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.

In this paper, we present counterexamples to maximal L^p-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal L²-regularity on H^-1 under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal L^p-regularity on H^-1(R^d) or L²-regularity on L²(R^d).

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,∞).

We show weighted non-autonomous L^q(L^p) maximal regularity for families of complex second-order systems in divergence form under a mixed regularity condition in space and time. To be more precise, we let p,q∈(1,∞) and we consider coefficient functions in C_t^β+ε with values in C_x^α+ε subject to the parabolic relation 2β+α=1. If [Formula presented], we can likewise deal with spatial [Formula presented] regularity. The starting point for this result is a weak (p,q)-solution theory with uniform constants. Further key ingredients are a commutator argument that allows us to establish higher a priori spatial regularity, operator-valued pseudo differential operators in weighted spaces, and a representation formula due to Acquistapace and Terreni. Furthermore, we show p-bounds for semigroups and square roots generated by complex elliptic systems under a minimal regularity assumption for the coefficients.