Singular behavior of the solution to the stochastic heat equation on a polygonal domain

Lindner, Felix

doi:10.1007/s40072-014-0030-x

Cited by 9 publications

(5 citation statements)

References 31 publications

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“…To mention an example, there exist bounded C 1 domains Ω ⊆ R d such that if we define S(Ω) to be the set of all solutions to the Poisson equation with zero Dirichlet boundary conditions and right hand sides f ∈ C ∞ (Ω), then s p (S(Ω)) = 1 + 1/p, see Section 3.1 for details. Similar results for (stochastic) evolution equations can be found, e.g., in [20,25]. At the same time, we know that the solution to most of the equations in the aforementioned references may have higher regularity α > s p in the scale ( * ), see, e.g., [3,5,7,8,10,11,12,15,16,21].…”

Section: Introductionsupporting

confidence: 76%

On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

Cioica-Licht

Weimar

2020

J Fourier Anal Appl

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We study the interrelation between the limit L p (Ω)-Sobolev regularity s p of (classes of) functions on bounded Lipschitz domains Ω ⊆ R d , d ≥ 2, and the limit regularity α p within the corresponding adaptivity scale of Besov spaces B α τ,τ (Ω), where 1/τ = α/d + 1/p and α > 0 (p > 1 fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best N -term approximation. We show how additional information on the Besov or Triebel-Lizorkin regularity may be used to deduce upper bounds for α p in terms of s p simply by means of classical embeddings and the extension of complex interpolation to suitable classes of quasi-Banach spaces due to Kalton, Mayboroda, and Mitrea (Contemp. Math. 445). The results are applied to the Poisson equation, to the p-Poisson problem, and to the inhomogeneous stationary Stokes problem. In particular, we show that already established results on the Besov regularity for the Poisson equation are sharp.

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Section: Introductionsupporting

confidence: 76%

On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

Cioica-Licht

Weimar

2020

J Fourier Anal Appl

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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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“…An analogous result holds for the deterministic heat equation, restricting the spatial L p -Sobolev regularity of the solution accordingly, see, e.g., [7,Section 5.2] and [8,Section 8]. For p = 2, it has recently been extended to the case of a semilinear stochastic heat equation in [22], where a framework of Sobolev spaces without weights and with possibly negative orders of smoothness in time has been used. In order to relate these results to our problem, we note that if p ≥ 2 and the inner integral O .…”

Section: Introductionmentioning

confidence: 99%

An $$L_{p}$$ L p -estimate for the stochastic heat equation on an angular domain in $$\mathbb {R}^2$$ R 2

Cioica-Licht

Kim

Lee

et al. 2017

Stoch PDE: Anal Comp

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We prove a weighted Lp-estimate for the stochastic convolution associated to the stochastic heat equation with zero Dirichlet boundary condition on a planar angular domain Dκ 0 ⊂ R 2 with angle κ 0 ∈ (0, 2π). Furthermore, we use this estimate to establish existence and uniqueness of a solution to the corresponding equation in suitable weighted Lp-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 0 .2010 Mathematics Subject Classification. 60H15; 35R60, 35K05.

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Singular behavior of the solution to the stochastic heat equation on a polygonal domain

Cited by 9 publications

References 31 publications

On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

On the Limit Regularity in Sobolev and Besov Scales Related to Approximation Theory

Stochastic maximal regularity for rough time-dependent problems

An $$L_{p}$$ L p -estimate for the stochastic heat equation on an angular domain in $$\mathbb {R}^2$$ R 2

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