arXiv subjects

Ryan Hynd

Publications and source records attributed to Ryan Hynd.

At least 37 records · Page 2Linked to original sources

arXiv2020

Newton's second law with a semiconvex potential

We make the elementary observation that the differential equation associated with Newton's second law $m\ddotγ(t)=-D V(γ(t))$ always has a solution for given initial conditions provided that the potential energy $V$ is semiconvex. That is, if $-D V$ satisfies a one-sided Lipschitz condition. We will then build upon this idea to verify the existence of solutions for the Jeans-Vlasov equation, the pressureless Euler equations in one spatial dimension and the equations of elastodynamics under appropriate semiconvexity assumptions.

math.AP↗

arXiv2020

On the symmetry and monotonicity of Morrey extremals

We employ Clarkson's inequality to deduce that each extremal of Morrey's inequality is axially symmetric and is antisymmetric with respect to reflection about a plane orthogonal to its axis of symmetry. We also use symmetrization methods to show that each extremal is monotone in the distance from its axis of symmetry and in the direction of its axis when restricted to spheres centered at the intersection of its axis and its antisymmetry plane.

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arXiv2020

A trajectory map for the pressureless Euler equations

We consider the dynamics of a collection of particles that interact pairwise and are restricted to move along the real line. Moreover, we focus on the situation in which particles undergo perfectly inelastic collisions when they collide. The equations of motion are a pair of partial differential equations for the particles' mass distribution and local velocity. We show that solutions of this system exist for given initial conditions by rephrasing these equations in Lagrangian coordinates and then by solving for the associated trajectory map.

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arXiv2019

Probability measures on the path space and the sticky particle system

We study collections of point masses which move freely along the real line and stick together when they collide via perfectly inelastic collisions. We quantify the way particles stick together and explain how to associate a probability measure on the space of continuous paths to such a collection of evolving point masses. These observations lead to a new method of designing solutions to the sticky particle system in one spatial dimension which have nonincreasing kinetic energy and satisfy an entropy inequality.

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arXiv2019

Lagrangian coordinates for the sticky particle system

The sticky particle system is a system of partial differential equations which assert the conservation of mass and momentum of a collection of particles that interact only via inelastic collisions. These equations arise in Zel'dovich's theory for the formation of large scale structures in the universe. We will show that this system of equations has a solution in one spatial dimension for given initial conditions by generating a trajectory mapping in Lagrangian coordinates.

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arXiv2018

Lipschitz regularity for a homogeneous doubly nonlinear PDE

We study the doubly nonlinear PDE $$ |\partial_t u|^{p-2}\,\partial_t u-\textrm{div}(|\nabla u|^{p-2}\nabla u)=0. $$ This equation arises in the study of extremals of Poincaré inequalities in Sobolev spaces. We prove spatial Lipschitz continuity and Hölder continuity in time of order $(p-1)/p$ for viscosity solutions. As an application of our estimates, we obtain pointwise control of the large time behavior of solutions.

math.AP↗

arXiv2018

Extremal functions for Morrey's inequality in convex domains

For a bounded domain $Ω\subset \mathbb{R}^n$ and $p>n$, Morrey's inequality implies that there is $c>0$ such that $$ c\|u\|^p_{\infty}\le \int_Ω|Du|^pdx $$ for each $u$ belonging to the Sobolev space $W^{1,p}_0(Ω)$. We show that the ratio of any two extremal functions is constant provided that $Ω$ is convex. We also explain why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this property. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the $p$-Laplacian.

math.AP↗

arXiv2018

Partial regularity for type two doubly nonlinear parabolic systems

We study weak solutions ${\bf v}:U\times (0,T)\rightarrow \mathbb{R}^m$ of the nonlinear parabolic system $$ Dψ({\bf v}_t)=\text{div}DF(D{\bf v}), $$ where $ψ$ and $F$ are convex functions. This is a prototype for more general doubly nonlinear evolutions which arise in the study of structural properties of materials. Under the assumption that the second derivatives of $F$ are Hölder continuous, we show that $D^2{\bf v}$ and ${\bf v}_t$ are locally Hölder continuous except for possibly on a lower dimensional subset of $U\times (0,T)$. Our approach relies on two integral identities, decay of the local space-time energy of solutions, and fractional time derivative estimates for $D^2{\bf v}$ and ${\bf v}_t$.

math.AP↗

arXiv2018

Partial regularity for doubly nonlinear parabolic systems of the first type

We study solutions ${\bf v}$ of the parabolic system of PDE $$ \partial_t\left(Dψ({\bf v})\right)=\text{div}DF(D{\bf v}). $$ Here $ψ$ and $F$ are convex functions, and this is a model equation for more general doubly nonlinear evolutions that arise in the study of phase transitions in materials. We show that if ${\bf v}$ is a weak solution, then $D{\bf v}$ is locally Hölder continuous except for possibly on a lower dimensional subset of the domain of ${\bf v}$. Our proof is based on compactness properties of solutions, two integral identities and a fractional time derivative estimate for $D{\bf v}$.

math.AP↗

arXiv2017

Large time behavior of solutions of Trudinger's equation

We study the large time behavior of solutions $v:Ω\times(0,\infty)\rightarrow \mathbb{R}$ of the PDE $\partial_t(|v|^{p-2}v)=Δ_pv.$ We show that $e^{\left(λ_p/(p-1)\right)t}v(x,t)$ converges to an extremal of a Poincaré inequality on $Ω$ with optimal constant $λ_p$, as $t\rightarrow \infty$. We also prove that the large time values of solutions approximate the extremals of a corresponding "dual" Poincaré inequality on $Ω$. Moreover, our theory allows us to deduce the large time asymptotics of related doubly nonlinear flows involving various boundary conditions and nonlocal operators.

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arXiv2016

A doubly nonlinear evolution for the optimal Poincaré inequality

We study the large time behavior of solutions of the PDE $|v_t|^{p-2}v_t=Δ_p v$. A special property of this equation is that the Rayleigh quotient $\int_Ω|Dv(x,t)|^pdx /\int_Ω|v(x,t)|^pdx$ is nonincreasing in time along solutions. As $t$ tends to infinity, this ratio converges to the optimal constant in Poincaré's inequality. Moreover, appropriately scaled solutions converge to a function for which equality holds in this inequality. An interesting limiting equation also arises when $p$ tends to infinity, which provides a new approach to approximating ground states of the infinity Laplacian.

math.AP↗

arXiv2016

Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution

The nonlinear and nonlocal PDE $$ |v_t|^{p-2}v_t+(-Δ_p)^sv=0 \, , $$ where $$ (-Δ_p)^s v\, (x,t)=2 \,\text{PV} \int_{\mathbb{R}^n}\frac{|v(x,t)-v(x+y,t)|^{p-2}(v(x,t)-v(x+y,t))}{|y|^{n+sp}}\, dy, $$ has the interesting feature that an associated Rayleigh quotient is non-increasing in time along solutions. We prove the existence of a weak solution of the corresponding initial value problem which is also unique as a viscosity solution. Moreover, we provide Hölder estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional $p$-Laplacian.

math.AP↗

arXiv2016

Approximation of the least Rayleigh quotient for degree $p$ homogeneous functionals

We present two novel methods for approximating minimizers of the abstract Rayleigh quotient $Φ(u)/ \|u\|^p$. Here $Φ$ is a strictly convex functional on a Banach space with norm $\|\cdot\|$, and $Φ$ is assumed to be positively homogeneous of degree $p\in (1,\infty)$. Minimizers are shown to satisfy $\partial Φ(u)- λ\mathcal{J}_p(u)\ni 0$ for a certain $λ\in \mathbb{R}$, where $\mathcal{J}_p$ is the subdifferential of $\frac{1}{p}\|\cdot\|^p$. The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfy $$ \partial Φ(u_k)- \mathcal{J}_p(u_{k-1})\ni 0 \quad (k\in \mathbb{N}). $$ The second method is based on the large time behavior of solutions of the doubly nonlinear evolution $$ \mathcal{J}_p(\dot v(t))+\partialΦ(v(t))\ni 0 \quad(a.e.\;t>0) $$ and more generally $p$-curves of maximal slope for $Φ$. We show that both schemes have the remarkable property that the Rayleigh quotient is nonincreasing along solutions and that properly scaled solutions converge to a minimizer of $Φ(u)/ \|u\|^p$. These results are new even for Hilbert spaces and their primary application is in the approximation of optimal constants and extremal functions for inequalities in Sobolev spaces.

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arXiv2015

An eigenvalue problem for fully nonlinear elliptic equations with gradient constraints

We consider the problem of finding $λ\in \mathbb{R}$ and a function $u:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfy the PDE $$ \max\left\{λ+ F(D^2u) -f(x),H(Du)\right\}=0, \quad x\in \mathbb{R}^n. $$ Here $F$ is elliptic, positively homogeneous and superadditive, $f$ is convex and superlinear, and $H$ is typically assumed to be convex. Examples of this type of PDE arise in the theory of singular ergodic control. We show that there is a unique $λ^*$ for which the above equation has a solution $u$ with appropriate growth as $|x|\rightarrow \infty$. Moreover, associated to $λ^*$ is a convex solution $u^*$ that has bounded second derivatives, provided $F$ is uniformly elliptic and $H$ is uniformly convex. It is unknown whether or not $u^*$ is unique up to an additive constant; however, we verify this is the case when $n=1$ or when $F, f,H$ are "rotational."

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arXiv2015

Compactness methods for doubly nonlinear parabolic systems

We study solutions of the system of PDE $Dψ({\bf v}_t)=\text{div}DF(D{\bf v})$, where $ψ$ and $F$ are convex functions. This type of system arises in various physical models for phase transitions. We establish compactness properties of solutions that allow us to verify partial regularity when $F$ is quadratic and characterize the large time limits of weak solutions. Special consideration is also given to systems that are homogeneous and their connections with nonlinear eigenvalue problems. While the uniqueness of weak solutions of such systems of PDE remains an open problem, we show scalar equations always have a preferred solution that is also unique as a viscosity solution.

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arXiv2015

Value functions in the Wasserstein spaces: finite time horizons

We study analogs of value functions arising in classical mechanics in the space of probability measures endowed with the Wasserstein metric $W_p$, for $1<p<\infty$. Our main result is that each of these generalized value functions is a type of viscosity solution of an appropriate Hamilton-Jacobi equation, completing a program initiated by Gangbo, Tudorascu, and Nguyen. Of particular interest is a formula we derive for a generalized value function when the associated potential energy is of the form ${\cal V}(μ)=\int_{\mathbb{R}^d}V(x)dμ(x)$. This formula allows us to make rigorous a well known heuristic connection between Euler-Poisson equations and classical Hamilton-Jacobi equations. Further results are presented which suggest there is a rich theory to be developed of deterministic control in the Wasserstein spaces.

math.AP↗

arXiv2015

Inverse iteration for $p$-ground states

We adapt the inverse iteration method for symmetric matrices to some nonlinear PDE eigenvalue problems. In particular, for $p\in (1,\infty)$ and a given domain $Ω\subset\mathbb{R}^n$, we analyze a scheme that allows us to approximate the smallest value the ratio $\int_Ω|Dψ|^p dx/\int_Ω|ψ|^p dx$ can assume for functions $ψ$ that vanish on $\partial Ω$. The scheme in question also provides a natural way to approximate minimizing $ψ$. Our analysis also extends in the limit as $p\rightarrow\infty$ and thereby fashions a new approximation method for ground states of the infinity Laplacian.

math.AP↗

arXiv2015

On Hamilton-Jacobi-Bellman equations with convex gradient constraints

We study PDE of the form $\max\{F(D^2u,x)-f(x), H(Du)\}=0$ where $F$ is uniformly elliptic and convex in its first argument, $H$ is convex, $f$ is a given function and $u$ is the unknown. These equations are derived from dynamic programming in a wide class of stochastic singular control problems. In particular, examples of these equations arise in mathematical finance models involving transaction costs, in queuing theory, and spacecraft control problems. The main aspects of this work are to identify conditions under which solutions are uniquely defined and have Lipschitz continuous gradients. We also generalize previous results known for the case where $M\mapsto F(M,x)$ is the maximum of finitely many linear functions.

math.AP↗