Hardy and Sobolev inequalities in the half space

2019 ◽  
Vol 161 (1) ◽  
pp. 230-244
Author(s):  
Y. Mizuta ◽  
T. Shimomura
Keyword(s):  
Half Space ◽  
Sobolev Inequalities

scholarly journals Sobolev inequalities for fractional Neumann Laplacians on half spaces

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Roberta Musina ◽  
Alexander I. Nazarov
Keyword(s):  
Half Space ◽  
Sobolev Inequalities ◽  
Neumann Laplacian ◽  
Sobolev Constants

Abstract We consider different fractional Neumann Laplacians of order {s\in(0,1)} on domains {\Omega\subset\mathbb{R}^{n}} , namely, the restricted Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{R}}}} , the semirestricted Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sr}}}} and the spectral Neumann Laplacian {{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sp}}}} . In particular, we are interested in the attainability of Sobolev constants for these operators when Ω is a half-space.

scholarly journals Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups

2020 ◽  
Vol 10 (03) ◽  
pp. 2050016
Author(s):  
Michael Ruzhansky ◽  
Bolys Sabitbek ◽  
Durvudkhan Suragan
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Hardy Inequality ◽  
Euclidean Distance ◽  
Sobolev Inequalities ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Hardy Inequalities ◽  
Angle Function ◽  
Distance To The Boundary

In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

scholarly journals New Sharp Gagliardo–Nirenberg–Sobolev Inequalities and an Improved Borell–Brascamp–Lieb Inequality

2018 ◽  
Vol 2020 (10) ◽  
pp. 3042-3083 ◽  
Author(s):  
François Bolley ◽  
Dario Cordero-Erausquin ◽  
Yasuhiro Fujita ◽  
Ivan Gentil ◽  
Arnaud Guillin
Keyword(s):  
Half Space ◽  
Minkowski Inequality ◽  
Point Of View ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Geometric Point ◽  
Functional Point

Abstract We propose a new Borell–Brascamp–Lieb inequality that leads to novel sharp Euclidean inequalities such as Gagliardo–Nirenberg–Sobolev inequalities in $ {\mathbb{R}}^n$ and in the half-space $ {\mathbb{R}}^n_+$. This gives a new bridge between the geometric point of view of the Brunn–Minkowski inequality and the functional point of view of the Sobolev-type inequalities. In this way we unify, simplify, and generalize results by S. Bobkov–M. Ledoux, M. del Pino–J. Dolbeault, and B. Nazaret.

Weighted Hardy–Littlewood–Sobolev inequalities on the upper half space

2016 ◽  
Vol 18 (05) ◽  
pp. 1550067 ◽  
Author(s):  
Jingbo Dou
Keyword(s):  
Half Space ◽  
Type Inequality ◽  
Boundary Term ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Biharmonic Operator ◽  
Trace Inequality ◽  
Hardy Type Inequality ◽  
Hardy Type

In this paper, we establish a weighted Hardy–Littlewood–Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. As an application, we can show a weighted Sobolev–Hardy trace inequality with [Formula: see text]-biharmonic operator.

On a technique for deriving explicit transfer matrices of orthotropic layers

2015 ◽  
Vol 37 (4) ◽  
pp. 303-315 ◽  
Author(s):  
Pham Chi Vinh ◽  
Nguyen Thi Khanh Linh ◽  
Vu Thi Ngoc Anh
Keyword(s):  
Wave Propagation ◽  
Explicit Form ◽  
Half Space ◽  
Transfer Matrix ◽  
Rayleigh Waves ◽  
Secular Equation ◽  
Transfer Matrices ◽  
Orthotropic Layer ◽  
Wave Propagation Problem ◽  
Arbitrary Thickness

This paper presents  a technique by which the transfer matrix in explicit form of an orthotropic layer can be easily obtained. This transfer matrix is applicable for both the wave propagation problem and the reflection/transmission problem. The obtained transfer matrix is then employed to derive the explicit secular equation of Rayleigh waves propagating in an orthotropic half-space coated by an orthotropic layer of arbitrary thickness.

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