A weightedLp-theory for divergence type parabolic PDEs with BMO coefficients onC1-domains

Abstract

In this paper we present a weighted Lp-theory of second-order parabolic partial differential equations defined on C 1 domains. The leading coefficients are assumed to be measurable in time variable and have VMO (vanishing mean oscillation) or small BMO (bounded mean oscillation) with respect to space variables, and lower order coefficients are allowed to be unbounded and to blow up near the boundary. Our BMO condition is slightly relaxed than the others in the literature.

Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces

“…Following the arguments in [24] (also see [15,17]), as an application one can obtain the corresponding L p -theory for SPDEs with the coefficients in this paper. We also note that our results can be extended to Cauchy problems with appropriate initial conditions.…”

“…dx. Since the work in [20], there has been much attention to the solvability theory for equations in the weighted Sobolev spaces H γ p,θ ; see [14,18,15,17]. The necessity of such theory came from stochastic partial differential equations (SPDEs) and is well explained in [19].…”

“…Coefficients with small mean oscillations were considered in [25] for SPDEs in the setting of H γ p,θ . Recently, in [15,17] the authors treated nondivergence and divergence type equations, respectively, with coefficients having small mean oscillations. For instance, in [17] the coefficients are assumed to have small mean oscillations in both the space and time variables.…”

“…In the divergence case, we take larger function spaces for data on the right-hand side of the equations than those in the previous results. For instance, in [25,17] the right-hand sides of the equations under consideration have the form D i g i + f with g = (g 1 , . .…”

“…Therefore, in our case the right-hand side of (1.1) is in a larger space M −1 H −1 p,θ + L p,θ when λ > 0. As in [14,25,15,17] we allow the lower-order coefficients to blow up at certain rates near the boundary. On the other hand, in the previous results, those coefficients need to approach zero far away from the boundary when equations are considered in a half space.…”