A full interpolation theory for Sobolev functions with smoothness between 0 and 1 and vanishing trace on a part of the boundary of an open set is established. Geometric assumptions are of mostly measure theoretic nature and reach beyond Lipschitz regular domains. Previous results were limited to regular geometric configurations or Hilbertian Sobolev spaces. Sets with porous boundary and their characteristic multipliers on smoothness spaces play a major role in the arguments.
We show L p -estimates for square roots of second order 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. Our range is optimal, and upper and lower endpoints are sharp if they do not coincide with 1 or ∞.
We construct whole-space extensions of functions in a fractional Sobolev space of order $$s\in (0,1)$$
s
∈
(
0
,
1
)
and integrability $$p\in (0,\infty )$$
p
∈
(
0
,
∞
)
on an open set O which vanish in a suitable sense on a portion D of the boundary $${{\,\mathrm{\partial \!}\,}}O$$
∂
O
of O. The set O is supposed to satisfy the so-called interior thickness condition in$${{\,\mathrm{\partial \!}\,}}O {\setminus } D$$
∂
O
\
D
, which is much weaker than the global interior thickness condition. The proof works by means of a reduction to the case $$D=\emptyset $$
D
=
∅
using a geometric construction.
We show non-autonomous L q (L p ) maximal regularity for families of complex second-order systems in divergence form under a mixed Hölder regularity condition in space and time. To be more precise, we let p, q ∈ (1, ∞) and we consider coefficient functions in C β+ε t with values in C α+ε x subject to the parabolic relation 2β + α = 1. To this end, we provide a weak (p, q)-solution theory with uniform constants and establish a priori higher spatial regularity. Furthermore, we show p-bounds for semigroups and square roots generated by complex elliptic systems under a minimal regularity assumption for the coefficients.
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