On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

We consider the problem of describing the traces of functions in $$H^2(\Omega )$$ H 2 ( Ω ) on the boundary of a Lipschitz domain $$\Omega $$ Ω of $$\mathbb R^N$$ R N , $$N\ge 2$$ N ≥ 2 . We provide a definition of those spaces, in particular of $$H^{\frac{3}{2}}(\partial \Omega )$$ H 3 2 ( ∂ Ω ) , by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space $$H^1(\Omega )$$ H 1 ( Ω ) , based on the classical second order Steklov problem.

On BMO and Carleson Measures on Riemannian Manifolds

Let $\mathcal{M}$ be a Riemannian $n$-manifold with a metric such that the manifold is Ahlfors regular. We also assume either non-negative Ricci curvature or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions $u: \mathcal{M} \to \mathbb{R}$ by a Carleson measure condition of their $\sigma $-harmonic extension $U: \mathcal{M} \times (0,\infty ) \to \mathbb{R}$. We make crucial use of a $T(b)$ theorem proved by Hofmann, Mitrea, Mitrea, and Morris. As an application, we show that the famous theorem of Coifman–Lions–Meyer–Semmes holds in this class of manifolds: Jacobians of $W^{1,n}$-maps from $\mathcal{M}$ to $\mathbb{R}^n$ can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by Lenzmann and the 2nd-named author using only harmonic extensions, integration by parts, and trace space characterizations.

Trace operators of the bi-Laplacian and applications

We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the bi-Laplace equation with homogeneous Dirichlet condition, but are also relevant to describe conformity of mixed approximations. Our aim is to have well-posed (ultraweak) formulations that assume low regularity under the condition of an $L_2$ right-hand side function. We pursue two ways of defining traces and corresponding integration-by-parts formulas. In one case one obtains a nonclosed space. This can be fixed by switching to the Kirchhoff–Love traces from Führer et al. (2019, An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation. Math. Comp., 88, 1587–1619). Using different combinations of trace operators we obtain two well-posed formulations. For both of them we report on numerical experiments with the discontinuous Petrov–Galerkin method and optimal test functions. In this paper we consider two and three space dimensions. However, with the exception of a given counterexample in an appendix (related to the nonclosedness of a trace space) our analysis applies to any space dimension larger than or equal to two.

KMS states on the C*-algebras of Fell bundles over groupoids

We consider fibrewise singly generated Fell bundles over étale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo–Martin–Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of states on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a description of the trace space of the cross-sectional algebra. We apply our result to generalise Neshveyev’s main theorem to twisted groupoid C*-algebras, and then apply this to twisted C*-algebras of strongly connected finite k-graphs.

Trace spaces of directed tori with rectangular holes

Raussen (2010) gave the trace space, which corresponds to the executions of parallel non-looped, non-branching processes as a prod-simplicial complex derived from a poset. The connected components represent equivalent executions. For looped processes, the state space is a torus and the trace space is a disjoint union of trace spaces of deloopings. In the current paper, we develop the index poset for the trace space of the deloopings from the once delooped case. When just one process is looped, the index poset is generated as words in a regular language. We also construct a corresponding automaton.