Foliation of an Asymptotically Flat End by Critical Capacitors

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace–Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the nonlocal nature of our problem.

Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

We consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$\mathrm {C}(\partial M)$$ C ( ∂ M ) of continuous functions on the boundary $$\partial M$$ ∂ M of a compact manifold $$\overline{M}$$ M ¯ with boundary. We prove that it generates an analytic semigroup of angle $$\frac{\pi }{2}$$ π 2 , generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $$\frac{\pi }{2}$$ π 2 on the space $$\mathrm {C}(\overline{M})$$ C ( M ¯ ) .

First-order evolution equations with dynamic boundary conditions

In this paper, we introduce a general framework to study linear first-order evolution equations on a Banach space X with dynamic boundary conditions, that is with boundary conditions containing time derivatives. Our method is based on the existence of an abstract Dirichlet operator and yields finally to equivalent systems of two simpler independent equations. In particular, we are led to an abstract Cauchy problem governed by an abstract Dirichlet-to-Neumann operator on the boundary space ∂ X . Our approach is illustrated by several examples and various generalizations are indicated. This article is part of the theme issue ‘Semigroup applications everywhere’.

Validation of the Reduced Vector Potential Formulation with the DtN Boundary Condition

This work describes the validation against experimental data of a Reduced Vector Potential Formulation combined with a boundary condition given by the Dirichlet-to-Neumann operator. One of the classic nondestructive testing problems is investigated: the differential bobbin coil scan inside a tube with a defect. Several defects are simulated, and the results are compared to the experimental data acquired at four frequencies.

Convexity properties of the normalized Steklov zeta function of a planar domain

We consider the zeta function \zeta_{\Omega} for the Dirichlet-to-Neumann operator of a simply connected planar domain Ω bounded by a smooth closed curve of perimeter 2\pi. We name the difference \zeta_{\Omega}-\zeta_{\mathbb{D}} the normalized Steklov zeta function of the domain Ω, where 𝔻 denotes the closed unit disk. We prove that (\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(0)\geq 0 with equality if and only if Ω is a disk. We also provide an elementary proof that, for a fixed real 𝑠 satisfying s\leq-1, the estimate (\zeta_{\Omega}-\zeta_{\mathbb{D}})^{\prime\prime}(s)\geq 0 holds with equality if and only if Ω is a disk. We then bring examples of domains Ω close to the unit disk where this estimate fails to be extended to the interval (0,2). Other computations related to previous works are also detailed in the remaining part of the text.