TY - JOUR

T1 - Sobolev space theory and Hölder estimates for the stochastic partial differential equations on conic and polygonal domains

AU - Kim, Kyeong Hun

AU - Lee, Kijung

AU - Seo, Jinsol

N1 - Funding Information:
The first and third authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1A2C1A01003354).The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1F1A1058988).
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/12/15

Y1 - 2022/12/15

N2 - We establish existence, uniqueness, and Sobolev and Hölder regularity results for the stochastic partial differential equation du=(∑i,j=1daijuxixj+f0+∑i=1dfxii)dt+∑k=1∞gkdwtk,t>0,x∈D given with non-zero initial data. Here {wtk:k=1,2,⋯} is a family of independent Wiener processes defined on a probability space (Ω,P), aij=aij(ω,t) are merely measurable functions on Ω×(0,∞), and D is either a polygonal domain in R2 or an arbitrary dimensional conic domain of the type [Formula presented] where M is an open subset of Sd−1 with C2 boundary. We measure the Sobolev and Hölder 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.

AB - We establish existence, uniqueness, and Sobolev and Hölder regularity results for the stochastic partial differential equation du=(∑i,j=1daijuxixj+f0+∑i=1dfxii)dt+∑k=1∞gkdwtk,t>0,x∈D given with non-zero initial data. Here {wtk:k=1,2,⋯} is a family of independent Wiener processes defined on a probability space (Ω,P), aij=aij(ω,t) are merely measurable functions on Ω×(0,∞), and D is either a polygonal domain in R2 or an arbitrary dimensional conic domain of the type [Formula presented] where M is an open subset of Sd−1 with C2 boundary. We measure the Sobolev and Hölder 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.

KW - Conic domains

KW - Mixed weight

KW - Parabolic equation

KW - Weighted Sobolev regularity

UR - http://www.scopus.com/inward/record.url?scp=85138039081&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2022.09.003

DO - 10.1016/j.jde.2022.09.003

M3 - Article

AN - SCOPUS:85138039081

VL - 340

SP - 463

EP - 520

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -