Abstract:
We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.

Abstract:
We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains, for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.

Abstract:
For smooth domains, Liu et al. (Comm. Pure Appl. Math. 60: 1443-1487, 2007) used optimal estimates for the commutator of the Laplacian and the Leray projection operator to establish well-posedness of an extended Navier-Stokes dynamics. In their work, the pressure is not determined by incompressibility, but rather by a certain formula involving the Laplace-Leray commutator. A key estimate of Liu et al. controls the commutator strictly by the Laplacian in energy norm at leading order. In this paper we show that this strict control fails in a large family of bounded planar domains with corners. However, when the domain is an infinite cone, we find that strict control may be recovered in certain power-law weighted norms.

Abstract:
We construct solutions to p-Laplace type equations in unbounded Lipschitz domains in the plane with prescribed boundary data in appropriate fractional Sobolev spaces. Our approach builds on a Cauchy integral representation formula for solutions.

Abstract:
Boundary integral equations are an efficient and accurate tool for the numerical solution of elliptic boundary value problems. The solution is expressed as a layer potential; however, the error in its evaluation grows large near the boundary if a fixed quadrature rule is used. Firstly, we analyze this error for Laplace's equation with analytic density and the global periodic trapezoid rule, and find an intimate connection to the complexification of the boundary parametrization. Our main result is then a simple and efficient scheme for accurate evaluation up to the boundary for single- and double-layer potentials for the Laplace and Helmholtz equations, using surrogate local expansions about centers placed near the boundary. The scheme---which also underlies the recent QBX Nystr\"om quadrature---is asymptotically exponentially convergent (we prove this in the analytic Laplace case), requires no adaptivity, generalizes simply to three dimensions, and has O(N) complexity when executed via a locally-corrected fast multipole sum. We give an example of high-frequency scattering from an obstacle with perimeter 700 wavelengths long, evaluating the solution at $2\times 10^5$ points near the boundary with 11-digit accuracy in 30 seconds in MATLAB on a single CPU core.

Abstract:
By introducing a weight function to the Laplace operator, Bakry and \'Emery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-\'Emery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize \emph{all the results in Riemannian geometry based on gradient estimates to Bakry-\'Emery manifolds}.

Abstract:
We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace-Beltrami operator on closed surfaces of unit area.

Abstract:
Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues is linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials.

Abstract:
A classical open problem involving the Laplace operator on symmetric domains in Rn is whether all its Dirichlet eigenvalues vary smoothly upon one-parameter C1 perturbations of the domain. We provide a fairly complete answer to this question in dimension n = 2 on disks and squares and also for the second eigenvalue on balls in Rn for any n >= 3. Our approach bases on a suitable setting of the problem and uses an appropriate degenerate version of the implicit function theorem on Banach spaces as central tool.