# Dirichlet-to-Neumann Maps: Spectral Theory, Inverse Problems and Applications (16w5083)

Arriving in Oaxaca, Mexico Sunday, May 29 and departing Friday June 3, 2016

## Organizers

(University College London)

(Université de Montréal)

The DtN map is a first order elliptic pseudodifferential operator defined on the boundary of a Riemannian manifold. It has a discrete spectrum, and its eigenvalues coincide with the eigenvalues of the Steklov boundary value problem. For this reason, the spectrum of the DtN map is often referred to as the Steklov spectrum. The Steklov problem appears in various applications to physics, notably to hydrodynamics, vibration theory and the study of heat diffusion. Recently, there has been a lot interest in the properties of the Steklov eigenvalues and eigenfunctions from the viewpoint of geometric spectral theory. In particular, significant progress was achieved in the study of eigenvalue inequalities, spectral invariants and nodal geometry. Isoperimetric inequalities for eigenvalues is an important topic in spectral geometry, going back to the classical Faber-Krahn and Szeg\H{o}--Weinberger inequalities for the first Dirichlet and Neumann eigenvalues. In 1954, R. Weinstock proved that the disk is a unique maximizer of the first nonzero Steklov eigenvalue among all simply-connected planar domains of given perimeter. In 2010, Girouard and Polterovich extended this result to higher eigenvalues, showing that under similar conditions, the $k$-th Steklov eigenvalue achieves its maximum in the limit on a disjoint union of $k$ identical disks. For non-simply connected domains and surfaces, the problem is much more difficult. It was recently proved by Fraser and Schoen that among all surfaces of genus zero with $\ell \ge 1$ boundary components of fixed total length, there exists a smooth maximizer of the first nonzero Steklov eigenvalue. Moreover, Fraser and Schoen exhibited a link between the extremal surfaces for Steklov eigenvalues and the theory of free boundary minimal surfaces in Euclidean balls, and described explicitly the "critical catenoid" which is the maximizer for the case $\ell=2$. Other important inequalities for Steklov eigenvalues on Riemannian manifolds were recently obtained by Colbois, El Soufi, Girouard, Kokarev and Hassannezhad. In particular, the importance of the isoperimetric ratio for the control on the Steklov spectrum was emphasized. Inverse spectral problems for the Dirichlet-to-Neumann operator is a topic that unifies the two major themes of the workshop. Rephrasing the celebrated question of Marc Kac "Can one hear the shape of a drum?", one may ask which geometric quantities of a Riemannian manifold with boundary can be recovered from the Steklov spectrum. It follows from the standard eigenvalue asymptotics for pseudodifferential operators that the dimension and the volume of the boundary are Steklov spectral invariants. In a recent paper by Sher and Polterovich, further spectral invariants were computed using the heat trace expansion for the Dirichlet-to-Neumann operator. As a corollary, it was shown that the three-dimensional ball is uniquely determined by its Steklov spectrum among domains in $R^3$ with connected boundary. It is conjectured that a similar result should hold in arbitrary dimension. The same authors, together with Girouard and Parnovski, have also shown that the Steklov spectrum of a surface uniquely determines the number of its boundary components, as well as their lengths. This result does not generalize in a straightforward way to higher dimensions. In particular, it is an open question whether in higher dimensions the number of boundary components is a Steklov spectral invariant. Some other interesting results and conjectures on inverse problems for the Steklov spectrum on surfaces were stated by Jollivet and Sharafutdinov. The study of nodal domains and nodal lines of eigenfunctions has fascinated mathematicians and physicists for more than two centuries, starting with the famous experiments of Chladni with vibrating plates. For Laplace eigenfunctions, it was shown by Courant that the $k$-th eigenfunction has at most $k$ nodal domains. The same is true for the eigenfunctions of the Steklov problem, considered as functions in the "interior". However, the nodal count for eigenfunctions of the Dirichlet-to-Neumann operator, defined on the boundary of a manifold, is far more difficult, and essentially no general results are known when the boundary is of dimension two or higher. Very recently, Zelditch proved a sharp upper bound (conjectured earlier by Bellova-Lin) on the size of the nodal set of DtN eigenfunctions on real-analytic Riemannian manifolds. A similar result for the Laplace eigenfunctions was earlier proved by Donnelly-Fefferman, as a partial solution of Yau's conjecture. Unlike the case of Laplace eigenfunctions, lower bounds on the volume of the nodal sets of DtN eigenfunctions remain unknown even in the analytic category. Apart from the usual DtN operator acting on functions, one may also study its counterparts acting on differential forms. Different constructions of such operators were introduced by Belishev-Sharafutdinov, Joshi-Lionheart and, more recently, by Raulot-Savo. In particular, the latter authors proved interesting geometric inequalities for eigenvalues of such operators involving the isoperimetric ratio as well as the principal curvatures. The DtN map can be interpreted to encode information on measurements on a boundary of an object, and this motivates the study of inverse problems of the form: given the DtN map, determine the coefficients of the direct problem, that is, the governing PDE. These inverse problems appear in a wide variety of applications such as Electrical Impedance Tomography, where the direct problem is elliptic, and Reflection Seismology, with a hyperbolic direct problem. Inverse scattering problems can also be reduced to recovery from the Dirichlet-Neumann map when the coefficients are known outside a bounded domain. It would be interesting to relate the Steklov spectrum to spectral properties of the Scattering Operator. A canonical inverse boundary value problem is to determine a compact Riemannian or Lorentzian manifold with boundary from the Dirichlet-to-Neumann map of the Laplace operator on the manifold. The Riemannian or elliptic version of the inverse boundary value problem is called Calderon's problem. In the two dimensional case the problem was solved by Nachman, Sylvester,Lassas and Uhlmann, and this workshop will focus on the three and higher dimensional cases. Recently, Dos Santos Ferreira, Kurylev, Lassas and Salo solved Calderon's problem on a class of product type manifolds, which they call transversally anisotropic, via a reduction to the hyperbolic version using Wick rotation.The Lorentzian or hyperbolic version of the problem was solved by Eskin in the time independent case by using ideas from the Boundary Control method. This method was introduced by Belishev, and he and Kurylev used it to solve Gelfand's inverse boundary spectral problem.
A variant of the above inverse boundary value problem is to determine the manifold in a known conformal class. This problem has been solved on a class of manifolds that are transversally anisotropic modulo conformal scaling. In the hyperbolic case, the proof uses geometrical optics solutions to reduce the inverse boundary value problem to the problem of inverting the geodesic ray transform. In the elliptic case, Dos Santos Ferreira, Kenig, Salo and Uhlmann performed a similar reduction using complex geometrical optics solutions, that originate from the now classical work of Sylvester and Uhlmann. Concerning inversion of the geodesic ray transform, the recent result by Uhlmann and Vasy is a breakthrough but still poses restrictions on the geometry. On the other hand, Eskin has solved the hyperbolic conformally transversally anisotropic case supposing that the dependence on time is analytic and under the mild assumption that the geometry is non-trapping. In a setting related to partial data, that is, when the Dirichlet-to-Neumann map is known only partially, Lassas and Oksanen replaced the non-trapping assumption by a strictly weaker assumption on the asymptotics of the Neumann traces of the Dirichlet eigenfunctions of the manifold. In all the solved cases discussed above, there is a non-trivial conformal Killing field on the manifold. The problem is open in the case of general geometry, and under active research. In the Lorentzian case, black holes are natural counterexamples to unique solvability of the inverse boundary value problem. In the Riemannian case, non-smooth counterexamples can be obtained by using transformation optics as first realized by Greenleaf, Lassas and Uhlmann. The study of transformation optics has become a highly active field motivated by the possibility of building actual cloaking devices using metamaterials. Partly motivated by the existence of non-smooth counter examples, the inverse boundary value problem has been widely studied when the coefficients of the direct problem are less smooth. In the two dimensional case, Astala and Päivärinta solved the elliptic problem with $L^\infty$ coefficients. In higher dimensions, the recent exciting result by Haberman and Tataru assumes $C^1$ coefficients (and conformally Euclidean geometry). There is ongoing research to generalize their result for Lipschitz coefficients.