Related papers: Gradient Flows for Dirichlet Eigenvalues
We consider a fully nonlinear parabolic equation with nonlinear Neumann type boundary condition, and show that the longtime existence and convergence of the flow. Finally we apply this study to the boundary value problem for minimal…
We study Brakke's mean curvature flow with obstacles and with a right-angle boundary condition. Assuming that the obstacles have $C^{1,1}$-boundaries we prove that a weak solution exists globally in time. To show the existence we apply the…
We are concerned with inverse boundary problems for first order perturbations of the Laplacian, which arise as model operators in the acoustic tomography of a moving fluid. We show that the knowledge of the Dirichlet--to--Neumann map on the…
In this paper, we show existence and uniqueness of Ricci flow whose initial condition is a compact Alexandrov surface with curvature bounded from below. This requires a weakening of the notion of initial condition which is able to deal with…
We establish a representation of the heat flow with Wentzell boundary conditions on smooth domains as gradient descent dynamics for the entropy in a suitably extended Otto manifold of probability measures with additional boundary parts. Yet…
We prove eigenvalue processes from dynamical random matrix theory including Dyson Brownian motion, Wishart process, and Dynkin's Brownian motion of ellipsoids are results of projecting Brownian motion through Riemannian submersions induced…
We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted $p$-operator \[ -\Delta_{p,\gamma}u=-\text{div}(|\nabla u|^{p-2} \nabla u)+(x,\nabla u)|\nabla u|^{p-2}, \] where $p>1$, in the class…
We review the first and second boundary value problems for the Stokes system posed in a bounded Lipschitz domain in $\mathbb{R}^n.$ Particular attention is given to the mixed boundary condition: a Dirichlet condition is imposed for the…
This paper presents existence and uniqueness results for a class of parabolic systems with non linear diffusion and nonlocal interaction. These systems can be viewed as regular perturbations of Wasserstein gradient flows. Here we extend…
The aim of this thesis is to derive new gradient estimates for parabolic equations. The gradient estimates found are independent of the regularity of the initial data. This allows us to prove the existence of solutions to problems that have…
In this work we establish a gradient bound and Liouville-type theorems for solutions to Quasi-linear elliptic equations on compact Riemannian Manifolds with nonnegative Ricci curvature. Also, we provide a local splitting theorem when the…
We introduce the the fractional Laplacian on a subgraph of a graph with Dirichlet boundary condition. For a lattice graph, we prove the upper and lower estimates for the sum of the first $k$ Dirichlet eigenvalues of the fractional…
We consider the first Robin eigenvalue $\l_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of…
The goal of this paper is to relax convexity assumption on some classical results in mean curvature flow. In the first half of the paper, we prove a generalized version of Hamilton's differential Harnack inequality which holds for mean…
We prove global second-order regularity for a class of quasilinear elliptic equations, both with homogeneous Dirichlet and Neumann boundary conditions. A condition on the integrability of the second fundamental form on the boundary of the…
We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity…
One of the most popular approaches for solving total variation-regularized optimization problems in the space of measures are Particle Gradient Flows (PGFs). These restrict the problem to linear combinations of Dirac deltas and then perform…
We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer that this functional is asymptotically…
Flows on surfaces are one of the most fundamental and classical objects in dynamical systems, and are studied from various areas (e.g. integrable systems, differential equations, fluid mechanics). Though hyperbolic flows and recurrent flows…
In this paper, we study the evolution of submannifold moving by mean curvature minus a external force field. We prove that the flow has a long-time smooth solution for all time under almost optimal conditions. Those conditions are that the…