Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications

Abstract

We study the homogenization of fully nonlinear degenerate second-order pde, with "ellipticity" of the same order as the space oscillations, in periodic and almost periodic. As a special case we consider the class of quasi-linear, degenerate elliptic pde. The results apply to level sets equations describing the evolution of fronts with prescribed normal velocity. We also discuss an application about the averaged properties of interfacial motions in periodic and almost periodic environments.

Hölder estimates for degenerate elliptic equations with coercive Hamiltonians

“…Thus we prove in Theorem 3.1 that viscosity solutions of (1.1) are locally Lipschitz continuous and that the gradient estimate |Du(x)| ≤ is the distance function from the boundary of Ω. Theorem 3.1 extends previous similar results obtained using the classical Bernstein method in [10], [13] for the nondegenerate case or in the recent paper [14] for the degenerate problem set in the whole space. Similar results have been established in [3] through the so-called weak Bernstein method applied to viscosity solutions.…”

“…This assumption (which, if one really takes ϕ = 1, could be further refined with only Σ 2 Σ x 2 in the right hand side) is analogous to the one used in [14] to get global Lipschitz estimates for solutions in the whole space. This shows that our approach can replace, in some situations, the classical Bernstein method of differentiating the equation.…”

“…The above method can be applied to different Hamiltonians H(x, ξ) and, correspondingly, different adapted choices of cutoff functions ϕ. Anyway, this approach requires the coercivity of the Hamiltonian (a somehow necessary ingredient for having estimates in the case of degenerate ellipticity; see also [14]). Indeed, if we rephrase our proof for general functions H, the optimal choice of the parameter t in (3.19) would read as…”

Curvature Effect in Shear Flow: Slowdown of Turbulent Flame Speeds with Markstein Number

“…Owing to [9], when δ is small eough, the cell problem (1.4) has a viscosity solution. Formally, we can write the solution as…”

A priori gradient bounds for fully nonlinear parabolic equations and applications to porous medium models

“…Let us notice that in [13], a uniform gradient estimate is given for bounded solutions of the elliptic equation…”