The $p$-Laplace eigenvalue problem as $p\rightarrow \infty $ in a Finsler metric

M. Belloni; Bernhard Kawohl; P. Juutinen

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Displaying similar documents to “The $p$ -Laplace eigenvalue problem as $p \to \infty$ in a Finsler metric”

Extension of smooth functions in infinite dimensions II: manifolds

C. J. Atkin (2002)

Studia Mathematica

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Let M be a separable $C^{\infty}$ Finsler manifold of infinite dimension. Then it is proved, amongst other results, that under suitable conditions of local extensibility the germ of a $C^{\infty}$ function, or of a $C^{\infty}$ section of a vector bundle, on the union of a closed submanifold and a closed locally compact set in M, extends to a $C^{\infty}$ function on the whole of M.

The Morse-Sard-Brown Theorem for Functionals on Bounded Fréchet-Finsler Manifolds

Kaveh Eftekharinasab (2015)

Communications in Mathematics

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In this paper we study Lipschitz-Fredholm vector fields on bounded Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if $M$ is a connected smooth bounded Fréchet-Finsler manifold endowed with a connection $𝒦$ and if $ξ$ is a smooth Lipschitz-Fredholm vector field on $M$ with respect to $𝒦$ which satisfies condition (WCV), then, for any smooth functional $l$ on $M$ which is associated to $ξ$ , the set of the critical values of $l$ is of first category...

On the tangency of sets in generalized metric spaces for certain functions of the class $F_{ρ}^{*}$

T. Konik (1991)

Matematički Vesnik

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Connections partially adapted to a metric $(J^{4} = 1)$ -structure

E. Reyes, A. Montesinos, P. M. Gadea (1987)

Colloquium Mathematicae

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On canonical constructions on connections

Jan Kurek, Włodzimierz Mikulski (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We study how a projectable general connection $Γ$ in a 2-fibred manifold $Y^{2} \to Y^{1} \to Y^{0}$ and a general vertical connection $Θ$ in $Y^{2} \to Y^{1} \to Y^{0}$ induce a general connection $A (Γ, Θ)$ in $Y^{2} \to Y^{1}$ .

On almost everywhere differentiability of the metric projection on closed sets in $l^{p} (ℝ^{n})$ , $2 < p < \infty$

Tord Sjödin (2018)

Czechoslovak Mathematical Journal

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Let $F$ be a closed subset of $ℝ^{n}$ and let $P (x)$ denote the metric projection (closest point mapping) of $x \in ℝ^{n}$ onto $F$ in $l^{p}$ -norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in $ℝ^{n}$ in the Euclidean case $p = 2$ . We consider the case $2 < p < \infty$ and prove that the $i$ th component $P_{i} (x)$ of $P (x)$ is differentiable a.e. if $P_{i} (x) \neq x_{i}$ and satisfies Hölder condition of order $1 / (p - 1)$ if $P_{i} (x) = x_{i}$ .

Isometric embeddings of a class of separable metric spaces into Banach spaces

Sophocles K. Mercourakis, Vassiliadis G. Vassiliadis (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $(M, d)$ be a bounded countable metric space and $c > 0$ a constant, such that $d (x, y) + d (y, z) - d (x, z) \geq c$ , for any pairwise distinct points $x, y, z$ of $M$ . For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $ℓ_{\infty}$ .

Estimates of the principal eigenvalue of the $p$ -Laplacian and the $p$ -biharmonic operator

Jiří Benedikt (2015)

Mathematica Bohemica

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We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$ -Laplacian and the Navier $p$ -biharmonic operator on a ball of radius $R$ in $ℝ^{N}$ and its asymptotics for $p$ approaching $1$ and $\infty$ . Let $p$ tend to $\infty$ . There is a critical radius $R_{C}$ of the ball such that the principal eigenvalue goes to $\infty$ for $0 < R \leq R_{C}$ and to $0$ for $R > R_{C}$ . The critical radius is $R_{C} = 1$ for any $N \in ℕ$ for the $p$ -Laplacian and $R_{C} = \sqrt{2 N}$ in the case of the $p$ -biharmonic operator. When $p$ approaches $1$ , the principal eigenvalue...

On the multiplicity of eigenvalues of conformally covariant operators

Yaiza Canzani (2014)

Annales de l’institut Fourier

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Let $(M, g)$ be a compact Riemannian manifold and $P_{g}$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$ . We prove that if $P_{g}$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f \in C^{\infty} (M, ℝ)$ for which $P_{e^{f} g}$ has only simple non-zero eigenvalues is a residual set in $C^{\infty} (M, ℝ)$ . As a consequence we prove that if $P_{g}$ has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics...

About $w c s$ -covers and $w c s^{*}$ -networks on the Vietoris hyperspace $ℱ (X)$

Luong Quoc Tuyen, Ong V. Tuyen, Phan D. Tuan, Nguzen X. Truc (2023)

Commentationes Mathematicae Universitatis Carolinae

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We study some generalized metric properties on the hyperspace $ℱ (X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $w c s$ -covers if and only if so does $ℱ (X)$ . Moreover, $X$ has a sequence of $w c s$ -covers with property $(P)$ which is a point-star network if and only if so does $ℱ (X)$ , where $(P)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable....

Inequalities for real number sequences with applications in spectral graph theory

Emina Milovanović, Şerife Burcu Bozkurt Altındağ, Marjan Matejić, Igor Milovanović (2022)

Czechoslovak Mathematical Journal

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Let $a = (a_{1}, a_{2}, ..., a_{n})$ be a nonincreasing sequence of positive real numbers. Denote by $S = {1, 2, ..., n}$ the index set and by $J_{k} = {I = {r_{1}, r_{2}, ..., r_{k}}$ , $1 \leq r_{1} < r_{2} < \dots < r_{k} \leq n}$ the set of all subsets of $S$ of cardinality $k$ , $1 \leq k \leq n - 1$ . In addition, denote by $a_{I} = a_{r_{1}} + a_{r_{2}} + \dots + a_{r_{k}}$ , $1 \leq k \leq n - 1$ , $1 \leq r_{1} < r_{2} < \dots < r_{k} \leq n$ , the sum of $k$ arbitrary elements of sequence $a$ , where $a_{I_{1}} = a_{1} + a_{2} + \dots + a_{k}$ and $a_{I_{n}} = a_{n - k + 1} + a_{n - k + 2} + \dots + a_{n}$ . We consider bounds of the quantities $R S_{k} (a) = a_{I_{1}} / a_{I_{n}}$ , $L S_{k} (a) = a_{I_{1}} - a_{I_{n}}$ and $S_{k, α} (a) = \sum_{I \in J_{k}} a_{I}^{α}$ in terms of $A = \sum_{i = 1}^{n} a_{i}$ and $B = \sum_{i = 1}^{n} a_{i}^{2}$ . Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

Displaying similar documents to “The p -Laplace eigenvalue problem as p → ∞ in a Finsler metric”

Displaying similar documents to “The $p$ -Laplace eigenvalue problem as $p \to \infty$ in a Finsler metric”