The Minkowski problem for the torsional rigidity

We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of the $n$-dimensional Euclidean space. For the existence part we apply the variational method introduced by Jerison for analogous problems concerning other variational functionals. Uniqueness follows from the Brunn–Minkowski inequality for the torsional rigidity and corresponding equality conditions.

💡 Research Summary

The paper addresses a Minkowski‑type problem for the torsional rigidity functional defined on open bounded convex subsets of ℝⁿ. Torsional rigidity, denoted T(Ω), is the integral of the solution u of the Poisson problem –Δu = 1 in Ω with zero Dirichlet boundary data, i.e. T(Ω)=∫_Ω u dx. Unlike volume or surface area, T captures both size and shape in a nonlinear way and scales as T(tΩ)=t^{n+2} T(Ω). The authors aim to determine, for a prescribed measure μ on the unit sphere S^{n‑1}, whether there exists a convex body K whose associated “torsional surface measure” coincides with μ, and whether such a body is unique up to translation.

The first major contribution is the establishment of a Brunn–Minkowski inequality for torsional rigidity:
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