Numerical solution of the simple Monge–Ampère equation with nonconvex dirichlet data on non-convex domains

The existence of a unique numerical solution of the semi-Lagrangian method for the simple Monge-Ampere equation is known independently of the convexity of the domain or Dirichlet boundary data - when the Monge-Ampere equation is posed as a Bellman problem. However, the convergence to the viscosity solution has only been proved on strictly convex domains. In this paper, we provide numerical evidence that convergence of numerical solutions is observed more generally without convexity assumptions. We illustrate how in the limit multivalued functions may be approximated to satisfy the Dirichlet conditions on the boundary as well as local convexity in the interior of the domai

The iterative properties of solutions for a singular k-Hessian system

In this paper, we focus on the uniqueness and iterative properties of solutions for a singular k-Hessian system involving coupled nonlinear terms with different properties. Unlike the existing work, instead of directly dealing with the system, we use a coupled technique to transfer the Hessian system to an integral equation, and then by introducing an iterative technique, the iterative properties of solution are derived including the uniqueness of solution, iterative sequence, the error estimation and the convergence rate as well as entire asymptotic behaviour

A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows

In this article we set up a splitting variant of the JKO scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge-Kantorovich distance, and then for the Hellinger/Fisher-Rao distance. Exploiting some inf-convolution structure of the metric we show convergence of the whole process for the standard class of energy functionals under suitable compactness assumptions, and investigate in details the case of internal energies. The interest is double: On the one hand we prove existence of weak solutions for a certain class of reaction-advection-diffusion equations, and on the other hand this process is constructive and well adapted to available numerical solvers.Comment: Final version, to appear in SIAM SIM

Ground States in the Diffusion-Dominated Regime

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric decreasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and $C^\infty$ inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness