arXiv subjects
Publications and source records attributed to Cale Rankin.
We give simple new proofs of two well-known results for the Schr\"odinger operator: first, the Brunn--Minkowski inequality for Dirichlet eigenvalues and, second, the log-concavity of the first Dirichlet eigenfunction. Our proof of the first applies to a class of domains including $C^{1,1}$ connected domains and convex potentials. In the special case of convex domains, the second result is a simple corollary.
The simplest genuinely multidimensional monopolist's problem involves minimizing a linearly perturbed Dirichlet energy among nonnegative convex functions $u$ on an open domain $X \subset [0, \infty)^2$. The geometry of the region of strict convexity $\Omega\subset X$ for the unique minimizer $u$ is of central interest. A relatively closed portion $X_1^0 \subset X$ of the domain is comprised of line segments starting and ending on $\partial X$ along which $u$ is affine. For convex polygons and potentially all domains $X \subset \mathbf{R}^2$, we build on results with Zhang to show that outside $X_1^0 \cup \{u=0\}$, the free boundary of $\Omega$ is a continuous curve of Hausdorff dimension one, and that $\Omega$ has density $1/2$ along it (and is $C^\alpha_{\mathrm{loc}}$ for all $0<\alpha<1$), except perhaps at a discrete set of singular points. We do this by showing that much of the free boundary solves an obstacle problem whose endogenous obstacle is $C^2$. From a slightly stronger conclusion, we deduce the free boundary becomes locally $C^\infty$ outside a closed set whose relative interior is empty. In response to the circulation of the present manuscript, we received a concurrent but independent work of Chen, Figalli and Zhang who verify a strengthening sufficient for this partial regularity result; (they show in particular that $\alpha=1$ and the discrete set mentioned above is empty).
We study the Monopolist's problem with a focus on the free boundary separating bunched from unbunched consumers, especially in the plane, and give a full description of its solution for the family of square domains $\{(a,a+1)^2\}_{a \ge 0}$. The Monopolist's problem is fundamental in economics, yet widely considered analytically intractable when both consumers and products have more than one degree of heterogeneity. Mathematically, the problem is to minimize a smooth, uniformly convex Lagrangian over the space of nonnegative convex functions. What results is a free boundary problem between the regions of strict and nonstrict convexity. Our work is divided into three parts: a study of the structure of the free boundary problem on convex domains in $\mathbf{R}^n$ showing that the product allocation map remains Lipschitz up to portions of the fixed boundary and that each bunch extends to this boundary; a proof in $\mathbf{R}^2$ that the interior free boundary can only fail to be smooth in one of four specific ways (cusp, high frequency oscillations, stray bunch, nontransversal bunch); and, finally, the first complete solution to Rochet and Chon\'e's example on the family of squares $\Omega = (a,a+1)^2$, where we discover bifurcations first to targeted and then to blunt bunching as the distance $a \ge 0$ to the origin is increased. To do this, we extend the localization for measures in convex-order to accommodate potential discontinuities in the product allocation map at the fixed boundary. We also employ techniques from the study of the Monge--Amp\`ere equation and the obstacle problem
We modify the JKO scheme, which is a time discretization of Wasserstein gradient flows, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a Riemannian metric, our modified JKO scheme converges under suitable conditions to the corresponding Riemannian Fokker--Planck equation. Thus on a Riemannian manifold one may replace the (squared) Riemannian distance with any cost function which induces the metric. Of interest is when the Riemannian distance is computationally intractable, but a suitable cost has a simple analytic expression. We consider the Fokker--Planck equation on compact submanifolds with the Neumann boundary condition and on complete Riemannian manifolds with a finite drift condition. As an application we consider Hessian manifolds, taking as a cost the Bregman divergence.
A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma--Trudinger--Wang condition is the Pogorelov second derivative bound. This translates to an apriori interior $C^1$ estimate for smooth optimal maps. Here we give a new derivation of this estimate which relies in part on Kim, McCann and Warren's observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost.
We prove the interior $C^{1,1}$ regularity of the indirect utilities which solve a subclass of principal-agent problems originally considered by Figalli, Kim, and McCann. Our approach is based on construction of a suitable comparison function which, essentially, allows one to pinch the solution between parabolas. The original ideas for this proof arise from an earlier, unpublished, result of Caffarelli and Lions for bilinear preferences which we extend here to general quasilinear benefit functions. We give a simple example which shows the $C^{1,1}$ regularity is optimal.
The Bregman-Wasserstein divergence is the optimal transport cost when the underlying cost function is given by a Bregman divergence, and arises naturally in fields such as statistics and machine learning. We establish fundamental properties of the Bregman-Wasserstein divergence and propose a novel generalized transport geometry that promotes the Bregman geometry to the space of probability distributions. We provide a probabilistic interpretation involving exponential families and define generalized displacement interpolations compatible with the Bregman geometry. These interpolations are used to derive a generalized Pythagorean inequality, which is of independent interest. Furthermore, we construct a generalized dualistic geometry that lifts the differential geometry of the Bregman divergence to an infinite-dimensional statistical manifold. On the computational side, we demonstrate how Bregman-Wasserstein optimal transport maps can be estimated using neural approaches, establish the well-posedness of Bregman-Wasserstein barycenters, and relate them to Bayesian learning. Finally, we introduce the Bregman-Wasserstein JKO scheme for discretizing Riemannian Wasserstein gradient flows.
We provide a geometric interpretation of the well known A3w condition for regularity of optimal transport maps.
We prove two H\"older regularity results for solutions of generated Jacobian equations. First, that under the A3 condition and the assumption of nonnegative $L^p$ valued data solutions are $C^{1,\alpha}$ for an $\alpha$ that is sharp. Then, under the additional assumption of positive Dini continuous data, we prove a $C^{2}$ estimate. Thus the equation is uniformly elliptic and when the data is H\"older continuous solutions are in $C^{2,\alpha}$.
This is a PhD thesis about generated Jacobian equations; our purpose is twofold. First, we provide an introduction to these equations, whilst, at the same time, collating some results scattered throughout the literature. The other goal is to present the author's own results on these equations. These results all concern solutions of generated Jacobian equations, usually paired with the second boundary value problem. We prove strict convexity and $C^1$ differentiability results under optimal hypothesis in two dimensions, and the same results in higher dimensions with some additional hypothesis. We also consider uniqueness results for the second boundary value problem, and the application of the uniqueness results to global regularity. We conclude with notes on the parabolic generated Jacobian equation. The arXiv version contains minor updates to the ANU open research repository version which is available from the listed DOI.
This article has two purposes. The first is to prove solutions of the second boundary value problem for generated Jacobian equations are strictly $g$-convex. The second is to prove the global $C^3$ regularity of Aleksandrov solutions to the same problem under stronger hypothesis. These are related because the strict $g$-convexity is essential for the proof of the global regularity. The assumptions for the strict $g$-convexity are the natural extension of those used by Chen and Wang in the optimal transport case. They improve the existing domain conditions though at the expense of requiring a $C^3$ generating function. We prove the global regularity under the hypothesis that Jiang and Trudinger recently used to obtain the existence of a globally smooth solution and an additional condition on the height of solutions. Our proof of global regularity is by modifying Jiang and Trudinger's existence result to construct a globally $C^3$ solution intersecting the Aleksandrov solution. Then the strict convexity yields the interior regularity to apply the author's uniqueness results.
We present a proof of strict $g$-convexity in 2D for solutions of generated Jacobian equations with a $g$-Monge-Amp\`ere measure bounded away from 0. Subsequently this implies $C^1$ differentiability in the case of a $g$-Monge-Amp\`ere measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge-Amp\`ere case. Thus, like theirs, our argument is local and yields a quantitative estimate on the $g$-convexity. As a result our differentiability result is new even in the optimal transport case: we weaken previously required domain convexity conditions. Moreover in the optimal transport case and the Monge-Amp\`ere case our key assumptions, namely A3w and domain convexity, are necessary.
We prove that if two $C^{1,1}(Ω)$ solutions of the second boundary value problem for the generated Jacobian equation intersect in $Ω$ then they are the same solution. In addition we extend this result to $C^{2}(\overlineΩ)$ solutions intersecting on the boundary, via an additional convexity condition on the target domain.