We use a Riemannnian approximation scheme to define a notion of sub-Riemannian Gaussian curvature for a Euclidean C 2 -smooth surface in the Heisenberg group H away from characteristic points, and a notion of sub-Riemannian signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carathéodory distance in H is provided.
In this paper, we deal with the composite plate problem, namely the following optimization eigenvalue problem: [Formula: see text] where [Formula: see text] is a class of admissible densities, [Formula: see text] for Dirichlet boundary conditions and [Formula: see text] for Navier boundary conditions. The associated Euler–Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator [Formula: see text]. In the spirit of [S. Chanillo, D. Grieser, M. Imai, K. Kurata and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys. 214 (2000) 315–337], we study qualitative properties of the optimal pairs [Formula: see text]. In particular, we prove existence and regularity and we find the explicit expression of [Formula: see text]. When [Formula: see text] is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of [Formula: see text] and radial symmetry of both [Formula: see text] and [Formula: see text].
Abstract. We prove a general magnetic Bourgain-Brezis-Mironescu formula which extends the one obtained in [37] in the Hilbert case setting. In particular, after developing a rather complete theory of magnetic bounded variation functions, we prove the validity of the formula in this class.
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