Seminar Analysis: Gyula Csató (Technische Universität Dortmund)

Results and conjectures about some isoperimetric problems with density

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. That is, written here for simplicity in dimension 2, the following infimum is attained by the ball

2πR= inf{∫_∂Ω 1 dσ(x) : Ω⊂R² and ∫_Ω 1 dx=πR²}.

The isoperimetric problem with density is a generalization of this question: given two positive functions f,g:R²→R² one studies the existence of minimizers of

I(C) = inf{∫_∂Ω g(x) dσ(x) : Ω⊂R² and ∫_Ω f(x) dx=C}.

I will mainly talk about the situation when f(x) =|x|q and g(x) =|x|p.This is a reach problem with strong variations in difficulties depending on the values of p and q. Some cases are still an open problem. One case has an interesting application related to the Moser-Trudinger imbedding. I will also mention the situation when f=g=eψ is strictly positive and radial, which leads to the log-convex density conjecture.

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