scholarly journals A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems

2020 ◽  
Vol 268 (10) ◽  
pp. 5996-6032 ◽  
Author(s):  
Quốc Anh Ngô ◽  
Van Hoang Nguyen
Keyword(s):  
Sobolev Spaces ◽  
Type Inequality ◽  
Elliptic Problems ◽  
Higher Order ◽  
Sobolev Type ◽  
Sobolev Type Inequality

A Sharp Adams-Type Inequality for Weighted Sobolev Spaces

2020 ◽  
Vol 71 (2) ◽  
pp. 517-538
Author(s):  
João Marcos do Ó ◽  
Abiel Costa Macedo ◽  
José Francisco de Oliveira
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Sobolev Type ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Classical Work

Abstract In a classical work (Ann. Math.128, (1988) 385–398), D. R. Adams proved a sharp Trudinger–Moser inequality for higher-order derivatives. We derive a sharp Adams-type inequality and Sobolev-type inequalities associated with a class of weighted Sobolev spaces that is related to a Hardy-type inequality.

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

scholarly journals A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

2017 ◽  
Vol 29 (3) ◽  
pp. 515-542 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
T. RICCIARDI ◽  
G. ZECCA
Keyword(s):  
Total Population ◽  
Critical Mass ◽  
Type System ◽  
Type Inequality ◽  
Chemical Substance ◽  
Sobolev Type ◽  
Lyapunov Functionals ◽  
Single Chemical ◽  
Chemotaxis System ◽  
Sobolev Type Inequality

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy–Littlewood–Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

Existence and isoperimetric characterization of steady spherical vortex rings in a uniform flow in RN

2004 ◽  
Vol 134 (3) ◽  
pp. 449-476 ◽  
Author(s):  
Geoffrey R. Burton ◽  
Luca Preciso
Keyword(s):  
Type Inequality ◽  
Fluid Flows ◽  
Semilinear Elliptic Equation ◽  
Vortex Rings ◽  
Sobolev Type ◽  
Semilinear Elliptic ◽  
Stream Functions ◽  
Spherical Vortex ◽  
Sobolev Type Inequality

We study a boundary-value problem for a particular semilinear elliptic equation on Rn (n ≥ 2), whose solutions represent generalized stream functions for steady axisymmetric ideal fluid flows. Solutions are shown to exist that generalize those already known in dimensions 2 and 3. An isoperimetric characterization is given for our solutions, which represent generalized spherical vortex-rings. As a corollary, a sharp Sobolev-type inequality is obtained.

scholarly journals A Higher-Order Hardy-Type Inequality in Anisotropic Sobolev Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Paolo Secchi
Keyword(s):  
Sobolev Spaces ◽  
Hyperbolic Systems ◽  
Type Inequality ◽  
Initial Boundary ◽  
Higher Order ◽  
Hardy Type Inequality ◽  
Characteristic Boundary ◽  
Hardy Type ◽  
Anisotropic Sobolev Spaces ◽  
Initial Boundary Value

We prove a higher-order inequality of Hardy type for functions in anisotropic Sobolev spaces that vanish at the boundary of the space domain. This is an important calculus tool for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary.

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