For fixed drift strength and fixed container volume, the ball, together with an inward drift aligned with the expected-lifetime gradient, maximizes the total gradient and Laplacian of the expected exit time.
Beck, The torsion function of convex domains of high ec centricity, Potential Analysis, to appear
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Optimal Trapping of Brownian Motion: A Nonlinear Analogue of the Torsion Function
For fixed drift strength and fixed container volume, the ball, together with an inward drift aligned with the expected-lifetime gradient, maximizes the total gradient and Laplacian of the expected exit time.