42 research outputs found
The Kato Square Root Problem for Mixed Boundary Conditions
We consider the negative Laplacian subject to mixed boundary conditions on a
bounded domain. We prove under very general geometric assumptions that slightly
above the critical exponent its fractional power domains still
coincide with suitable Sobolev spaces of optimal regularity. In combination
with a reduction theorem recently obtained by the authors, this solves the Kato
Square Root Problem for elliptic second order operators and systems in
divergence form under the same geometric assumptions.Comment: Inconsistencies in Section 6 remove
Recommended from our members
Maximal parabolic regularity for divergence operators on distribution spaces
We show that elliptic second order operators A of divergence type
fulfill maximal parabolic regularity on distribution spaces, even if the
underlying domain is highly non-smooth, the coefficients of A are
discontinuous and A is complemented with mixed boundary conditions.
Applications to quasilinear parabolic equations with non-smooth data are
presente
Maximal parabolic regularity for divergence operators on distribution spaces
We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented
The Kato Square Root Problem follows from an extrapolation property of the Laplacian
On a domain Ω ⊆ _ Rd we consider second-order elliptic systems in divergence-form with bounded complex coefficients, realized via a sesquilinear form with domain H1/0 (Ω) ⊆ V ⊆ H1 (Ω). Under very mild assumptions on Ω and V we show that the solution to the Kato Square Root Problem for such systems can be deduced from a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends earlier results of McIntosh [25] and Axelsson-Keith-McIntosh [6] to non-smooth coefficients and domains
Recommended from our members
Hölder continuity for second order elliptic problems with nonsmooth data
The well known De Giorgi result on Hölder continuity for solutions of
the Dirichlet problem is re-established for mixed boundary value problems,
provided that the underlying domain is a Lipschitz domain and the border
between the Dirichlet and the Neumann boundary part satisfies a very general
geometric condition. Implications of this result for optimal control theory
are presented
Hölder continuity for second order elliptic problems with nonsmooth data
The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented
Direct computation of elliptic singularities across anisotropic, multi-material edges
We characterise the singularities of elliptic div-grad operators at points or edges where several materials meet on a Dirichlet or Neumann part of the boundary of a two- or three-dimensional domain. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several classes of three- and four-material edges. These bounds can be used to prove optimal regularity results for elliptic div-grad operators on three-dimensional, heterogeneous, polyhedral domains with mixed boundary conditions. We demonstrate this for the benchmark L--shape problem