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.

We establish existence, uniqueness and higher order weighted Lp-Sobolev regularity for the stochastic heat equation with zero Dirichlet boundary condition on angular domains and on polygonal domains in R 2 . We use a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary to measure the regularity with respect to the space variable. In this way we can capture the influence of both main sources for singularities: the incompatibility between noise and boundary condition on the one hand and the singularities of the boundary on the other hand. The range of admissible powers of the distance to the vertexes is described in terms of the maximal interior angle and is sharp.2010 Mathematics Subject Classification. 60H15; 35R60, 35K05.

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.

We use the scale B s τ (L τ (Ω)), 1/τ = s/d + 1/2, s > 0, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains Ω ⊂ R d , d ≥ 3, with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. By using Banach's fixed point theorem, we extend this analysis to the stationary Navier-Stokes equation with suitable Reynolds number and data, respectively.