Asymptotic Analysis

In this paper, a mathematical modeling of the physical mechanism underlying the concept of tunable metasurfaces is provided. The scattering properties of two metasurfaces are analyzed. The considered metasurfaces are designed by placing either a Helmholtz resonator or a pair of Helmholtz resonators in a periodic lattice. The subwavelength resonant properties and the concept of hybridization of Helmholtz resonators are exploited in order to shape the scattered waves in unusual way by such metasurfaces.

This paper is concerned with wave propagation inside a cavity with a tunable boundary condition. It is a follow-up of [Asympt. Anal. (2019), to appear]. Cavities, because they trap waves for long times due to their reflecting walls, are used in a vast number of scientific domains. Indeed, in these closed media and due to interferences, the free space continuum of solutions becomes a discrete set of stationary eigenmodes. These enhanced stationary fields are commonly used in fundamental physics to increase wave-matter interactions. The eigenmodes and associated eigenfrequencies of a cavity are imposed by its geometrical properties through the boundary conditions. In this paper, we show the effect of a small change of boundary condition on the Green’s function of the cavity. This is achieved through the use of a tunable reflecting metasurface. The boundary condition can be switched from Dirichlet to Neumann at some specific resonant frequencies.

We consider stochastic differential equations with a drift term of gradient type and driven by Gaussian white noise on R d . Particular attention is given to the kernel p t , t > 0 of the transition semigroup associated with the solution process.

Under some rather strong regularity and growth assumptions on the coefficients, we adapt previous work by Thierry Hargé on Schrödinger operators and prove that the small time asymptotic expansion of p t , t > 0 is Borel summable.

We also briefly indicate some extensions and applications.

We introduce a new capacity associated to a non negative function V. We apply this notion to the study of a necessary and sufficient condition to ensure the existence and uniqueness of a Schrödinger type equation with measure data and with an operator whose coefficients are discontinuous. Namely, for a potential V, f a bounded Radon measure on Ω, then the equation L V u = − Δ u + U · ∇ u + V u = f has a solution in L 1 ( V ) ∩ L 0 1 ( Ω ) = { g   measurable , ∫ Ω | g | V d x   is finite and  lim ε → 0 ∫ { x : dist ( x ; ∂ Ω ) ⩽ ε } | g | d x = 0 } if and only if f does not charge “irregular points” of V, provided that the set of “irregular points” have a zero potential capacity. As a byproduct of our results, we have the non existence of a Green operator for some L V .

Our method is also based on a new topology and density of C c 2 ( Ω ∖ K ) in C 0 2 ( Ω ‾ ) whenever K has a zero potential-capacity.