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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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