Refined upper bounds on the coarsening rate of discrete, ill-posed diffusion equations

"We study coarsening phenomena observed in discrete, ill-posed diffusion equations that arise in a variety of applications, including computer vision, population dynamics and granular flow. Our results provide rigorous upper bounds on the coarsening rate in any dimension. Heuristic arguments and the numerical experiments we perform indicate that the bounds are in agreement with the actual rate of coarsening."http://deepblue.lib.umich.edu/bitstream/2027.42/64211/1/non8_12_002.pd

On the evolution of subcritical regions for the Perona-Malik equation

The Perona-Malik equation is a celebrated example of forward-backward parabolic equation. The forward behavior takes place in the so-called subcritical region, in which the gradient of the solution is smaller than a fixed threshold. In this paper we show that this subcritical region evolves in a different way in the following three cases: dimension one, radial solutions in dimension greater than one, general solutions in dimension greater than one. In the first case subcritical regions do not shrink, that is, that they expand with a nonnegative rate. In the second case they expand with a positive rate and always spread over the whole domain after a finite time, depending only on the (outer) radius of the domain. As a by-product, we obtain a nonexistence result for global-in-time classical radial solutions with large enough gradient. In the third case we show an example where subcritical regions do not expand. Our proofs exploit comparison principles for suitable degenerate and nonsmooth free boundary problems

The monopolist's problem: existence, relaxation, and approximation

We study a variational problem arising from a generalization of an economic model introduced by Rochet and Chone ́. In this model a monopolist proposes a set Y of products with pricelist p.Each rational consumer chooses which product to buy by solving a personal minimum problem, taking into account his/her tastes and economic possibilities. The monopolist looks for the optimal price list which minimizes costs, hence maximizes the profit. This leads to a minimum problem for functionals F(p) (the “pessimistic cost expectation”) and G(p) (the “optimistic cost expectation”), which are in turn defined through two nested variational problems. We prove that the minimum of G exists and coincides with the infimum of F. We also provide a variational approximation of G by smooth functionals defined in finite dimensional Euclidean spaces

Hyperbolic-parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation

We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 at the infinity. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t goes to infinity and as the parameter goes to 0

Gradient estimates for the Perona-Malik equation

We consider the Cauchy problem for the Perona–Malik equation in a bounded open set, with Neumann boundary conditions. In the one dimensional case, we prove some a priori estimates. Then we consider the semi-discrete scheme obtained by replacing the space derivatives by finite differences. Extending the previous estimates to the discrete setting we prove a compactness result for this scheme and we characterize the possible limits in some cases. Finally we give examples to show that the corresponding estimates are in general false in the more dimensional case

Global Existence and Asymptotic Behaviour for a Mildly Degenetate Dissipative Hyperbolic Equation of Kirchhoff Type

We consider the mildly degenerate Kirchhoff equation with a dissipative term. We prove that if the nonlinear term is locally Lipschitz continuous and the initial data are small enough then the Cauchy problem is globally well-posed for positive times. Moreover, we study the asymptotic behavior of the solutions

Global in time uniform convergences for linear hyperbolic-parabolic singular perturbations

We consider a linear hyperbolic-parabolic singular perturbation problem and we estimate the convergence rate of the solutions. Moreover we prove that our regularity requirement is sharp for our estimates

Optimal derivative loss for abstract wave equations

We consider an abstract wave equation with a propagation speed that depends only on time. We assume that the propagation speed is differentiable for positive times, continuous up to the origin, but with first derivative that is potentially singular at the origin. We examine the derivative loss of solutions, and in particular we investigate which conditions on the modulus of continuity and on the behavior of the derivative in the origin yield, respectively, no derivative loss, an arbitrarily small derivative loss, a finite derivative loss, or an infinite derivative loss. As expected, we obtain that stronger assumptions on the modulus of continuity can compensate weaker assumptions on the growth of the derivative, and viceversa. Suitable counterexamples show that our results are sharp. We prove indeed that, for every set of conditions, the class of propagation speeds that satisfy the given conditions, and for which the corresponding equation exhibits a derivative loss as large as possible, is nonempty and actually also residual in the sense of Baire category

A class of local classical solutions for the one dimensional Perona - Malik equation

We consider the Cauchy problem for the one-dimensional Perona- Malik equation with homogeneous Neumann boundary conditions. We prove that the set of initial data for which this equation has a local- in-time classical solution is dense in C^1

Unstable simple modes for a class of Kirchhoff equations

It is well known that Kirchhoff equations admits infinitely many simple modes, i. e. time periodic solutions with only one Fourier component in the space variable. We prove that, for some choices of the nonlinearity, these simple modes are unstable provided that their energy is large enough