On the wave equation with a large rough potential

On the Yamabe Equation with Rough Potentials

We study the existence of non–trivial solutions to the Yamabe equation: −∆u+ a(x) = μu|u| 4 n−2 μ > 0, x ∈ Ω ⊂ R with n ≥ 4, u(x) = 0 on ∂Ω under weak regularity assumptions on the potential a(x). More precisely in dimension n ≥ 5 we assume that: (1) a(x) belongs to the Lorentz space L n 2 (Ω) for some 1 ≤ d < ∞, (2) a(x) ≤ M < ∞ a.e. x ∈ Ω, (3) the set {x ∈ Ω|a(x) < 0} has positive measure, (4...

Dispersive Estimates for a Linear Wave Equation with Electromagnetic Potential

We consider radial solutions to the Cauchy problem for a linear wave equation with a small short-range electromagnetic potential (depending on space and time) and zero initial data. We present two dispersive estimates that provide, in particular, an optimal decay rate in time t−1 for the solution. Also, we apply these estimates to obtain similar results for the linear massless Dirac equation pe...

Decay Estimates for the Wave Equation with Potential*

A High Order Approximation of the Two Dimensional Acoustic Wave Equation with Discontinuous Coefficients

This paper concerns with the modeling and construction of a fifth order method for two dimensional acoustic wave equation in heterogenous media. The method is based on a standard discretization of the problem on smooth regions and a nonstandard method for nonsmooth regions. The construction of the nonstandard method is based on the special treatment of the interface using suitable jump conditio...

Boltzmann Equation with a Large Potential in a Periodic Box

The stability of the Maxwellian of the Boltzmann equation with a large amplitude external potential Φ has been an important open problem. In this paper, we resolve this problem with a large C−potential in a periodic box T, d ≥ 3. We use [1] in Lp−L∞ framework to establish the well-posedness and the L∞−stability of the Maxwellian μE(x, v) = exp { − |v| 2 2 − Φ(x) } .