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Robert L. Jerrard

Publications and source records attributed to Robert L. Jerrard.

At least 19 recordsLinked to original sources
arXiv2024

Alexandrov's estimate revisited

Alexandrov's estimate states that if $Ω$ is a bounded open convex domain in ${\mathbb R}^n$ and $u:\bar Ω\to {\mathbb R}$ is a convex solution of the Monge-Ampere equation $\det D^2 u = f$ that vanishes on $\partial Ω$, then \[ |u(x) - u(y)| \le ω(|x-y|)(\int_Ωf)^{1/n} \qquad \mbox{for }ω(δ) = C_n\,\mbox{diam}(Ω)^{\frac{n-1}n} δ^{1/n}. \] We establish a variety of improvements of this, depending on the geometry of $\partial Ω$. For example, we show that if the curvature is bounded away from $0$, then the estimate remains valid if $ω(δ)$ is replaced by $C_Ωδ^{\frac 12 + \frac 1{2n}}$. We determine the sharp constant $C_Ω$ when $n=2$, and when $n\ge 3$ and $\partial Ω$ is $C^2$, we determine the sharp asymptotics of the optimal modulus of continuity $ω_Ω(δ)$ as $δ\to 0$. For arbitrary convex domains, we characterize the scaling of the optimal modulus $ω_Ω$. Under very mild nondegeneracy conditions, our results yield the improved Holder estimate, $ω_Ω(δ) \le C δ^α$ for some $α>1/n$.

math.AP↗
arXiv2019

Local minimizers with unbounded vorticity for the $2$d Ginzburg-Landau functional

A central focus of Ginzburg-Landau theory is the understanding and characterization of vortex configurations. On a bounded domain $Ω\subseteq \mathbb{R}^2,$ global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit $ε\to 0,$ where $ε>0$ is the inverse of the Ginzburg-Landau parameter. The presence of an applied magnetic field of strength $h_{ex}\gg 1$ makes possible the existence of stable vortex states. A notable open problem is whether there are solutions of the Ginzburg-Landau equation for any number of vortices below $ h_{ex} |Ω| /2 π,$ for external fields of up to super-heating field strength. The best earlier partial results give, for every $0 0,$ the existence of local minimizers of the Ginzburg-Landau functional with a prescribed number of vortices in the range $1 \leq N \leq \min \{ K | \log ε|, c ( h_{ex} |Ω| /2 π) \}$ and for values of $1\ll_εh_{ex}$ smaller than a power of the Ginzburg-Landau parameter. In this paper, we prove that there are constants $K_1, α>0$ such that given natural numbers satisfying \[1\leq N \leq \frac{h_{ex}}{2π}(|Ω|-h_{ex}^{-1/4}),\] local minimizers of the Ginzburg-Landau functional with this many vortices exist, for fields such that $K_1\leq h_{ex} \leq 1/ε^α.$ Our strategy consists in combining: the minimization over a subset of configurations for which we can obtain a very precise localization of vortices; expansion of the energy in terms of a modified vortex interaction energy that allows for a reduction to a potential theory problem; and a quantitative vortex separation result for admissible configurations. Our results provide detailed information about the vorticity and refined asymptotics of the local minimizers that we construct.

math.AP↗
arXiv2019

Renormalized energy between vortices in some Ginzburg-Landau models on 2-dimensional Riemannian manifolds

We study a variational Ginzburg-Landau type model depending on a small parameter $\varepsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian manifold $S$. As $\varepsilon\to 0$, these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus $\mathfrak{g}$ of $S$ is different than $1$. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorticity-dependent lattice in the $2\mathfrak{g}$-dimensional space of harmonic $1$-forms on $S$ if $\mathfrak{g}\geq 1$. Our second main result determines the interaction energy (called renormalized energy) between vortex points as a $Γ$-limit (at the second order) as $\varepsilon\to 0$. The renormalized energy governing the optimal location of vortices depends on the Gauss curvature of $S$ as well as on the quantized flux. The coupling between flux quantization constraints and vorticity, and its impact on the renormalized energy, are new phenomena in the theory of Ginzburg-Landau type models. We also extend this study to two other (extrinsic) models for embedded hypersurfaces $S\subset \mathbb{R}^3$, in particular, to a physical model for non-tangent maps to $S$ coming from micromagnetics.

math.AP↗
arXiv2019

Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_c(M)$ of compactly supported diffeomorphisms of a manifold $M$, and show that it vanishes for the critical Sobolev norms $W^{s,n/s}$, where $n$ is the dimension of $M$ and $s\in(0,1)$. This completes the proof that the geodesic distance induced by $W^{s,p}$ vanishes if $sp\le n$ and $s<1$, and is positive otherwise. The proof is achieved by combining the techniques of two recent papers --- [JM19] by the authors, which treated the sub-critical case, and [BHP18] of Bauer, Harms and Preston, which treated the critical 1-dimensional case.

math.DG↗
arXiv2018

Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_\text{c}(M)$ of compactly supported diffeomorphisms, for various Sobolev norms $W^{s,p}$. Our main result is that the geodesic distance vanishes identically on every connected component whenever $s<\min\{n/p,1\}$, where $n$ is the dimension of $M$. We also show that previous results imply that whenever $s > n/p$ or $s \ge 1$, the geodesic distance is always positive. In particular, when $n\ge 2$, the geodesic distance vanishes if and only if $s<1$ in the Riemannian case $p=2$, contrary to a conjecture made in Bauer et al. [BBHM13].

math.DG↗
arXiv2017

Nearly Parallel Vortex Filaments in the 3D Ginzburg-Landau Equations

We introduce a framework to study the occurrence of vortex filament concentration in $3D$ Ginzburg-Landau theory. We derive a functional that describes the free-energy of a collection of nearly-parallel quantized vortex filaments in a cylindrical $3$-dimensional domain, in certain scaling limits; it is shown to arise as the $Γ$-limit of a sequence of scaled Ginzburg-Landau functionals. Our main result establishes for the first time a long believed connection between the Ginzburg-Landau functional and the energy of nearly parallel filaments that applies to many mathematically and physically relevant situations where clustering of filaments is expected. In this setting it also constitutes a higher-order asymptotic expansion of the Ginzburg-Landau energy, a refinement over the arclength functional approximation. Our description of the vorticity region significantly improves on previous studies and enables us to rigorously distinguish a collection of multiplicity one vortex filaments from an ensemble of fewer higher multiplicity ones. As an application, we prove the existence of solutions of the Ginzburg-Landau equation that exhibit clusters of vortex filaments whose small-scale structure is governed by the limiting free-energy functional.

math.AP↗
arXiv2017

A refined description of evolving interfaces in certain nonlinear wave equations

We improve on recent results that establish the existence of solutions of certain semilinear wave equations possessing an interface that roughly sweeps out a timelike surface of vanishing mean curvature in Minkowski space. Compared to earlier work, we present sharper estimates, in stronger norms, of the solutions in question.

math.AP↗
arXiv2017

Interaction energy between vortices of vector fields on Riemannian surfaces

We study a variational Ginzburg-Landau type model depending on a small parameter $ε>0$ for (tangent) vector fields on a $2$-dimensional Riemannian surface. As $ε\to 0$, the vector fields tend to be of unit length and will have singular points of a (non-zero) index, called vortices. Our main result determines the interaction energy between these vortices as a $Γ$-limit (at the second order) as $ε\to 0$.

math.AP↗
arXiv2016

Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii equation

Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation.

math.AP↗
arXiv2016

On the vortex filament conjecture for Euler flows

In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in $\mathbb{R}^3$, we prove that the curve evolves to leading order by binormal curvature flow. Our approach combines new estimates on the distance of the corresponding Hamiltonian-Possion structures with stability estimates recently developed in Ref. 15.

math.AP↗
arXiv2015

Weighted TV minimization and applications to vortex density models

Motivated in part by models arising from mathematical descriptions of Bose-Einstein condensation, we consider total variation minimization problems in which the total variation is weighted by a function that may degenerate near the domain boundary, and the fidelity term contains a weight that may be both degenerate and singular. We develop a general theory for a class of such problems, with special attention to the examples arising from physical models.

math-ph↗
arXiv2014

Sobolev spaces of isometric immersions of arbitrary dimension and codimension

We prove the $C^{1}$ regularity and developability of $W^{2,p}$ isometric immersions of $n$-dimensional flat domains into ${\mathbb R}^{n+k}$ where $p\ge \min\{2k, n\}$. Another parallel consequence of our methods is a similar regularity and rigidity result for the $W^{2,n}$ solutions of the degenerate Monge-Ampère equations in $n$ dimensions. The analysis also applies to the situations when the degeneracy is extended to $(k+1)\times (k+1)$ minors of the Hessian matrix and the solution is $W^{2,p}$, with $p\ge \min\{2k, n\}$.

math.AP↗
arXiv2013

Hydrodynamic limit of the Gross-Pitaevskii equation

We study dynamics of vortices in solutions of the Gross-Pitaevskii equation $i \partial_t u = Δu + \varepsilon^{-2} u (1 - |u|^2)$ on $\mathbb{R}^2$ with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter $\varepsilon$. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [21].

math.AP↗
arXiv2013

Existence and uniqueness of minimizers of general least gradient problems

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems - under certain sharp conditions - for minimizers of the general least gradient problem \[\inf_{u\in BV_f(Ω)} \int_Ωφ(x,Du),\] where $f:\partial Ω\to \R$ is continuous, \[ BV_f(Ω):=\{v\in BV(Ω): \ \ \forall x\in \partial Ω, \ \ \lim_{r\to 0} \ \esssup_{y\in Ω, |x-y|<r} |f(x) - v(y)| = 0 \ \} %BV_f(Ω)=\{u\in BV(Ω): {0.1cm} u|_{\partial Ω}=f {0.1cm} \hbox{and} {0.1cm} {0.1cm} u {0.1cm} \hbox{is continuous at} {0.1cm} \partial Ω\}. \] and $φ(x,ξ)$ is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the $ξ$ variable. In particular we prove that if $a\in C^{1,1}(Ω)$ is bounded away from zero, then minimizers of the weighted least gradient problem $\inf_{u \in BV_f}\int_Ω a|Du|$ are unique in $BV_f(Ω)$. We construct counterexamples to show that the regularity assumption $a\in C^{1,1}$ is sharp, in the sense that it can not be replaced by $a\in C^{1,α}(Ω)$ with any $α<1$.

math.FA↗
arXiv2013

Accelerating fronts in semilinear wave equations

We study dynamics of interfaces in solutions of the equation $ε\Box u + \frac 1 εf_ε(u)=0$, for $f_ε$ of the form $f_ε(u) = (u^2-1)(2u- εκ)$, for $κ\in {\mathbb R}$, as well as more general, but qualitatively similar, nonlinearities. We consider equations of this form both in $(1+n)$-dimensional Minkowski space, $n\ge 1$, and on certain more general Lorentzian manifolds, and we prove that for suitable initial data, solutions exhibit interfaces that sweep out timelike hypersurfaces of mean curvature proportional to $κ$. In particular, in 1 dimension these interfaces behave like a relativistic point particle subject to constant acceleration.

math.AP↗
arXiv2012

On the regularity of timelike extremal surfaces

We study a class of timelike weakly extremal surfaces in flat Minkowski space $\mathbb R^{1+n}$, characterized by the fact that they admit a $C^1$ parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class $C^k$, then the surface is regularly immersed away from a closed singular set of euclidean Hausdorff dimension at most $1+1/k$, and that this bound is sharp. We also show that, generically with respect to a natural topology, the singular set of a timelike weakly extremal cylinder in $\mathbb R^{1+n}$ is 1-dimensional if $n=2$, and it is empty if $n \ge 4$. For $n=3$, timelike weakly extremal surfaces exhibit an intermediate behavior.

math.AP↗
arXiv2012

Topological defects in the abelian Higgs model

We give a rigorous description of the dynamics of the Nielsen-Olesen vortex line. In particular, given a worldsheet of a string, we construct initial data such that the corresponding solution of the abelian Higgs model will concentrate near the evolution of the string. Moreover, the constructed solution stays close to the Nielsen-Olesen vortex solution.

math.AP↗