Towards Optimal Gradient Bounds for the Torsion Function in the Plane
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Towards Optimal Gradient Bounds for the Torsion Function in the Plane Jeremy G. Hoskins1 · Stefan Steinerberger2 Received: 17 May 2020 / Accepted: 6 October 2020 © Mathematica Josephina, Inc. 2020
Abstract Let ⊂ R2 be a bounded, convex domain and let u be the solution of −u = 1 vanishing on the boundary ∂. The estimate ∇u L ∞ () ≤ c||1/2 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 < (2π )−1/2 . We also give a numerical construction showing that the optimal constant satisfies c ≥ 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. Keywords Torsion · Saint Venant theory · Gradient estimate · Brownian motion · Potential theory · Hermite–Hadamard inequality · Maximum shear stress · Subharmonic functions Mathematics Subject Classification 28A75 · 31A05 · 31B05 · 35B50 · 60J65 · 60J70 · 74B05
S.S. is supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.
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Stefan Steinerberger [email protected] Jeremy G. Hoskins [email protected]
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Department of Statistics, The University of Chicago, Chicago, IL 60637, USA
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Department of Mathematics, University of Washington, Seattle, WA 98195, USA
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J. G. Hoskins, S. Steinerberger
1 Introduction and Result We study a basic question for elliptic partial differential equations in the plane that arises naturally in a variety of contexts. Let ⊂ R2 be a convex domain and let −u = 1 u=0
inside on ∂.
The purpose of this paper is to study a shape optimization problem and to establish bounds on the gradient. More precisely, our main result is as follows. Theorem 1 (Main result) Let ⊂ R2 be a convex, bounded domain and let u denote the solution of −u = 1 with Dirichlet boundary conditions. For some universal √ c < 1/ 2π ∼ 0.398, ∇u L ∞ () ≤ c · ||1/2 and the constant c cannot be replaced by 0.358.
√ Though this is only a very slight improvement over the bound 1/ 2π obtained in [12], our proof uses very different arguments, which is somewhat robust and suggestive of future directions. One could presumably make the improvement explicit, though it would result in a very small number (something like ∼ 10−10 or possibly even less as it depends on other constants for which only non-optimal estimates are available). Small improvements of this nature are not unusual, we refer to Bonk [17] or Bourgain [18] for other examples. As in those cases, the main advance is not the new constant but the new ideas that allow for an improvement. We also have high precision numerical results that imply an improved lower bound and possibly point the way to a better understanding of the optimal shape (Fig. 1). We note that, in contrast to many other shape optimization p
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