Sobolev space theory and H\"older estimates for the stochastic partial differential equations on conic and polygonal domains

TL;DR

This paper proves existence, uniqueness, and regularity results for stochastic PDEs on conic and polygonal domains, extending Sobolev and H"older estimates with sharp weight ranges.

Contribution

It introduces new Sobolev and H"older regularity results for stochastic PDEs on conic and polygonal domains with sharp weight ranges.

Findings

Established existence and uniqueness of solutions.

Derived Sobolev and H"older regularity estimates.

Identified sharp ranges for weight parameters.

Abstract

We establish existence, uniqueness, and Sobolev and H\"older regularity results for the stochastic partial differential equation $$ du=\left(\sum_{i,j=1}^d a^{ij}u_{x^ix^j}+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt+\sum_{k=1}^{\infty}g^kdw^k_t, \quad t>0, \,x\in \mathcal{D} $$ given with non-zero initial data. Here $\{w^k_t: k=1,2,\cdots\}$ is a family of independent Wiener processes defined on a probability space $(\Omega, \mathbb{P})$, $a^{ij}=a^{ij}(\omega,t)$ are merely measurable functions on $\Omega\times (0,\infty)$, and $\mathcal{D}$ is either a polygonal domain in $\mathbb{R}^2$ or an arbitrary dimensional conic domain of the type \begin{equation} \label{conic} \mathcal{D}(\mathcal{M}):=\left\{x\in \mathbb{R}^d :\,\frac{x}{|x|}\in \mathcal{M}\right\}, \quad \quad \mathcal{M}\in S^{d-1}, \quad (d\geq 2) \end{equation} where $\mathcal{M}$ is an open subset of $S^{d-1}$ with $C^2$ boundary. We measure the Sobolev and H\"older regularities of arbitrary order derivatives of the solution using a system of mixed weights consisting of appropriate powers of the distance to the vertices and of the distance to the boundary. The ranges of admissible powers of the distance to the vertices and to the boundary are sharp.