# [Seminario] Seminario CAPDE Gyula Csato

Duvan Henao dhenao en mat.puc.cl
Jue Jun 2 09:12:57 CLT 2016

Estimados todos,

los invitamos cordialmente al seminario CAPDE de este

lunes 6 de junio, 5:00pm

Sala 5 Facultad de Matemáticas, P.U.C.

Gyula Csató (Universidad de Concepción )

" About Hardy-Sobolev, Moser-Trudinger and isoperimetric inequalities with densities "

The standard isoperimetric inequality states that among all sets with a given fixed volume (or area in dimension 2) the ball has the smallest perimeter. [See the attached pdf version of the abstract for a more precise statement.] The isoperimetric problem with density is a generalization of this question: given two positive functions f and g from R^2 to R, one studies the existence of minimizers of $\int_{\partial \Omega} g(x)$ among all domains $\Omega$ such that $\int_\Omega f(x)$ equals a fixed given constant. I will talk about some results when f(x)=|x|^q and g(x)=|x|^p, where p,q are real numbers. This is a rich problem with strong variations in difficulties depending on the values of p and q. I will first give an overview on Sobolev, Hardy-Sobolev and Moser-Trudinger inequalities and establish a different kind of connections to isoperimetric inequalities with densities. Finally I will present some of the results appearing in the following references:

1. Csató G., An isoperimetric problem with density and the Hardy-Sobolev inequality in R^2, Differential Integral Equations, 28, Number 9/10 (2015), 971-988.
2. Csató G. and Roy P., Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var. Partial Differential Equations, 54, Issue 2 (2015), 2341-2366.
3. Csató G. and Roy P., The singular Moser-Trudinger inequality on simply connected domains, Comm. Partial Differential Equations, to appear.

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Esperamos contar con su presencia,

Núcleo Milenio Centro para el Análisis de Ecuaciones en Derivadas Parciales

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