scholarly journals On the Existence of Extremals for Moser-Type Inequalities in Gauss Space

2020 ◽  
Author(s):  
Andrea Cianchi ◽  
Vít Musil ◽  
Luboš Pick
Keyword(s):  
Euclidean Space ◽  
Type Inequality ◽  
Sobolev Type ◽  
Optimal Constant ◽  
Classical Inequality ◽  
Gauss Space ◽  
Sobolev Type Inequality

Abstract The existence of an extremal in an exponential Sobolev-type inequality, with optimal constant, in Gauss space is established. A key step in the proof is an augmented version of the relevant inequality, which, by contrast, fails for a parallel classical inequality by Moser in the Euclidean space.

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.

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