arXiv subjects
Publications and source records attributed to Ben Weinkove.
We consider the setting of two disks in a domain in $\mathbb{R}^2$ which are almost touching and have finite and positive conductivities, giving rise to a divergence form elliptic equation with discontinuous coefficients. We use the maximum principle to give a new proof of a gradient bound of Li-Vogelius.
Dimension reduction, widely used in science, maps high-dimensional data into low-dimensional space. We investigate a basic mathematical model underlying the techniques of stochastic neighborhood embedding (SNE) and its popular variant t-SNE. Distances between points in high dimensions are used to define a probability distribution on pairs of points, measuring how similar the points are. The aim is to map these points to low dimensions in an optimal way so that similar points are closer together. This is carried out by minimizing the relative entropy between two probability distributions. We consider the gradient flow of the relative entropy and analyze its long-time behavior. This is a self-contained mathematical problem about the behavior of a system of nonlinear ordinary differential equations. We find optimal bounds for the diameter of the evolving sets as time tends to infinity. In particular, the diameter may blow up for the t-SNE version, but remains bounded for SNE.
The $p$-Laplacian evolution equation and the $α$-Gauss curvature flow with a flat side are degenerate parabolic equations with evolving free boundaries. We give proofs of smooth short-time existence, up to the free boundaries, using a result of the authors on linear degenerate equations on a fixed domain.
We show that the porous medium equation does not in general preserve $α$-concavity of the pressure for $0\leα<1/2$ or $1/2<α\le 1$. In particular, this resolves an open problem of Vázquez on whether concavity of pressure is preserved by the porous medium equation. Our results strengthen an earlier work of Ishige-Salani, who considered the case of small $α>0$. Since Daskalopoulos-Hamilton-Lee showed that $1/2$-concavity is preserved, our result is sharp. Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For $0\leα<1/2$, we give another set of examples to show that concavity can be broken at a boundary point.
We prove existence of smooth solutions to linear degenerate parabolic equations on bounded domains assuming a structure condition of Fichera. We use this to give a proof of a smooth short time existence result for the porous medium equation $u_t = Δu^m$ for $1<m\le 2$.
The perfect conductivity problem concerns optimal bounds for the magnitude of an electric field in the presence of almost touching perfect conductors. This reduces to obtaining gradient estimates for harmonic functions with Dirichlet boundary conditions in the narrow region between the conductors. In this paper we extend estimates of Bao-Li-Yin to deal with the case when the boundaries of the conductors are given by graphs with arbitrary vanishing orders. Our estimates allow us to deal with globally defined narrow regions with possibly non-trivial topology. We also prove the sharpness of our estimates in terms of the distance between the perfect conductors. The precise optimality statement we give is new even in the setting of Bao-Li-Yin.
We consider a well-known quasi-static model for the shape of a liquid droplet. The solution can be described in terms of time-evolving domains in $\mathbb{R}^n$. We give an example to show that convexity of the domain can be instantaneously broken.
We consider the Dirichlet problem for a class of semilinear equations on two dimensional convex domains. We give a sufficient condition for the solution to be concave. Our condition uses comparison with ellipses, and is motivated by an idea of Kosmodem'yanskii. We also prove a result on propagation of concavity of solutions from the boundary, which holds in all dimensions.
We consider the insulated conductivity problem with two unit balls as insulating inclusions, a distance of order $\varepsilon$ apart. The solution $u$ represents the electric potential. In dimensions $n \ge 3$ it is an open problem to find the optimal bound on the gradient of $u$, the electric field, in the narrow region between the insulating bodies. Li-Yang recently proved a bound of order $\varepsilon^{-(1-γ)/2}$ for some $γ>0$. In this paper we use a direct maximum principle argument to sharpen the Li-Yang estimate for $n \ge 4$. Our method gives effective lower bounds on the best constant $γ$, which in particular approach $1$ as $n$ tends to infinity.
We give a survey on the Chern-Ricci flow, a parabolic flow of Hermitian metrics on complex manifolds. We emphasize open problems and new directions.
We prove local strong convexity of the space-time level sets of the heat equation on convex rings for zero initial data, strengthening a result of Borell. Our proof introduces a parabolic version of a two-point maximum principle of Rosay-Rudin.
We consider convex solutions of nonlinear elliptic equations which satisfy the structure condition of Bian-Guan. We prove a weak Harnack inequality for the eigenvalues of the Hessian of these solutions. This can be viewed as a quantitative version of the constant rank theorem.
We construct examples for the one-phase Stefan problem which show that $α$-concavity of the solution is in general not preserved in time, for $0 \le α<1/2$. In particular, this shows that, in contrast to the case of the heat equation for a fixed convex domain, log concavity is not preserved for solutions of the Stefan problem.
We consider the complex Monge-Ampère equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on compact Hermitian manifolds.
We extend the continuity equation of La Nave-Tian to Hermitian metrics and establish its interval of maximal existence. The equation is closely related to the Chern-Ricci flow, and we illustrate this in the case of elliptic bundles over a curve of genus at least two.
We construct solutions to the heat equation on convex rings showing that quasiconcavity may not be preserved along the flow, even for smooth and subharmonic initial data.
We establish a maximum principle for a two-point function in order to analyze the convexity of level sets of harmonic functions. We show that this can be used to prove a strict convexity result involving the smallest principal curvature of the level sets.
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.