An L p -estimate for the stochastic heat equation on an angular domain in R 2 - Cioica-Licht, Petru A.; Kim, Kyeong Hun; Lee, Kijung; Lindner, Felix (Stochastics and Partial Differential Equations: Analysis and Computations, 2018)
We prove a weighted L p -estimate for the stochastic convolution associated with the stochastic heat equation with zero Dirichlet boundary condition on a planar angular domain Dκ0⊂R2 with angle κ∈ (0 , 2 π). Furthermore, we use this estimate to establish existence and uniqueness of a solution to the corresponding equation in suitable weighted L p -Sobolev spaces. In order to capture the singular behaviour of the solution and its derivatives at the vertex, we use powers of the distance to the vertex as weight functions. The admissible range of weight parameters depends explicitly on the angle κ.
Keywords: Angular domain; Corner singularity; Non-smooth domain; Stochastic heat equation; Stochastic partial differential equation; Weighted L p -estimate; Weighted Sobolev regularity