Parabolic Lp Dirichlet boundary value problem and VMO-type time-varying domains

Abstract

We prove the solvability of the parabolic L p Dirichlet boundary value problem for 1 < p ≤ ∞ for a PDE of the form ut = div(A∇u) + B • ∇u on time-varying domains where the coefficients A = [a ij (X, t)] and B = [b i ] satisfy a certain natural small Carleson condition. This result brings the state of affairs in the parabolic setting up to the elliptic standard.Furthermore, we establish that if the coefficients of the operator A, B satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic L p Dirichlet boundary value problem is solvable for all 1 < p ≤ ∞. This result is related to results in papers by Maz'ya, Mitrea and Shaposhnikova, and Hofmann, Mitrea and Taylor where the fact that boundary of domain has normal in VMO or near VMO implies invertibility of certain boundary operators in L p for all 1 < p ≤ ∞ which then (using the method of layer potentials) implies solvability of the L p boundary value problem in the same range for certain elliptic PDEs.Our result does not use the method of layer potentials since the coefficients we consider are too rough to use this technique, but remarkably we recover L p solvability in the full range of p's as the two papers mentioned above.

An Integralgeometric Approach to Dorronsoro Estimates

“…Acknowledgements. I thank Katrin Fässler for many useful discussions during the preparation of this paper, and in particular for pointing out the references [6,11]. I am also thankful to Martí Prats for pointing out the reference [9].…”

“…Having said that, since the very definition of being parabolic Lipschitz contains the Fourier transform, literally nothing can be proven about these functions without some reference to Fourier analysis. The derivation of (3.4) from Definition 3.1 in [6] consists of verifying that the parabolic derivative Dψ of ψ is in BMO (as explained on [6, p. 5-6], this not quite the same object as D n ψ), and then inferring condition (3.4) from the classical work of Strichartz [18] (as detailed in the appendix of [6], more precisely [6, (8.4)-(8.5)]).…”

Sub-elliptic boundary value problems in flag domains

“…For precise definitions, see [37]. As a more recent reference where (more general) parabolic equations are solved by means of layer potentials in time-varying domains, see the paper [16] of Dindoš, Dyer, and Hwang. 1.4.2.…”