In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation ∂ t u=Δ(Δu) −3 . This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solution u with Δu≥0 is obtained via a suitable substitution. Our investigations reveal the close connection between this problem and the equation ∂ t ρ+ρ 2 Δ 2 ρ 3 =0 , another crystal surface model first proposed by H. Al Hajj Shehadeh, R. V. Kohn and J. Weare in [1].

It is well-known that in Banach spaces with finite cotype, the $R$ -bounded and $\gamma $ -bounded families of operators coincide. If in addition $X$ is a Banach lattice, then these notions can be expressed as square function estimates. It is also clear that $R$ -boundedness implies $\gamma $ -boundedness. In this note we show that all other possible inclusions fail. Furthermore, we will prove that $R$ -boundedness is stable under taking adjoints if and only if the underlying space is $K$ -convex.

We show that positive$α$-stable densities are hyperbolically completely monotone if and only if$α$≤1/2. This gives a positive answer to a question raised by L. Bondesson in 1977.

We study the homogenization of a stationary conductivity problem in a random heterogeneous medium with highly oscillating conductivity coefficients and an ensemble of simply closed conductivity resistant membranes. This medium is randomly deformed and then rescaled from a periodic one with periodic membranes, in a manner similar to the random medium proposed by Blanc, Le Bris, and Lions (2006). Across the membranes, the flux is continuous but the potential field itself undergoes a jump of Robin-type. We prove that, for almost all realizations of the random deformation, as the small scale of variations of the medium goes to zero, the random conductivity problem is well approximated by that of an effective medium which has deterministic and constant coefficients and contains no membrane. The effective coefficients are explicitly represented. One of our main contributions is to provide a solution to the associated auxiliary problem that is posed on the whole space with infinitely many interfaces, a setting that is out of the standard stationary ergodic framework.

Light bullets are spatially localized ultra-short optical pulses in more than one space dimensions. They contain only a few electromagnetic oscillations under their envelopes and propagate long distances without essentially changing shapes. Light bullets of femtosecond durations have been observed in recent numerical simulation of the full Maxwell systems. The sineGordon (SG) equation comes as an asymptotic reduction of the two level dissipationless MaxwellBloch system. We derive a new and complete nonlinear Schrdinger (NLS) equation in two space dimensions for the SG pulse envelopes so that it is globally well-posed and has all the relevant higher order terms to regularize the collapse of the standard critical NLS (CNLS). We perform a modulation analysis and found that SG pulse envelopes undergo focusingdefocusing cycles. Numerical results are in qualitative agreement with asymptotics and