arXiv subjects

David Jerison

Publications and source records attributed to David Jerison.

24 records · Page 2Linked to original sources

arXiv2011

Logarithmic fluctuations for internal DLA

Let each of n particles starting at the origin in Z^2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high probability) close to a disk B_r of radius r=\sqrt{n/π}. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant C such that the following holds with probability one: B_{r - C \log r} \subset A(πr^2) \subset B_{r+ C \log r} for all sufficiently large r.

math.PR↗

arXiv2011

Internal DLA in Higher Dimensions

Let A(t) denote the cluster produced by internal diffusion limited aggregation (internal DLA) with t particles in dimension d > 2. We show that A(t) is approximately spherical, up to an O(\sqrt{\log t}) error.

math.PR↗

arXiv2011

Internal DLA and the Gaussian free field

In previous works, we showed that the internal DLA cluster on \Z^d with t particles is a.s. spherical up to a maximal error of O(\log t) if d=2 and O(\sqrt{\log t}) if d > 2. This paper addresses "average error": in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when d=2 and of order r^{1-d/2} (for a radius r cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.

math.PR↗

arXiv2010

Singularities of the wave trace near cluster points of the length spectrum

Let $-λ_j$ be the eigenvalues of the Laplace operator on the unit disk with Dirichlet conditions. The distribution $h(t) = \sum_j e^{i\sqrtλ_j t}$ is the trace of the solution operator of the wave equation on the disk. It is well known that $h$ has isolated singularities at the lengths of the reflecting geodesics. In particular, $h$ is singular at $t_k$, the perimeter of the regular inscribed polygon with $k$ sides. Evidently, $t_k < 2π$, the perimeter of the circle, and $t_k$ tends to $2π$. In this paper, we show that $h(t)$ is infinitely differentiable as $t$ tends to $2π$ from the right.

math.AP↗

arXiv2010

A gradient bound for free boundary graphs

We prove an analogue for a one-phase free boundary problem of the classical gradient bound for solutions to the minimal surface equation. It follows, in particular, that every energy-minimizing free boundary that is a graph is also smooth. The method we use also leads to a new proof of the classical minimal surface gradient bound.

math.AP↗

arXiv2007

Asymptotics of eigenfunctions on plane domains

We consider a family of domains $(Ω_N)_{N>0}$ obtained by attaching an $N\times 1$ rectangle to a fixed set $Ω_0 = \{(x,y): 0<y<1, -ϕ(y)<x<0\}$, for a Lipschitz function $ϕ\geq 0$. We derive full asymptotic expansions, as $N\to\infty$, for the $m$th Dirichlet eigenvalue (for any fixed $m$) and for the associated eigenfunction on $Ω_N$. The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain $Ω_\infty$. We determine the first variation of this scattering phase, with respect to $ϕ$, at $ϕ\equiv 0$. This is then used to prove sharpness of results, obtained previously by the same authors, about the location of extrema and nodal line of eigenfunctions on convex domains.

math.SP↗