Poincaré inequalities in punctured domains.(English)Zbl 1048.26012

Summary: The classic Poincaré inequality bounds the $$L^q$$-norm of a function $$f$$ in a bounded domain $$\Omega\subset\mathbb{R}^n$$ 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 $$\nabla+iA(x)$$ (geometrically, a connection on a $$U(1)$$ bundle). Unlike the $$A=0$$ case, the infimum of $$\|(\nabla+iA)f \|_p$$ over all $$f$$ with a given $$\| f\|_q$$ 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.

MSC:

 26D10 Inequalities involving derivatives and differential and integral operators 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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