A refined Green's function estimate of the time measurable parabolic operators with conic domains

Abstract

We present a refined Green's function estimate of the time measurable parabolic operators on conic domains that involves mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary.

An $L_p$-theory for the stochastic heat equation on angular domains in $\mathbb{R}^2$ with mixed weights

“…The key ingredient for the proof of this assertion is the following Green function estimate, recently proven in [11], see, in particular, Theorem 2.6 and Remark 2.2 therein as well as [12,Section 2], in particular, Theorem 2.4(3) therein. Moreover, we need some subtle estimates, which we prove in Section 4.…”

“…Theorem 2.2 ( [11]). For all 0 < λ 1 , λ 2 < π κ0 , there exist finite constants C, σ > 0 depending only on κ 0 , λ 1 , and λ 2 , such that…”