The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript

Abstract

How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander's approach and its applications, with an emphasis on eigenvalue estimates and spectral asymptotics. In particular, we obtain results for the DtN maps on non-smooth boundaries in the Riemannian setting, the DtN operators for the Helmholtz equation and the DtN operators on differential forms.

Weyl asymptotics for Poincaré–Steklov eigenvalues in a domain with Lipschitz boundary

“…It is well known that for the Laplacian, the operator equals, up to lower order terms, . † / 1 2 , for a smooth boundary; an interesting discussion can be found in [18]. For general elliptic operators, in [2,3], M. S. Agranovich presented an explanation of the coefficient ˇ.x; 0 / for the case of a smooth (or 'almost smooth') surface †: We discuss now the latter formula.…”

“…We just mention here that in a series of papers, starting with [29], for the case of a piecewise smooth Lipschitz boundary, the authors succeeded in describing how the corners influence the deviation of the eigenvalues from their behavior in the smooth case. Meanwhile, in the Lipschitz case, many important properties of the P-S (or the Neumann-to-Dirichlet, N-to-D, operator) were established, see, e.g., [8][9][10]18]; a number of important Steklov-type problems arising in hydrodynamics were considered recently in [40].…”

“…Thus, the eigenvalue asymptotics problem for Lipschitz boundary remained unresolved. In the recent fundamental review [17] and in [18], this problem is listed among unsolved and challenging.…”

Generic properties of Steklov eigenfunctions

“…And it could have applications in other areas, like the inverse conductivity problems ( [2], [19]) and cloaking ( [11]) where the Dirichlet-to-Neumann operator has been used. See [14] for more geometric questions related to Steklov eigenvalues and [13] for recent results about Steklov eigenvalues.…”

Weyl's law for the Steklov problem on surfaces with rough boundary

“…where ω d is the volume of the d-dimensional unit ball. We refer also to [San55,Sha71,Sus99b,AA96] for earlier results on this topic, as well as to [Roz79,Edw93,GPPS14,GKLP22] for improvements of the error estimate under stronger regularity assumptions. Extending the asymptotic formula (1.2) to domains with Lipschitz boundaries is a well known open problem, see e.g.…”