arXiv subjects

Manfred Sauter

Publications and source records attributed to Manfred Sauter.

8 recordsLinked to original sources

arXiv2025

Variational solutions of the Dirichlet problem, Lebesgue's cusp and non-local properties

A recent result from [AtES24] allows one to define variational solutions of the Dirichlet problem for general continuous boundary data. We establish basic properties of this notion of solution and show that it coincides with the Perron solution. Variational solutions can elegantly be characterised in terms of the given boundary function when the variational solution has finite energy. However, it is impossible to decide in terms of the regularity of the given boundary function when a classical solution exists. We demonstrate this by analysing Lebesgue's cusp, and more precisely Lebesgue's domain which is associated with the potential of a thin rod with mass density going to zero at one end. We also show that the non-continuity of the Perron solution at a singular point is a generic and non-local property.

math.AP↗

arXiv2025

Elliptic systems generating a positive semigroup are decoupled

We show that the semigroup associated to a second-order elliptic system is positive if and only if the differential equations are essentially decoupled and the coefficients are real-valued. This means the system can be replaced by an equivalent decoupled system (which defines the same semigroup). Instead of matrix coefficients we more generally consider operator-valued coefficients, which leads to interesting additional difficulties with respect to Banach space valued integration.

math.AP↗

arXiv2025

Maximal regularity for generalized boundary conditions in time

We consider autonomous and non-autonomous evolution equations on a time interval $[0,\tau]$ in a Banach space $X$ with the non-standard time-boundary condition $u(0)=\Phi u(\tau)$, where $\Phi$ is a linear map on $X$. If $\Phi=0$, this is an initial value problem, whereas $\Phi=I$ corresponds to periodic boundary conditions, and $\Phi=-I$ to antiperiodic boundary conditions. Our main point is to establish maximal $L^p$-regularity. In the non-autonomous case we consider two situations. The first concerns time-dependent operators with a fixed domain. In the second one we take $X=H$ a Hilbert space and consider evolution equations associated with non-autonomous forms. Of special interest is then maximal regularity in $H$ with a non-standard time-boundary condition.

math.AP↗

arXiv2022

The Wentzell Laplacian via forms and the approximative trace

We use form methods to define suitable realisations of the Laplacian on a domain $\Omega$ with Wentzell boundary conditions, i.e. such that $\partial_{\mathrm{n}}u + \beta u + \Delta u = 0$ holds in a suitable sense on the boundary of $\Omega$. For those realisations, we study their semigroup generation properties. Using the approximative trace, we give a unified treatment that in part allows irregular and even fractal domains. Moreover, we admit $\beta$ to be merely essentially bounded and complex-valued. If the domain is Lipschitz, we obtain a kernel continuous up to the boundary.

math.AP↗

arXiv2016

Uniqueness of the approximative trace

We study the approximative trace for individual elements in the Sobolev space $W^{1,p}(Ω)$ for $1\le p\le\infty$. This notion of a trace was introduced for $p=2$ in [AtE11] in the setting of general open sets $Ω\subset\mathbb{R}^d$. The approximative trace exhibits a curious nonuniqueness phenomenon. We provide a detailed analysis of this phenomenon based on methods of geometric measure theory and are able to give very weak geometric conditions that are sufficient for the uniqueness of the approximative trace. In particular, we prove that the approximative trace is unique on open sets with continuous boundary and on arbitrary connected domains in $\mathbb{R}^2$. Furthermore, we provide an example which shows that the uniqueness of the approximative trace depends on $p$. These results answer several open questions.

math.AP↗

arXiv2015

Nonseparability and von Neumann's theorem for domains of unbounded operators

A classical theorem of von Neumann asserts that every unbounded self-adjoint operator $A$ in a separable Hilbert space $H$ is unitarily equivalent to an operator $B$ in $H$ such that $D(A)\cap D(B)=\{0\}$. Equivalently this can be formulated as a property for nonclosed operator ranges. We will show that von Neumann's theorem does not directly extend to the nonseparable case. In this paper we prove a characterisation of the property that an operator range $\mathcal{R}$ in a general Hilbert space $H$ admits a unitary operator $U$ such that $U\mathcal{R}\cap\mathcal{R}=\{0\}$. This allows us to study stability properties of operator ranges with the aforementioned property.

math.FA↗

arXiv2014

A generalisation of the form method for accretive forms and operators

The form method as popularised by Lions and Kato is a successful device to associate m-sectorial operators with suitable elliptic or sectorial forms. McIntosh generalised the form method to an accretive setting, thereby allowing to associate m-accretive operators with suitable accretive forms. Classically, the form domain is required to be densely embedded into the Hilbert space. Recently, this requirement was relaxed by Arendt and ter Elst in the setting of elliptic and sectorial forms. Here we study the prospects of a generalised form method for accretive forms to generate accretive operators. In particular, we work with the same relaxed condition on the form domain as used by Arendt and ter Elst. We give a multitude of examples for many degenerate phenomena that can occur in the most general setting. We characterise when the associated operator is m-accretive and investigate the class of operators that can be generated. For the case that the associated operator is m-accretive, we study form approximation and Ouhabaz type invariance criteria.

math.FA↗

arXiv2013

The regular part of second-order differential sectorial forms with lower-order terms

We present a formula for the regular part of a sectorial form that represents a general linear second-order differential expression that may include lower-order terms. The formula is given in terms of the original coefficients. It shows that the regular part is again a differential sectorial form and allows to characterise when also the singular part is sectorial. While this generalises earlier results on pure second-order differential expressions, it also shows that lower-order terms truly introduce new behaviour.

math.AP↗