernard Dacorogna, Swarnendu Sil (2015)

Journal of the European Mathematical Society

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We study integrals of the form $\int_{Ω} f (d ω)$ , where $1 \leq k \leq n$ , $f : Λ^{k} \to ℝ$ is continuous and $ω$ is a $(k - 1)$ -form. We introduce the appropriate notions of convexity, namely ext. one convexity, ext. quasiconvexity and ext. polyconvexity. We study their relations, give several examples and counterexamples. We finally conclude with an application to a minimization problem.

Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a function $f \in L_{l o c}^{p} (ℝ ⁿ)$ the notion of p-mean variation of order 1, $₁^{p} (f, ℝ ⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^{1, p} (ℝ ⁿ)$ in terms of $₁^{p} (f, ℝ ⁿ)$ is directly related to the characterisation of $W^{1, p} (ℝ ⁿ)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

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Displaying similar documents to “A limit involving functions in $W_{0}^{1, p} (Ω)$ ”