On the regularity of the stochastic heat equation on polygonal domains in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>

Cioica-Licht, Petru A.; Kim, Kyeong Hun; Lee, Kijung

doi:10.1016/j.jde.2019.06.027

Cited by 14 publications

(10 citation statements)

References 17 publications

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“…[35,[55][56][57][58]60,61,63,74] where also the case of smooth domains has been considered, and later to e.g. [18][19][20]59,81] where the case of non-smooth domains is investigated. In the above mentioned results one uses L p -integrability in space, time and .…”

Section: Spdes Of Second Ordermentioning

confidence: 99%

Stochastic maximal regularity for rough time-dependent problems

Portal

Veraar

2019

Stoch PDE: Anal Comp

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We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain L p (L q) estimates for all p > 2 and q ≥ 2, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition, and obtain L p (L p) estimates together with optimal space-time regularity. For linear second order equations in divergence form with random coefficients that are merely measurable in both space and time, we obtain estimates in the tent spaces T p,2 σ of Coifman-Meyer-Stein. This is done in the deterministic case under no extra assumption, and in the stochastic case under the assumption that the coefficients are divergence free. Keywords Stochastic PDEs • Maximal regularity • VMO coefficients • Measurable coefficients • Higher order equations • Sobolev spaces • A p-weights

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Section: Spdes Of Second Ordermentioning

confidence: 99%

Stochastic maximal regularity for rough time-dependent problems

Portal

Veraar

2019

Stoch PDE: Anal Comp

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“…Other works on maximal temporally weighted L p -regularity are [51,55] for quasilinear parabolic evolution equations and [66] for parabolic problems with inhomogeneous boundary conditions. Concerning the use of spatial weights in applications to (S)PDEs, we would like to mention [1,7,8,14,15,25,30,50,52,53,59,57,60,63,62,64,72,85].…”

Section: Introductionmentioning

confidence: 99%

Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

Hummel¹,

Lindemulder²

2019

Preprint

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In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted L q -maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A ∞class. In Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the A p -range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

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“…Remark 2.4. The norm of the space V l,p β,δ (R 3 , S) is defined similarly as the norm of V l,p β,δ (K, S) replacing the integral domain K by R 3 . In order to study inhomogeneous boundary conditions g j = 0 later on, we introduce the trace spaces V l−1/p,p β,δ…”

Section: Preliminariesmentioning

confidence: 99%

“…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 , .…”

Section: Introductionmentioning

confidence: 99%

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

Schneider¹,

Szemenyei²

2021

Preprint

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We study the regularity of solutions to Dirichlet, Neumann and mixed boundary value problems in polyhedral cones K ⊂ R 3 in the specific scale B α τ,τ , 1 τ = α d + 1 p of Besov spaces. The regularity of the solution in these spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We show that the solutions are much smoother in this specific Besov scale compared to the ususal L 2 -Sobolev scale, which justifies the use of adaptive schemes. Our findings considerably extend [12, Thms. 2.1, 3.1]. The proofs are performed by combining Sobolev estimates with mixed weights -which measure the distance to the vertex and the edges of the cone, respectively -with characterizations of Besov spaces by wavelet expansions.

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On the regularity of the stochastic heat equation on polygonal domains in R2

Cited by 14 publications

References 17 publications

Stochastic maximal regularity for rough time-dependent problems

Stochastic maximal regularity for rough time-dependent problems

Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

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

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