The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$

We show that, under general conditions, the operator $bigl (- abla cdot mu abla +1bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(Omega)$ and $L^p(Omega)$, for $p in {]1,2[}$ if one presupposes that this isomorphism holds true for $p=2$. The domain $Omega$ is assumed to be bounded, the Dirichlet part $D$ of the boundary has to satisfy the well-known Ahlfors-David condition, whilst for the points from $overline {partial Omega setminus D}$ the existence of bi-Lipschitzian boundary charts is required.

A Meyers type regularity result for approximations of second order elliptic operators by Galerkin schemes

We prove a Meyers type regularity estimate for approximate solutions of second order elliptic equations obtained by Galerkin methods. The proofs rely on interpolation results for Sobolev spaces on graphs. Estimates for second order elliptic operators on rather general graphs are also obtained.

Sobolev spaces on multiple cones

The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $RR^n$. The analysis interestingly combines use of Poincare inequalities and of some Hardy type inequalities.

An atomic decomposition of the Haj{l}asz Sobolev space $Mone$ on manifolds

Several possible notions of Hardy-Sobolev spaces on a Riemannian manifold with a doubling measure are considered. Under the assumption of a Poincare inequality, the space $Mone$, defined by Haj{l}asz, is identified with a Hardy-Sobolev space defined in terms of atoms. Decomposition results are proved for both the homogeneous and the nonhomogeneous spaces.