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

Abstract

We prove a refined Lp-estimate (p ≥ 2) for the stochastic heat equation on angular domains in R 2 with mixed weights based on both, the distance to the boundary and the distance to the vertex. This way we can capture both causes for singularities of the solution: the incompatibility of noise and boundary condition on the one hand and the influence of boundary singularities (here, the vertex) on the other hand. Higher order Lp-Sobolev regularity with mixed weights is also established.

An extension operator for Sobolev spaces with mixed weights

“…Suppose that the boundary 𝜕 K consists of the vertex 𝑥 = 0, the edges (half lines) 𝑀 1 , … , 𝑀 𝑛 , and smooth faces Γ 1 , … , Γ 𝑛 . Hence, Ω ∩ 𝑆 2 is a domain of polygonal type on the unit sphere 𝑆 2 with sides Γ 𝑘 ∩ 𝑆 2 . Moreover, we consider the bounded polyhedral cone K obtained via truncation…”

“…[9], but can be compensated with suitable weight functions. In particular, spaces with mixed weights are needed when studying stochastic PDEs as is demonstrated in [2,3]. The weighted Sobolev spaces we are interested in can be seen as generalizations of the so-called Kondratiev spaces which appeared in the 60s in [10,11] and were studied in detail in [7].…”

An extension operator for Sobolev spaces with mixed weights

“…[9], but can be compensated with suitable weight functions. In particular, spaces with mixed weights are needed when studying stochastic PDEs as is demonstrated in [3,2]. The weighted Sobolev spaces we are interested in can be seen as generalizations of the so-called Kondratiev spaces which appeared in the 60s in [10,11] and were studied in detail in [6].…”

Sobolev meets Besov: Regularity for the Poisson equation with Dirichlet, Neumann and mixed boundary values

“…2.2]. Hence, it can be seen from ( 3) that the achievable order of adaptive algorithms depends on the regularity of the target function in the specific scale of Besov spaces (2).…”

“…For this we use Sobolev spaces with mixed weights from Maz'ya, Rossmann [22], sometimes denoted as V -spaces in the sequel. In particular, spaces with mixed weights are needed when studying stochastic PDEs as is demonstrated in [3,2]. To be precise, the weighted Sobolev spaces V l,p β,δ (K) with parameters l ∈ N 0 , 1 ≤ p < ∞, β ∈ R, and δ = (δ 1 , .…”