arXiv subjects

W. Arendt

Publications and source records attributed to W. Arendt.

15 recordsLinked to original sources

arXiv2023

Nittka's invariance criterion and Hilbert space valued parabolic equations in $L_p$

Nittka gave an efficient criterion on a form defined on $L_2(Ω)$ which implies that the associated semigroup is $L_p$-invariant for some given $p \in (1,\infty)$. We extend this criterion to the Hilbert space valued~$L_2(Ω,H)$. As an application we consider elliptic systems of pure second order. Our main result shows that the induced semigroup is $L_p$-contractive for all $p \in [p_-,p_+]$ for some $1 < p_- < 2 < p_+ < \infty$.

math.AP↗

arXiv2023

The Perron solution for elliptic equations without the maximum principle

In this article we consider the Dirichlet problem on a bounded domain $\Omega \subset {\bf R}^d$ with respect to a second-order elliptic differential operator in divergence form. We do not assume a divergence condition as in the pioneering work by Stampacchia, but merely assume that $0$ is not a Dirichlet eigenvalue. The purpose of this article is to define and investigate a solution of the Dirichlet problem, which we call Perron solution, in a setting where no maximum principle is available. We characterise this solution in different ways: by approximating the domain by smooth domains from the interior, by variational properties, by the pointwise boundary behaviour at regular boundary points and by using the approximative trace. We also investigate for which boundary data the Perron solution has finite energy. Finally we show that the Perron solution is obtained as an $H^1_0$-perturbation of a continuous function on $\overline \Omega$. This is new even for the Laplacian and solves an open problem.

math.AP↗

arXiv2023

Semigroups generated by multivalued operators and domain convergence for parabolic problems

The following version of the Lumer-Phillips is proved: a surjective dissipative operator is m-dissipative and invertible. The result remains true if dissipative linear relations (i.e multivalued operators) are considered. The main purpose of this article is to study relations which generate semigroups. We consider m-dissipative relations and also the holomorphic estimate for relations. Such relations are very useful if domain perturbations for the Laplacian are studied.

math.FA↗

arXiv2021

Spectral properties of weighted composition operators on $\Hol(\D)$ induced by rotations

In this article we study the spectrum $σ(T)$ and Waelbroeck spectrum $σ_W(T)$ of a weighted composition operator $T$ induced by a rotation on $\Hol(\D)$ and given by $$Tf(z)=m(z)f(βz) \ \ \ (z\in \D)$$ where $m\in \Hol(\D)$, $β\in \C$, $|β| = 1$. If $β^n\neq 1$ for all $n\in \N$ we show that $σ_W(T)$ is a disc if $m(z_0)=0$ for some $z_0\in \D$ and it is the circle $\{λ\in \C : |λ|=|m(0)|\}$ if $m(z)\neq 0$ for all $z\in \D$. We find examples of $m\in A(\D)$ (the disc algebra) such that $λ\Id-T$ is invertible in $\Hol(\D)$ (the Fréchet space of all holomorphic functions on $\D$), but $(λ\Id-T)^{-1}A(\D)\not\subset A(\D)$. Inspired by Bonet \cite{Bonet} we show that $\{β^n : n\in \N\}\subset σ(T)\neq \T$ when the weight is $m\equiv 1$ and $β$ a diophantine number. This shows that the spectrum is not closed in general.

math.FA↗

arXiv2020

Essentially coercive forms and asympotically compact semigroups

Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e. that dist(S(t),K(H)) tends to 0 as t tends to infinity, where K(H) denotes the space of all compact operators on the underlying Hilbert space.

math.FA↗

arXiv2019

Operators with continuous kernels

Let $Ω\subset {\bf R}^d$ be open. We investigate conditions under which an operator $T$ on $L_2(Ω)$ has a continuous kernel $K \in C(\overline Ω\times \overline Ω)$. In the centre of our interest is the condition $T L_2(Ω) \subset C(\overline Ω)$, which one knows for many semigroups generated by elliptic operators. This condition implies that $T^3$ has a kernel in $C(\overline Ω\times \overline Ω)$ if $T$ is self-adjoint and $Ω$ is bounded, and the power $3$ is best possible. We also analyse Mercer's theorem in our context.

math.AP↗

arXiv2018

The Dirichlet problem without the maximum principle

Consider the Dirichlet problem with respect to an elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, a_{kl} \, \partial_l - \sum_{k=1}^d \partial_k \, b_k + \sum_{k=1}^d c_k \, \partial_k + c_0 \] on a bounded Wiener regular open set $Ω\subset R^d$, where $a_{kl}, c_k \in L_\infty(Ω,R)$ and $b_k,c_0 \in L_\infty(Ω,C)$. Suppose that the associated operator on $L_2(Ω)$ with Dirichlet boundary conditions is invertible. Then we show that for all $φ\in C(\partial Ω)$ there exists a unique $u \in C(\overline Ω) \cap H^1_{\rm loc}(Ω)$ such that $u|_{\partial Ω} = φ$ and $A u = 0$. In the case when $Ω$ has a Lipschitz boundary and $φ\in C(\overline Ω) \cap H^{1/2}(\overline Ω)$, then we show that $u$ coincides with the variational solution in $H^1(Ω)$.

math.AP↗

arXiv2018

Generators of semigroups on Banach spaces inducing holomorphic semiflows

Let $A$ be the generator of a $C_0$-semigroup $T$ on a Banach space of analytic functions on the open unit disc. If $T$ consists of composition operators, then there exists a holomorphic function $G:{\mathbb D}\to{\mathbb C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study of the reciprocal implication.

math.FA↗

arXiv2017

The Dirichlet-to-Neumann operator on $C(\partial Ω)$

Let $Ω\subset {\bf R}^d$ be an open bounded set with Lipschitz boundary $Γ$. Let $D_V$ be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous coefficients and a positive potential $V$. We show that the semigroup generated by $-D_V$ leaves $C(Γ)$ invariant and that the restriction of this semigroup to $C(Γ)$ is a $C_0$-semigroup. We investigate positivity and spectral properties of this semigroup. We also present results where $V$ is allowed to be negative. Of independent interest is a new criterium for semigroups to have a continuous kernel.

math.AP↗

arXiv2016

Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator

In the very influential paper \cite{CS07} Caffarelli and Silvestre studied regularity of $(-Δ)^s$, $0<s<1$, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea \cite{ST10} and Galé, Miana and Stinga \cite{GMS13} gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the fractional power.

math.AP↗

arXiv2013

Analytical aspects of isospectral drums

We reexamine the proofs of isospectrality of the counterexample domains to Kac' question `Can one hear the shape of a drum?' from an analytical viewpoint. We reformulate isospectrality in a more abstract setting as the existence of a similarity transform intertwining two operators associated with elliptic forms, and give several equivalent characterizations of this property as intertwining the forms and form domains, the associated operators and operator domains, and the semigroups they generate. On a representative pair of counterexample domains, we use these criteria to show that the similarity transform intertwines not only the Laplacians with Neumann (or Dirichlet) boundary conditions but also any two appropriately defined elliptic operators on these domains, even if they are not self-adjoint. However, no such transform can intertwine these operators if Robin boundary conditions are imposed instead of Neumann or Dirichlet. We also remark on various operator-theoretic properties of such intertwining similarity transforms.

math.AP↗

arXiv2013

The Dirichlet-to-Neumann operator via hidden compactness

We show that to each symmetric elliptic operator of the form \[ \mathcal{A} = - \sum \partial_k \, a_{kl} \, \partial_l + c \] on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ one can associate a self-adjoint Dirichlet-to-Neumann operator on $L_2(\partial Ω)$, which may be multi-valued if 0 is in the Dirichlet spectrum of $\mathcal{A}$. To overcome the lack of coerciveness in this case, we employ a new version of the Lax--Milgram lemma based on an indirect ellipticity property that we call hidden compactness. We then establish uniform resolvent convergence of a sequence of Dirichlet-to-Neumann operators whenever their coefficients converge uniformly and the second-order limit operator in $L_2(Ω)$ has the unique continuation property. We also consider semigroup convergence.

math.AP↗

arXiv2011

Diffusion determines the compact manifold

We provide a short proof for the theorem that two compact Riemannian manifolds are isomorphic if and only there exists an order isomorphism which intertwines between the heat semigroups on the manifolds.

math.AP↗

arXiv2010

Sectorial forms and degenerate differential operators

If $a$ is a densely defined sectorial form in a Hilbert space which is possibly not closable, then we associate in a natural way a holomorphic semigroup generator with $a$. This allows us to remove in several theorems of semigroup theory the assumption that the form is closed or symmetric. Many examples are provided, ranging from complex sectorial differential operators, to Dirichlet-to-Neumann operators and operators with Robin or Wentzell boundary conditions.

math.AP↗

arXiv2010

The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set $Ω\subset {\rm R}^d$ whose boundary $Γ$ has a finite $(d-1)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2(Γ)$ by form methods. The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on $L_2(Γ)$. We show that the asymptotic behaviour of $S_t$ as $t \to \infty$ is related to properties of the trace of functions in $H^1(Ω)$ which $Ω$ may or may not have.

math.AP↗