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

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

doi:10.1007/s40072-017-0102-9

Cited by 8 publications

(10 citation statements)

References 24 publications

(17 reference statements)

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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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“…Proof. For simplicity of notation we only treat the case A = F. Let N be as in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) for…”

Section: Decomposition and Anisotropymentioning

confidence: 99%

“…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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“…Stochastic heat equation on a wedge. Our next application is an L p (L q )-version of the stochastic maximal regularity result in [CLKLL18] for the stochastic heat equation on an angular domain. The deterministic setting was considered in [Sol01, Theorem 1.1] and later improved in [Naz01, Theorem 1.1] and [PS07, Corollary 5.2].…”

Section: Applications To Stochastic Maximal Regularitymentioning

confidence: 99%

“…At the moment it is unclear whether the Dirichlet Laplacian −∆ on an angular domain has a bounded H ∞calculus, and how to characterize D((−∆) 1/2 ) in terms of weighted Sobolev spaces. Therefore, we can not apply [NVW12b] and instead we will use [CLKLL18] and extrapolation theory to derive L p (L q )-regularity results.…”

Section: Applications To Stochastic Maximal Regularitymentioning

confidence: 99%

Singular stochastic integral operators

Lorist,

Veraar

2019

Preprint

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In this paper we introduce Calderón-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove L p -extrapolation results under a Hörmander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the A2-conjecture. The results are applied to obtain pindependence and weighted bounds for stochastic maximal L p -regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on R d and smooth and angular domains. Contents 1. Introduction 1 2. Preliminaries 8 3. Stochastic integral operators 13 4. Singular γ-kernels of Hörmander and Dini type 20 5. Extrapolation for γ-integral operators 24 6. Sparse domination for γ-integral operators 32 7. Extension to spaces of homogeneous type 37 8. Applications to stochastic maximal regularity 38 9. p-Independence of the R-boundedness of convolutions 52 Appendix A. Technical estimates 54 References 60

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

Cited by 8 publications

References 24 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

Singular stochastic integral operators

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