SEMINAR @ SISSSA, prof. F. Robert - 13 Feb at 16 hrs.

[Barbara Corzani]

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FUNCTIONAL ANALYSIS SECTOR'S ACTIVITIES
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SEMINAR 

Prof. 
Frédéric Robert
(Université de Nice Sophia Antipolis)


Title:
"Extremals for Hardy­Sobolev type inequalities: the influence of the cur­vature." (Joint work with N.Ghoussoub)

Wed.  13 Feb. 2008  @  04.00 p.m.

SISSA - Main Building - ground floor - room B

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ABSTRACT:

We consider the optimal Hardy­Sobolev inequality on a smooth bounded domain of the Euclidean space. Roughly speaking, this inequality lies between the Hardy inequality and the Sobolev inequality. We address the questions of the value of the optimal constant and the existence of 
non­trivial extremals attached  to this inequality. When the singularity of the Hardy part is located on the  boundary of the domain, the geometry of the domain plays a crucial role: in particular, the convexity and the mean curvature are involved in these questions. The main difficulty to 
encounter is the possible bubbling phenomenon. We describe precisely this bubbling through refined concentration estimates. An offshot of these 
techniques allow us to provide general compactness properties for nonlinear equations, still under curvature conditions for the boundary of the domain.