Flow approach on the monotonicity of shape functionals

Abstract: We develop a geometric flow framework to investigate the following two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by stretching flows, we prove several new monotonicity properties of the torsional rigidity and the first eigenvalue along the evolutions on triangles and rhombuses. These results also lead to new and simpler proofs of some known results, without using the Steiner symmetrization argument. Second, utilizing the mean curvature flow, we give a new proof of the Saint-Venant inequality for smooth convex bodies. Third, by discovering a gradient norm inequality for the sides of rectangles, we prove monotonicity and rigidity results of the torsional rigidity on rectangles.