Let Ω ⊆ R^d be open and D ⊆ ∂Ω be a closed part of its boundary. Under very mild assumptions on Ω, we construct a bounded Sobolev extension operator for the Sobolev space W_D ^k,p(Ω), 1 ⩽ p < ∞, which consists of all functions in W^k,p(Ω) that vanish in a suitable sense on D. In contrast to earlier work, this construction is global and does not use a localization argument, which allows to work with a boundary regularity that is sharp at the interface dividing D and ∂Ω \ D. Moreover, we provide homogeneous and local estimates for the extension operator. Also, we treat the case of Lipschitz function spaces with a vanishing trace condition on D.

In this paper we consider the variational setting for SPDE on a Gelfand triple (V,H,V∗). Under the standard conditions on a linear coercive pair (A, B), and a symmetry condition on A we manage to extrapolate the classical L2-estimates in time to Lp-estimates for some p>2 without any further conditions on (A, B). As a consequence we obtain several other a priori regularity results of the paths of the solution. Under the assumption that V embeds compactly into H, we derive a universal compactness result quantifying over all (A, B). As an application of the compactness result we prove global existence of weak solutions to a system of second order quasi-linear equations.

We show L^p-estimates for square roots of second order complex elliptic systems L in divergence form on open sets in R^d subject to mixed boundary conditions. The underlying set is supposed to be locally uniform near the Neumann boundary part, and the Dirichlet boundary part is Ahlfors–David regular. The lower endpoint for the interval where such estimates are available is characterized by p-boundedness properties of the semigroup generated by −L, and the upper endpoint by extrapolation properties of the Lax–Milgram isomorphism. Also, we show that the extrapolation range is relatively open in (1,∞).

This book establishes a comprehensive theory to treat square roots of elliptic systems incorporating mixed boundary conditions under minimal geometric assumptions. To lay the groundwork, the text begins by introducing the geometry of locally uniform domains and establishes theory for function spaces on locally uniform domains, including interpolation theory and extension operators. In these introductory parts, fundamental knowledge on function spaces, interpolation theory and geometric measure theory and fractional dimensions are recalled, making the main content of the book easier to comprehend. The centerpiece of the book is the solution to Kato's square root problem on locally uniform domains. The Kato result is complemented by corresponding Lp bounds in natural intervals of integrability parameters.
This book will be useful to researchers in harmonic analysis, functional analysis and related areas