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.

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 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 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.

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.

scholarly journals A Sharp Multidimensional Hermite–Hadamard Inequality

2020 ◽  
Author(s):  
Simon Larson
Keyword(s):  
Recent Result ◽  
Convex Domain ◽  
Subharmonic Function ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Convex Domains ◽  
Hadamard Inequality ◽  
Bounded Convex Domain ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Bounded Convex

Abstract Let $\Omega \subset {\mathbb{R}}^d $, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to{\mathbb{R}}$ be a non-negative subharmonic function. In this paper, we prove the inequality $$\begin{equation*} \frac{1}{|\Omega|}\int_{\Omega} f(x)\, \textrm{d}x \leq \frac{d}{|\partial\Omega|}\int_{\partial\Omega} f(x)\, \textrm{d}\sigma(x)\,. \end{equation*}$$Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $\Omega \subset{\mathbb{R}}^d$ is a bounded convex domain and $u$ is the solution of $-\Delta u =1$ with homogeneous Dirichlet boundary conditions, then $$\begin{equation*} \|\nabla u\|_{L^\infty(\Omega)} &lt; d\frac{|\Omega|}{|\partial\Omega|}\,. \end{equation*}$$Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al. [2].

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