arXiv subjects
Publications and source records attributed to Lawford Hatcher.
We prove a variant of Rauch's hot spots conjecture for hyperbolic planar domains with small Neumann or mixed Dirichlet-Neumann eigenvalues. We conclude, for instance, that on bounded convex domains in the hyperbolic plane with sufficiently large area, second Neumann Laplace eigenfunctions have no interior critical points.
We study extrema of solutions to the heat equation (i.e. hot spots) on a class of warped product manifolds of the form $([0,L]\times M,dr^2+f(r)^2h)$ where $(M,h)$ is a closed Riemannian manifold. We prove that, under certain conditions on the warping function $f$, the statement of Rauch's hot spots conjecture holds for the corresponding warped product. We then go on to study the long-time behavior of hot spots on infinite cones over closed Riemannian manifolds. In this case, under appropriate hypotheses on the initial condition, there are four possible long-time behaviors depending only on the spectral gap of the fiber $(M,h)$.
We prove constant-curvature analogues of several results regarding the hot spots conjecture in dimension two. Our main theorem shows that the hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature. We also prove that, under certain circumstances, on constant (positive or negative) curvature triangles, first mixed Dirichlet-Neumann Laplace eigenfunctions have no non-vertex critical points. Moreover, we show that each of these eigenfunctions is monotonic with respect to some Killing field. Finally, we show that for general simply connected polygons of non-zero constant curvature--with exactly one family of exceptions--second Neumann eigenfunctions of the Laplacian have at most finitely many critical points.
Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $\Omega\subseteq\Rbb^n$ is a topic that goes back at least to the work of P\'olya \cite{polya}. We study the effect of the isoperimetric ratio of $\Omega$ on the number $N(\Omega)$ of Neumann eigenvalues that do not exceed the first Dirichlet eigenvalue, proving that $N(\Omega)$ is bounded above and below by a constant multiple of the isoperimetric ratio in the case of convex domains. We also show that these estimates do not hold in the non-convex setting, addressing questions of Cox-MacLachlan-Steeves \cite{coxetal} and Freitas \cite{freitas}. Despite these counterexamples, we find similar estimates for polygonal domains in $\Rbb^2$ as well as certain families of fiber bundles that asymptotically collapse onto their base spaces, the motivating examples being tubular neighborhoods of submanifolds.
We prove that on convex domains, first mixed Laplace eigenfunctions have no interior critical points if the Dirichlet region is connected and sufficiently small. We also find two seemingly new estimates on the first mixed eigenvalue to give explicit examples of when the Dirichlet region is sufficiently small.
The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in $\mathbb{R}^n$ attains its extrema only on the boundary of the domain. We present an analogous problem for domains with mixed Dirichlet-Neumann boundary conditions. We then solve this problem for Euclidean triangles and a class of planar domains bounded by the graphs of certain piecewise smooth functions.
We give an elementary new proof of the hot spots conjecture for L-shaped domains. This result, in addition to a new eigenvalue inequality, allows us to locate the hot spots in Swiss cross translation surfaces. We then prove, in several cases, that first mixed Dirichlet-Neumann eigenfunctions of the Laplacian on L-shaped domains also have no interior critical points. As a combination of these results, we prove the hot spots conjecture for five classes of domains tiled by L-shaped domains, including a class of non-simply connected domains. An interesting feature of the proofs is that we make positive use of the lack of regularity of eigenfunctions on non-convex polygons.