Conormal problem of higher-order parabolic systems with time irregular coefficients

Abstract

The paper is a comprehensive study of the Lp and the Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients in the half space and cylindrical domains with conormal derivative boundary condition. For the Lp estimates, we assume that the leading coefficients are only bounded measurable in the t variable and V M O with respect to x. We also prove the Schauder estimates in two situations: the coefficients are Hölder continuous only in the x variable; the coefficients are Hölder continuous in both variables.2010 Mathematics Subject Classification. 35J58, 35R05, 35B45.

Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces

“…Although the coefficients in [20] are called VMO x coefficients, their mean oscillations in x do not have to vanish as the radii of cylinders go to zero. For more related work about L p theory with BMO x or partially BMO x coefficients for parabolic systems and higher-order parabolic systems, we refer the reader to [7,8,6,10] and the references therein.…”

On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients

“…Therefore, D 2 u may not be continuous up to the boundary unless g 2 /a 2 is continuous. We also note that similar to [12,Theorem 2.4], from the proof below we can see that regularity assumptions on the coefficients and data with respect to the time variable are only required near the boundary.…”

Convergence rates in homogenization of higher-order parabolic systems

“…For f ∈ L 2 (0, T ; H −m (Ω)), h ∈ L 2 (Ω), problem (1.1) with homogeneous Neumann boundary data admits a unique weak solution u ε ∈ L 2 (0, T ; H m (Ω)) ∩ L ∞ (0, T ; L 2 (Ω)) [5]. Similar to the initial-Dirichlet problem, under the periodicity condition (1.4) u ε converges strongly in L 2 (0, T ; H m−1 (Ω)) to the solution u 0 of the following initial-Neumann problem…”