Abstract We establish a quantitative version of the isoperimetric inequality for the torsional rigidity of multiply connected domains, among sets with given area and with given joint area of the holes. Since the optimal shape is the annulus, we study how a domain approaches an annular configuration when its torsional rigidity is close to optimal. Our result shows that when the torsional rigidity is nearly optimal, the domain $$\Omega $$ Ω must be close to an annulus.

Abstract We study the length-preserving elastic flow of curves in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We prove global existence and subconvergence to critical points. The proof strategy involves a careful treatment of short-time existence, uniqueness, and parabolic energy estimates.

Abstract We prove the local Lipschitz regularity of the minimizers of functionals of the form $$ \mathcal {I}(u)=\int _\Omega f(\nabla u(x))+g(x)u(x)\,dx\qquad u\in \phi +W^{1,1}_0(\Omega ) $$ I ( u ) = ∫ Ω f ( ∇ u ( x ) ) + g ( x ) u ( x ) d x u ∈ ϕ + W 0 1 , 1 ( Ω ) where g is bounded and $$\phi $$ ϕ satisfies the Lower Bounded Slope Condition. The function f is assumed to be convex but not uniformly convex everywhere. As a byproduct, we also prove the existence of a locally Lipschitz minimizer for a class of functionals of the type above but allowing the function f to be nonconvex.

Abstract This paper studies the multi-marginal Monge problem in the setting of compact metric spaces proving existence and uniqueness of solutions when the cost function is Lipschitz. We apply the results obtained to solve an optics problem involving metalenses, that is, we design a refracting-reflecting metalens that preserves given energy distributions.

Abstract Given a compact surface with boundary, we introduce a family of functionals on the space of its Riemannian metrics, defined via eigenvalues of a Steklov-type problem. We prove that each such functional is uniformly bounded from above, and we characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions.

Abstract We prove lower bounds for the first non-trivial eigenvalue of the drift Laplacian on manifolds with Wentzell-type boundary condition in terms of some Cheeger-type constants for bulk-boundary interactions. Our results are in the spirit of Cheeger’s classical inequality.