scholarly journals Sharp numerical inclusion of the best constant for embeddingH01(Ω)↪Lp(Ω)on bounded convex domain

2017 ◽  
Vol 311 ◽  
pp. 306-313 ◽  
Author(s):  
Kazuaki Tanaka ◽  
Kouta Sekine ◽  
Makoto Mizuguchi ◽  
Shin’ichi Oishi
Keyword(s):  
Convex Domain ◽  
Best Constant ◽  
Bounded Convex Domain ◽  
Bounded Convex

scholarly journals REFINED GEOMETRIC Lp HARDY INEQUALITIES

2003 ◽  
Vol 05 (06) ◽  
pp. 869-881 ◽  
Author(s):  
G. BARBATIS ◽  
S. FILIPPAS ◽  
A. TERTIKAS
Keyword(s):  
Finite Number ◽  
Convex Domain ◽  
Hardy Potential ◽  
Hardy Inequalities ◽  
Best Constant ◽  
Bounded Convex Domain ◽  
Logarithmic Corrections ◽  
Bounded Convex ◽  
Distance To The Boundary

For a bounded convex domain Ω in RN we prove refined Hardy inequalities that involve the Hardy potential corresponding to the distance to the boundary of Ω, the volume of Ω, as well as a finite number of sharp logarithmic corrections. We also discuss the best constant of these inequalities.

On the monotonicity of the principal frequency of the p-Laplacian

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Marian Bocea ◽  
Mihai Mihăilescu
Keyword(s):  
Real Number ◽  
Convex Domain ◽  
Distance Function ◽  
Smooth Boundary ◽  
Monotone Function ◽  
Fixed Integer ◽  
Bounded Convex Domain ◽  
Principal Frequency ◽  
Bounded Convex ◽  
Increasing Function

Abstract For any fixed integer {D>1} we show that there exists {M\in[e^{-1},1]} such that for any open, bounded, convex domain {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary for which the maximum of the distance function to the boundary of Ω is less than or equal to M, the principal frequency of the p-Laplacian on Ω is an increasing function of p on {(1,\infty)} . Moreover, for any real number {s>M} there exists an open, bounded, convex domain {\Omega\subset{\mathbb{R}}^{D}} with smooth boundary which has the maximum of the distance function to the boundary of Ω equal to s such that the principal frequency of the p-Laplacian is not a monotone function of {p\in(1,\infty)} .

scholarly journals Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$

2017 ◽  
Vol 120 (2) ◽  
pp. 305
Author(s):  
Željko Čučković ◽  
Sönmez Şahutoğlu
Keyword(s):  
Convex Domain ◽  
Hankel Operator ◽  
Smooth Boundary ◽  
Essential Norm ◽  
Hankel Operators ◽  
Convex Domains ◽  
Norm Estimates ◽  
Bounded Convex Domain ◽  
Derivatives Of ◽  
Bounded Convex

Let $\Omega \subset \mathbb{C}^2$ be a bounded convex domain with $C^1$-smooth boundary and $\varphi \in C^1(\overline{\Omega})$ such that $\varphi $ is harmonic on the non-trivial disks in the boundary. We estimate the essential norm of the Hankel operator $H_{\varphi }$ in terms of the $\overline{\partial}$ derivatives of $\varphi$ “along” the non-trivial disks in the boundary.

scholarly journals Interior regularity of the degenerate Monge-Ampère equation

2003 ◽  
Vol 68 (1) ◽  
pp. 81-92 ◽  
Author(s):  
Zbigniew Błocki
Keyword(s):  
Convex Domain ◽  
Main Tool ◽  
Degenerate Case ◽  
Strongly Convex ◽  
Bounded Convex Domain ◽  
Interior Regularity ◽  
Monge Ampere ◽  
Bounded Convex

We study interior C1,1 regularity of generalised solutions of the Monge-Ampére equation det D2u = ψ, ψ ≥ 0, on a bounded convex domain Ω in ℝn with u = ϕ on ∂Ω. We prove in particular that u ∈ C1,1(Ω) if either i) ϕ = 0 and ψ1/(n − 1) ∈ C1,1 (Ω) or ii) Ω is C1,1 strongly convex, ϕ ∈ C1,1 (Ω̅), ψ1/(n − 1) ∈ C1,1(Ω̅) and ψ > 0 on U ∩ Ω, where U is a neighbourhood of ∂Ω. The main tool is an improvement of Pogorelov's well known C1,1 estimate so that it can be applied to the degenerate case.

A hypersurface containing the support of a Radon transform must be an ellipsoid. II: The general case

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jan Boman
Keyword(s):  
Convex Domain ◽  
Radon Transform ◽  
Symmetric Case ◽  
Compactly Supported ◽  
Bounded Convex Domain ◽  
Bounded Convex ◽  
Special Case

Abstract If the Radon transform of a compactly supported distribution f ≠ 0 {f\neq 0} in ℝ n {\mathbb{R}^{n}} is supported on the set of tangent planes to the boundary ∂ ⁡ D {\partial D} of a bounded convex domain D, then ∂ ⁡ D {\partial D} must be an ellipsoid. The special case of this result when the domain D is symmetric was treated in [J. Boman, A hypersurface containing the support of a Radon transform must be an ellipsoid. I: The symmetric case, J. Geom. Anal. 2020, 10.1007/s12220-020-00372-8]. Here we treat the general case.

scholarly journals The power concavity of solutions of some semilinear elliptic boundary-value problems

1985 ◽  
Vol 31 (2) ◽  
pp. 181-184 ◽  
Author(s):  
Grant Keady
Keyword(s):  
Boundary Value Problems ◽  
Convex Domain ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Bounded Convex Domain ◽  
Semilinear Elliptic ◽  
Image Position ◽  
Bounded Convex

Let Ω be a bounded convex domain in R2 with a smooth boundary. Let 0 < γ < 1. Let be a solution, positive in Ω, ofThen the function uα is concave for α = (l–γ)/2.

scholarly journals Some Weighted Norm Estimates for the Composition of the Homotopy and Green’s Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Huacan Li ◽  
Qunfang Li
Keyword(s):  
Convex Domain ◽  
Upper Bound ◽  
Integral Inequality ◽  
Weighted Norm ◽  
Norm Inequalities ◽  
Norm Estimates ◽  
Bounded Convex Domain ◽  
Green’S Operator ◽  
Weighted Integral ◽  
Bounded Convex

We establish theAr(D)-weighted integral inequality for the composition of the HomotopyTand Green’s operatorGon a bounded convex domain and also motivated it to the global domain by the Whitney cover. At the same time, we also obtain some(p,q)-type norm inequalities. Finally, as applications of above results, we obtain the upper bound for theLpnorms ofT(G(u))or(T(G(u)))Bin terms ofLqnorms ofuordu.

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