Non-local Gel'fand problem in higher dimensions

Tosiya Miyasita; Takashi Suzuki

Displaying similar documents to “Non-local Gel'fand problem in higher dimensions”

On connectivity properties of hyperspaces of $R_{0}$ spaces

M. Mršević (1979)

Matematički Vesnik

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Local connectedness, connectedness im kleinen, and another properties of hyperspaces of $R_{0}$ spaces

C. Dorsett (1979)

Matematički Vesnik

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La Queste del Saint $G R_{a}$ (AL): a Computational Approach to Local Algebra

Teo Mora (1989)

Publications mathématiques et informatique de Rennes

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The Bohr inequality for ordinary Dirichlet series

R. Balasubramanian, B. Calado, H. Queffélec (2006)

Studia Mathematica

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We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f (s) = \sum_{n = 1}^{\infty} a ₙ n^{- s}$ with ${| | f | |}_{\infty} : = s u p_{ℜ s > 0} | f (s) | < \infty$ , then $\sum_{n = 1}^{\infty} | a ₙ | n^{- 2} {\leq | | f | |}_{\infty}$ and even slightly better, and $\sum_{n = 1}^{\infty} | a ₙ | n^{- 1 / 2} {\leq C | | f | |}_{\infty}$ , C being an absolute constant.

Continuous pluriharmonic boundary values

Per Åhag, Rafał Czyż (2007)

Annales Polonici Mathematici

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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

The comparison principle and Dirichlet problem in the class $_{p} (f)$ , p > 0

Pham Hoang Hiep (2006)

Annales Polonici Mathematici

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We establish the comparison principle in the class $_{p} (f)$ . The result obtained is applied to the Dirichlet problem in $_{p} (f)$ .

On Dirichlet type spaces on the unit ball of $ℂ^{n}$

Małgorzata Michalska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we discuss characterizations of Dirichlet type spaces on the unit ball of $ℂ^{n}$ obtained by P. Hu and W. Zhang [2], and S. Li [4].

On Dirichlet-Schrödinger operators with strong potentials

Gabriele Grillo (1995)

Studia Mathematica

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We consider Schrödinger operators H = -Δ/2 + V (V≥0 and locally bounded) with Dirichlet boundary conditions, on any open and connected subdomain $D \subset ℝ^{n}$ which either is bounded or satisfies the condition $d (x, D^{c}) \to 0$ as |x| → ∞. We prove exponential decay at the boundary of all the eigenfunctions of H whenever V diverges sufficiently fast at the boundary ∂D, in the sense that $d {(x, D^{C})}^{2} V (x) \to \infty$ as $d (x, D^{C}) \to 0$ . We also prove bounds from above and below for Tr(exp[-tH]), and in particular we give criterions for the finiteness of...

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

S. Topa (1976)

Annales Polonici Mathematici

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On some $L^{p}$ -estimates for solutions of elliptic equations in unbounded domains

Sara Monsurrò, Maria Transirico (2015)

Mathematica Bohemica

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In this review article we present an overview on some a priori estimates in $L^{p}$ , $p > 1$ , recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^{p}$ -bounds, $p > 2$ , for...

$H^{p}$ -spaces of conjugate systems on local fields

Jia Chao (1974)

Studia Mathematica

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On discrete mean values of Dirichlet $L$ -functions

Ertan Elma (2021)

Czechoslovak Mathematical Journal

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Let $χ$ be a nonprincipal Dirichlet character modulo a prime number $p ⩾ 3$ and let $𝔞_{χ} : = \frac{1}{2} (1 - χ (- 1))$ . Define the mean value $ℳ_{p} (- s, χ) : = \frac{2}{p - 1} \sum ψ (mod p) ψ (- 1) = - 1 L (1, ψ) L (- s, χ \bar{ψ}) (σ : = ℜ s > 0) .$ We give an identity for $ℳ_{p} (- s, χ)$ which, in particular, shows that $ℳ_{p} (- s, χ) = L (1 - s, χ) + 𝔞_{χ} 2 p^{s} L (1, χ) ζ (- s) + o (1) (p \to \infty)$ for fixed $0 < σ < \frac{1}{2}$ and $| t : = ℑ s | = o (p^{(1 - 2 σ) / (3 + 2 σ)})$ .

Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed

Marek Fila, Michael Winkler (2008)

Journal of the European Mathematical Society

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We consider a reaction-diffusion-convection equation on the halfline (0,1) with the zero Dirichlet boundary condition at $x = 0$ . We find a positive selfsimilar solution $u$ which blows up in a finite time $T$ at $x = 0$ while $u (x, T)$ remains bounded for $x > 0$ .

The Dirichlet-Bohr radius

Daniel Carando, Andreas Defant, Domingo A. Garcí, Manuel Maestre, Pablo Sevilla-Peris (2015)

Acta Arithmetica

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Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial $\sum_{n \leq x} a_{n} n^{- s}$ we have $\sum_{n \leq x} | a_{n} | r^{Ω (n)} \leq s u p_{t \in ℝ} | \sum_{n \leq x} a_{n} n^{- i t} |$ . We prove that the asymptotically correct order of L(x) is ${(l o g x)}^{1 / 4} x^{- 1 / 8}$ . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results...

Characterizations of $H_{x} 1$ -sufficiency of jets

Le Tien Tam (1986)

Annales Polonici Mathematici

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Estimates near the boundary for second order derivatives of solutions of the Dirichlet problem for the biharmonic equation

Vladimir A. Kondratiev, Olga A. Oleinik (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Per ogni soluzione della (1) nel dominio limitato $\Omega$ ,, appartenente a $H_{0}^{2}(\Omega)$ e soddisfacente le condizioni (2), si dimostra la maggiorazione (5), valida nell'intorno di ogni punto $x^{0}$ del contorno; si consente a $\partial\Omega$ di essere singolare in $x^{0}$ .

Weak and strong density results for the Dirichlet energy

Mariano Giaquinta, Domenico Mucci (2004)

Journal of the European Mathematical Society

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Let $𝒴$ be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in $B^{n} \times 𝒴$ with equibounded Dirichlet energies, $B^{n}$ being the unit ball in $ℝ^{n}$ . More precisely, weak limits of graphs of smooth maps $u_{k} : B^{n} \to 𝒴$ with equibounded Dirichlet integral give rise to elements of the space ${cart}^{2, 1} (B^{n} \times 𝒴)$ (cf. [4], [5], [6]). In this paper we prove that every element $T$ in ${cart}^{2, 1} (B^{n} \times 𝒴)$ is the...

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p, q)$ -growth with exponents $p \leq q < \infty$ and show for the scalar case that locally bounded local minimizers are of class $C^{1, μ}$ . Note that to our knowledge the only $C^{1, μ}$ -results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators

Dachun Yang, Sibei Yang

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Let Φ be a concave function on (0,∞) of strictly critical lower type index $p_{Φ} \in (0, 1]$ and $ω \in A_{\infty}^{l o c} (ℝ ⁿ)$ (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space $h_{ω}^{Φ} (ℝ ⁿ)$ via the local grand maximal function. Let $ρ (t) \equiv t^{- 1} / Φ^{- 1} (t^{- 1})$ for all t ∈ (0,∞). We also introduce the BMO-type space $b m o_{ρ, ω} (ℝ ⁿ)$ and establish the duality between $h_{ω}^{Φ} (ℝ ⁿ)$ and $b m o_{ρ, ω} (ℝ ⁿ)$ . Characterizations of $h_{ω}^{Φ} (ℝ ⁿ)$ , including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...

A study of various results for a class of entire Dirichlet series with complex frequencies

Niraj Kumar, Garima Manocha (2018)

Mathematica Bohemica

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Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_{k} e^{〈 λ^{k}, z 〉}$ for which $(| λ^{k} {| / e)}^{| λ^{k} |} k! | a_{k} |$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.

Hölder continuity of bounded generalized solutions for some degenerated quasilinear elliptic equations with natural growth terms

Salvatore Bonafede (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation $\sum_{i = 1}^{m} \frac{\partial}{\partial x_{i}} a_{i} (x, u, \nabla u) - c_{0} {| u |}^{p - 2} u = f (x, u, \nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $λ (| u |) \sum_{i = 1}^{m} a_{i} (x, u, η) η_{i} \geq ν (x) {| η |}^{p},$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$ .