2020/05/04 by Simon Larson, Larson, Simon
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #math.CA #math.FA
paper · pdf · doi:10.48550/arxiv.2005.01853
13 pages
arxiv created 2020/05/22 · arxiv updated 2020/05/25
Let Ω⊂ ℝd, d ≥ 2, be a bounded convex domain and f\colon Ω→ ℝ be a non-negative subharmonic function. In this paper we prove the inequality (1)/(|Ω|)∫Ωf(x) dx ≤ (d)/(|∂Ω|)∫∂Ω f(x) dσ(x) . Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if Ω⊂ ℝd is a bounded convex domain and u is the solution of -Δu =1 with homogeneous Dirichlet boundary conditions, then ‖∇ u‖L^∞(Ω) < d(|Ω|)/(|∂Ω|) . Moreover, both inequalities are sharp in the sense that if the constant d is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by d3/2 due to Beck et al.