On the Torsion Function for Simply Connected, Open Sets in $$\mathbb {R}^2$$

Abstract

For an open set $$\Omega \subset \mathbb {R}^2$$ Ω ⊂ R 2 let $$\lambda (\Omega )$$ λ ( Ω ) denote the bottom of the spectrum of the Dirichlet Laplacian acting in $$L^2(\Omega )$$ L 2 ( Ω ) . Let $$w_\Omega $$ w Ω be the torsion function for $$\Omega $$ Ω , and let $$\Vert .\Vert _p$$ ‖ . ‖ p denote the $$L^p$$ L p norm. It is shown that there exist $$\eta _1>0,\eta _2>0$$ η 1 > 0 , η 2 > 0 such that (i) $$\Vert w_{\Omega }\Vert _{\infty } \lambda (\Omega )\ge 1+\eta _1$$ ‖ w Ω ‖ ∞ λ ( Ω ) ≥ 1 + η 1 for any non-empty, open, simply connected set $$\Omega \subset \mathbb {R}^2$$ Ω ⊂ R 2 with $$\lambda (\Omega ) >0$$ λ ( Ω ) > 0 , (ii) $$\Vert w_{\Omega }\Vert _1\lambda (\Omega )\le {(1-\eta _2)}|\Omega |$$ ‖ w Ω ‖ 1 λ ( Ω ) ≤ ( 1 - η 2 ) | Ω | for any non-empty, open, simply connected set $$\Omega \subset \mathbb {R}^2$$ Ω ⊂ R 2 with finite measure $$|\Omega |$$ | Ω | .