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- W3106236473 abstract "Let $$Omega subset mathbb {R}^2$$ be a bounded, convex domain and let u be the solution of $$-Delta u = 1$$ vanishing on the boundary $$partial Omega $$ . The estimate $$begin{aligned} Vert nabla uVert _{L^{infty }(Omega )} le c |Omega |^{1/2} end{aligned}$$ is classical. We use the P-functional, the stability theory of the torsion function and Brownian motion to establish the estimate for a universal $$c < (2pi )^{-1/2}$$ . We also give a numerical construction showing that the optimal constant satisfies $$c ge 0.358$$ . The problem is important in different settings: (1) as the maximum shear stress in Saint Venant Elasticity Theory, (2) as an optimal control problem for the constrained maximization of the lifetime of Brownian motion started close to the boundary, and (3) optimal Hermite–Hadamard inequalities for subharmonic functions on convex domains." @default.
- W3106236473 created "2020-11-23" @default.
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- W3106236473 date "2020-11-09" @default.
- W3106236473 modified "2023-10-18" @default.
- W3106236473 title "Towards Optimal Gradient Bounds for the Torsion Function in the Plane" @default.
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- W3106236473 doi "https://doi.org/10.1007/s12220-020-00553-5" @default.
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