Poincaré inequalities in punctured domains

Abstract

The classic Poincaré inequality bounds the $L^q$-norm of a function $f$ in a bounded domain $\Omega \subset \mathrm{R}^n$ in terms of some $L^p$-norm of its gradient in $\Omega$. We generalize this in two ways: In the first generalization we remove a set $\Gamma$ from $\Omega$ and concentrate our attention on $\Lambda = \Omega \setminus \Gamma$. This new domain might not even be connected and hence no Poincaré inequality can generally hold for it, or if it does hold it might have a very bad constant. This is so even if the volume of $\Gamma$ is arbitrarily small. A Poincaré inequality does hold, however, if one makes the additional assumption that $f$ has a finite $L^p$ gradient norm on the whole of $\Omega$, not just on $\Lambda$. The important point is that the Poincaré inequality thus obtained bounds the $L^q$-norm of $f$ in terms of the $L^p$ gradient norm on $\Lambda$ (not $\Omega$) plus an additional term that goes to zero as the volume of $\Gamma$ goes to zero. This error term depends on $\Gamma$ only through its volume. Apart from this additive error term, the constant in the inequality remains that of the ‘nice’ domain $\Omega$. In the second generalization we are given a vector field $A$ and replace $\nabla $ by $\nabla +i A(x)$ (geometrically, a connection on a $U(1)$ bundle). Unlike the $A=0$ case, the infimum of $\Vert (\nabla +i A)f \Vert_p $ over all $f$ with a given $\Vert f \Vert_q$ is in general not zero. This permits an improvement of the inequality by the addition of a term whose sharp value we derive. We describe some open problems that arise from these generalizations.

Authors

Elliott H. Lieb

Department of Mathematics and Physics
Princeton University
Princeton, NJ 08544
United States

Robert Seiringer

Department of Physics
Princeton University
Princeton, NJ 08544
United States

Jakob Yngvason

Institute for Theoretical Physics
University of Vienna
Vienna
Austria