On the Poincaré Inequality on Open Sets in Rn

We show that the Poincaré inequality holds on an open set D⊂Rn if and only if D admits a smooth, bounded function whose Laplacian has a positive lower bound on D. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on D is equivalent to the finiteness of the strict...

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Bibliographic Details
Published inComputational methods and function theory Vol. 25; no. 1; pp. 213 - 238
Format Journal Article
LanguageEnglish
Published Heidelberg Springer Nature B.V 01.03.2025
Subjects
Eigenvalues
Lower bounds
Upper bounds
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Summary:We show that the Poincaré inequality holds on an open set D⊂Rn if and only if D admits a smooth, bounded function whose Laplacian has a positive lower bound on D. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on D is equivalent to the finiteness of the strict inradius of D measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet–Laplacian.
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ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-024-00550-7