We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or more generally Euclidean surfaces with conic singularities) by analysis of the wave operator on the flat Euclidean cone $C(mathbb{S}^1_rho) := mathbb{R}_+ times left(mathbb{R} bi
g/ 2pirho mathbb{Z}right)$ of radius $rho > 0$ equipped with the metric $h(r,theta) = d r^2 + r^2 , dtheta^2$. Using explicit oscillatory integrals and relying on the fundamental solution to the wave equation in geometric regions related to flat wave propagation and diffraction by the cone point, we can prove spectral cluster estimates equivalent to those in works on smooth Riemannian manifolds.
Using a new local smoothing estimate of the first and third authors, we prove local-in-time Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and global-in-time Stric
hartz estimates without a loss exterior to a large class of polygonal obstacles with Dirichlet boundary conditions. In addition, we prove a global-in-time local smoothing estimate in exterior wedge domains with Dirichlet boundary conditions and discuss some nonlinear applications.
We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrodinger equations. We examine two constrained minimization pro
blems, which give rise to such solutions. One yields what we call $F_lambda$-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.
We prove Strichartz estimates with a loss of derivatives for the Schrodinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon follow from tho
se on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a result of the second author regarding the Schrodinger equation on the Euclidean cone.
In this result, we develop the techniques of cite{KS1} and cite{BW} in order to determine a class of stable perturbations for a minimal mass soliton solution of a saturated, focusing nonlinear Schrodinger equation {c} i u_t + Delta u + beta (|u|^2) u
= 0 u(0,x) = u_0 (x), in $reals^3$. By projecting into a subspace of the continuous spectrum of $mathcal{H}$ as in cite{S1}, cite{KS1}, we are able to use a contraction mapping similar to that from cite{BW} in order to show that there exist solutions of the form e^{i lambda_{min} t} (R_{min} + e^{i mathcal{H} t} phi + w(x,t)), where $e^{i mathcal{H} t} phi + w(x,t)$ disperses as $t to infty$. Hence, we have long time persistance of a soliton of minimal mass despite the fact that these solutions are shown to be nonlinearly unstable in cite{CP1}.
We develop the techniques of cite{KS1} and cite{ES1} in order to derive dispersive estimates for a matrix Hamiltonian equation defined by linearizing about a minimal mass soliton solution of a saturated, focussing nonlinear Schrodinger equation {c} i
u_t + Delta u + beta (|u|^2) u = 0 u(0,x) = u_0 (x), in $reals^3$. These results have been seen before, though we present a new approach using scattering theory techniques. In further works, we will numerically and analytically study the existence of a minimal mass soliton, as well as the spectral assumptions made in the analysis presented here.
We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.
In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in cite{M1}. There, the methods developed in Burq-Zworski cite{BZ3} to study eigenfunctions for billiards which have rectangular components
were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ int_U |u|^2 geq c int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.
