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 $\frac{1}{2}$ 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

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

Coercivity for elliptic operators and positivity of solutions on Lipschitz domains

We show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of non-zero boundary measure or if a suitable Robin boundary condition is posed. Moreover, we prove the positivity of solutions in a general, abstract setting, provided that the right hand side is a positive functional. Finally, positive elements from $W^-1,2$ are identified as positive measures

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

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