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Morrey's classical inequality implies the H\"older continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality $$ \lambda\biggl\|\frac{u}{d_\Omega^{1-n/p}}\biggr\|_{\infty}^p\le \int_\Omega |Du|^p \,dx $$ for any open set $\Omega\subsetneq \mathbb{R}^n$. This inequality is valid for functions supported in $\Omega$ and with $\lambda$ a positive constant independent of $u$. The crucial hypothesis is that the exponent $p$ exceeds the dimension $n$. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of $\Omega$, sharp constants, and the existence of a nontrivial $u$ which saturates the inequality.

We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the Sticky Particles dynamics is confined to a compact set.

A ball polyhedron is the intersection of a finite number of closed balls in $\mathbb{R}^3$ with the same radius. In this note, we study ball polyhedra in which the set of centers defining the balls have the maximum possible number of diametric pairs. We explain how to compute the perimeter and volume of these shapes by employing the Gauss-Bonnet theorem and another integral formula. In addition, we show how to adapt this method to approximate the volume of Meissner polyhedra, which are constant width bodies constructed from ball polyhedra.

We study a non-linear eigenvalue problem for vector-valued eigenfunctions and give a succinct uniqueness proof for minimizers of the associated Rayleigh quotient.

We study the decay (at infinity) of extremals of Morrey's inequality in $\mathbb{R}^n$. These are functions satisfying $$ \displaystyle \sup_{x\neq y}\frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}}= C(p,n)\|\nabla u\|_{L^p(\mathbb{R}^n)} , $$ where $p>n$ and $C(p,n)$ is the optimal constant in Morrey's inequality. We prove that if $n \geq 2$ then any extremal has a power decay of order $\beta$ for any $$ \beta<-\frac13+\frac{2}{3(p-1)}+\sqrt{\left(-\frac13+\frac{2}{3(p-1)}\right)^2+\frac13}. $$

We consider Meissner polyhedra in $\mathbb{R}^3$. These are constant width bodies whose boundaries consist of pieces of spheres and spindle tori. We define these shapes by taking appropriate intersections of congruent balls and show that they are dense within the space of constant width bodies in the Hausdorff topology. This density assertion was essentially proved by Sallee. However, we offer a modern viewpoint taking into consideration the recent progress in understanding ball polyhedra and in constructing constant width bodies based on these shapes.

We revisit a classic proof of the Blaschke-Lebesgue theorem. It is based on the support function of a convex curve and the approximation of constant width curves by Reuleaux polygons.

We consider the basic problem of approximating Nash equilibria in noncooperative games. For monotone games, we design continuous time flows which converge in an averaged sense to Nash equilibria. We also study mean field equilibria, which arise in the large player limit of symmetric noncooperative games. In this setting, we will additionally show that the approximation of mean field equilibria is possible under a suitable monotonicity hypothesis.

We introduce a flow in the space of constant width bodies in three-dimensional Euclidean space that simultaneously increases the volume and decreases the circumradius of the shape as time increases. Starting from any initial constant width figure, we show that the flow exists for all positive times and converges to a closed ball as time tends to plus infinity. We also anticipate that this flow is interesting to study for negative times and that it would provide a mechanism to decrease the volume and increase the circumradius of a constant width body.

We consider the SIR model and study the first time the number of infected individuals begins to decrease and the first time this population is below a given threshold. We interpret these times as functions of the initial susceptible and infected populations and characterize them as solutions of a certain partial differential equation. This allows us to obtain integral representations of these times and in turn to estimate them precisely for large populations.

We consider the problem of approximating Nash equilibria of $N$ functions $f_1,\dots, f_N$ of $N$ variables. In particular, we deduce conditions under which systems of the form $$ \dot u_j(t)=-\nabla_{x_j}f_j(u(t)) $$ $(j=1,\dots, N)$ are well posed and in which the large time limits of their solutions $u(t)=(u_1(t),\dots, u_N(t))$ are Nash equilibria for $f_1,\dots, f_N$. To this end, we will invoke the theory of maximal monotone operators. We will also identify an application of these ideas in game theory and show how to approximate equilibria in some game theoretic problems in function spaces.

We consider a susceptible, infected, and recovered infectious disease model which incorporates a vaccination rate. In particular, we study the problem of choosing the vaccination rate in order to reduce the number of infected individuals to a given threshold as quickly as possible. This is naturally a problem of time-optimal control. We interpret the optimal time as a solution of two dynamic programming equations and give necessary and sufficient conditions for a vaccination rate to be optimal.

We consider the PDE $-\Delta_pu=\rho$, where $\rho$ is a signed Borel measure on $\mathbb{R}^n$. For each $p>n$, we characterize solutions as extremals of a generalized Morrey inequality determined by $\rho$.

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.

We study the limiting behavior as $|x|\rightarrow \infty$ of extremal functions $u$ for Morrey's inequality on $\mathbb{R}^n$. In particular, we compute the limit of $u(x)$ as $|x|\rightarrow \infty$ and show $|x||Du(x)|$ tends to $0$. To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form $-\Delta_pu=c(\delta_{x_0}-\delta_{y_0})$ for some $c\in \mathbb{R}$ and distinct $x_0,y_0\in \mathbb{R}^n$. More generally, we explain how to quantitatively deduce the asymptotic flatness of bounded $p$-harmonic functions on exterior domains of $\mathbb{R}^n$ for $p>n$.

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.

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.

We make the elementary observation that the differential equation associated with Newton's second law $m\ddot\gamma(t)=-D V(\gamma(t))$ 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.

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 Poincar\'e inequalities in Sobolev spaces. We prove spatial Lipschitz continuity and H\"older 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.

We give a qualitative description of extremals for Morrey's inequality. Our theory is based on exploiting the invariances of this inequality, studying the equation satisfied by extremals and the observation that extremals are optimal for a related convex minimization problem.

We consider the dynamics of finite systems of point masses which move along the real line. We suppose the particles interact pairwise and undergo perfectly inelastic collisions when they collide. In particular, once particles collide, they remain stuck together thereafter. Our main result is that if the interaction potential is semi-convex, this sticky particle property can quantified and is preserved upon letting the number of particles tend to infinity. This is used to show that solutions of the pressureless Euler equations exist for given initial conditions and satisfy an entropy inequality.

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.

We study weak solutions ${\bf v}:U\times (0,T)\rightarrow \mathbb{R}^m$ of the nonlinear parabolic system $$ D\psi({\bf v}_t)=\text{div}DF(D{\bf v}), $$ where $\psi$ 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 H\"older continuous, we show that $D^2{\bf v}$ and ${\bf v}_t$ are locally H\"older 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 ${\bf v}_t$.

We study the large time behavior of solutions $v:\Omega\times(0,\infty)\rightarrow \mathbb{R}$ of the PDE $\partial_t(|v|^{p-2}v)=\Delta_pv.$ We show that $e^{\left(\lambda_p/(p-1)\right)t}v(x,t)$ converges to an extremal of a Poincar\'e inequality on $\Omega$ with optimal constant $\lambda_p$, as $t\rightarrow \infty$. We also prove that the large time values of solutions approximate the extremals of a corresponding "dual" Poincar\'e inequality on $\Omega$. Moreover, our theory allows us to deduce the large time asymptotics of related doubly nonlinear flows involving various boundary conditions and nonlocal operators.

We study solutions ${\bf v}$ of the parabolic system of PDE $$ \partial_t\left(D\psi({\bf v})\right)=\text{div}DF(D{\bf v}). $$ Here $\psi$ 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 H\"older 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 $D{\bf v}$.

For a bounded domain $\Omega\subset \mathbb{R}^n$ and $p>n$, Morrey's inequality implies that there is $c>0$ such that $$ c\|u\|^p_{\infty}\le \int_\Omega|Du|^pdx $$ for each $u$ belonging to the Sobolev space $W^{1,p}_0(\Omega)$. We show that the ratio of any two extremal functions is constant provided that $\Omega$ 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., Comment: 23 pages, 7 figures

We present two novel methods for approximating minimizers of the abstract Rayleigh quotient $\Phi(u)/ \|u\|^p$. Here $\Phi$ is a strictly convex functional on a Banach space with norm $\|\cdot\|$, and $\Phi$ is assumed to be positively homogeneous of degree $p\in (1,\infty)$. Minimizers are shown to satisfy $\partial \Phi(u)- \lambda\mathcal{J}_p(u)\ni 0$ for a certain $\lambda\in \mathbb{R}$, 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 $$ \partial \Phi(u_k)- \mathcal{J}_p(u_{k-1})\ni 0 \quad (k\in \mathbb{N}). $$ The second method is based on the large time behavior of solutions of the doubly nonlinear evolution $$ \mathcal{J}_p(\dot v(t))+\partial\Phi(v(t))\ni 0 \quad(a.e.\;t>0) $$ and more generally $p$-curves of maximal slope for $\Phi$. 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 $\Phi(u)/ \|u\|^p$. 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., Comment: 43 pages

The nonlinear and nonlocal PDE $$ |v_t|^{p-2}v_t+(-\Delta_p)^sv=0 \, , $$ where $$ (-\Delta_p)^s v\, (x,t)=2 \,\text{PV} \int_{\mathbb{R}^n}\frac{|v(x,t)-v(x+y,t)|^{p-2}(v(x,t)-v(x+y,t))}{|y|^{n+sp}}\, dy, $$ 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 H\"older estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional $p$-Laplacian.

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 $\Omega\subset\mathbb{R}^n$, we analyze a scheme that allows us to approximate the smallest value the ratio $\int_\Omega|D\psi|^p dx/\int_\Omega|\psi|^p dx$ can assume for functions $\psi$ that vanish on $\partial \Omega$. The scheme in question also provides a natural way to approximate minimizing $\psi$. 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.

We consider the problem of finding $\lambda\in \mathbb{R}$ and a function $u:\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfy the PDE $$ \max\left\{\lambda + F(D^2u) -f(x),H(Du)\right\}=0, \quad x\in \mathbb{R}^n. $$ 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 $\lambda^*$ for which the above equation has a solution $u$ with appropriate growth as $|x|\rightarrow \infty$. Moreover, associated to $\lambda^*$ 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."

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.

We study solutions of the system of PDE $D\psi({\bf v}_t)=\text{div}DF(D{\bf v})$, where $\psi$ 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.

We study the large time behavior of solutions of the PDE $|v_t|^{p-2}v_t=\Delta_p v$. A special property of this equation is that the Rayleigh quotient $\int_{\Omega}|Dv(x,t)|^pdx /\int_{\Omega}|v(x,t)|^pdx$ is nonincreasing in time along solutions. As $t$ tends to infinity, this ratio converges to the optimal constant in Poincar\'{e}'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.

We perform a systematic study of optimization problems in the Wasserstein spaces that are analogs of infinite horizon, deterministic control problems. We derive necessary conditions on action minimizing paths and present a sufficient condition for their existence. We also verify that the corresponding generalized value functions are a type of viscosity solution of a time independent, Hamilton-Jacobi equation in the space of probability measures. Finally, we prove a special case of a conjecture involving the subdifferential of generalized value functions and their relation to action minimizing paths., Comment: We corrected several typographical errors

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

In this paper, we construct a dumbbell domain for which the associated principle $\infty$-eigenvalue is not simple. This gives a negative answer to the outstanding problem posed by Juutinen-Lindquivst-Manfredi ("The $\infty$-eigenvalue problem", Arch. Ration. Mech. Anal. 148, 1999, no.2, 89-105). It remains a challenge to determine whether simplicity holds for convex domains., Comment: 11 pages

We characterize the price of a European option on several assets for a very risk averse seller, in a market with small transaction costs as a solution of a nonlinear diffusion equation. This problem turns out to be one of asymptotic analysis of parabolic PDE, and the interesting feature is the role of a nonlinear PDE eigenvalue problem. In particular, we generalize previous work of Guy Barles and H. Mete Soner who studied this problem for a European option on a single asset.