On the torsion function for simply connected, open sets in \R²

2024/02/22 by M. van den Berg, Berg, Michiel van den, Dorin Bucur +1
Engineering · Mathematics · Computer Science · #Advanced Numerical Analysis Techniques #Fixed Point Theorems Analysis #Digital Image Processing Techniques

paper · pdf · doi:10.48550/arxiv.2402.14448

Abstract

For an open set \Om ⊂ \R² let λ(\Om) denote the bottom of the spectrum of the Dirichlet Laplacian acting in L²(\Om). Let w_\Om be the torsion function for \Om, and let ‖.‖_p denote the L^p norm. It is shown that there exist η₁>0,η₂>0 such that (i) ‖w_\Om‖_∞ λ(\Om)≥ 1+η₁ for any non-empty, open, simply connected set \Om⊂ \R² with \lb(\Om) >0, (ii) ‖w_\Om‖₁λ(\Om)≤ (1-η₂)|\Om| for any non-empty, open, simply connected set \Om⊂\R² with finite measure |\Om|.

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