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| Let $n \geq 2$, let $\Omega \subset \mathbf{R}^n$ be a bounded domain with a smooth boundary, and let $1 \leq p \leq \frac{2n}{n-2}$. The Sobolev embedding theorem and Rellich compactness together imply that
$$\mathcal{C}_p(\Omega) = \inf \left \{ \frac{\int_\Omega |\nabla u|^2 d\mu}{ \left ( \int_\Omega |u|^p d\mu \right )^{2/p}} : u \in W^{1,2}_0(\Omega) \right \}$$
is a finite, positive number, and it is realized by a nontrivial function $\phi$. I will discuss some new results linking the number $\mathcal{C}_p(\Omega)$ and its associated extremal function $\phi$ to other, more classical quantities, such as volume, perimeter, principal frequency, and torsional rigidity. The inequalities we can prove generalize some theorems of Polya, Szego, Payne, Rayner, Kohler-Jobin, Chiti, and others. Moreover, we can classify which realize equality in our bounds. This is all joint work with Tom Carroll of the University of Cork, in Ireland. |
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