arXiv subjects

Wolfgang Arendt

Publications and source records attributed to Wolfgang Arendt.

At least 19 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 $[0,τ]$ in a Banach space $X$ with the non-standard time-boundary condition $u(0)=Φu(τ)$, where $Φ$ is a linear map on $X$. If $Φ=0$, this is an initial value problem, whereas $Φ=I$ corresponds to periodic boundary conditions, and $Φ=-I$ 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↗

arXiv2024

Semi-linear evolution equations via positive semigroups

We study semi-linear evolutionary problems where the linear part is the generator of a positive $C_0$-semigroup. The non-linear part is assumed to be quasi-increasing. Given an initial value in between a sub- and a super-solution of the stationary problem we find a solution of the semi-linear evolutionary problem. Convergence as $t\to\infty$ is also studied for the solutions. Our results are applied to the logistic equation with diffusion, to a Lotka-Volterra competition model and the Fisher equation from population genetics.

math.AP↗

arXiv2023

Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions

We first give a general error estimate for the nonconforming approximation of a problem for which a Banach-Ne{\v c}as-Babu{\v s}ka (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types of space-time discretizations, both based on a conforming Galerkin method in space. The first one is the Euler $θ$--scheme. In this case, we show that the BNB inequality is always satisfied, and may require an extra condition on the time step for $θ$ $\le$ 1 2. The second one is the time discontinuous Galerkin method, where the BNB condition holds without any additional condition.

math.NA↗

arXiv2022

The Wentzell Laplacian via forms and the approximative trace

We use form methods to define suitable realisations of the Laplacian on a domain $Ω$ with Wentzell boundary conditions, i.e. such that $\partial_{\mathrm{n}}u + βu + Δu = 0$ holds in a suitable sense on the boundary of $Ω$. 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 $β$ to be merely essentially bounded and complex-valued. If the domain is Lipschitz, we obtain a kernel continuous up to the boundary.

math.AP↗

arXiv2022

Extensions of derivations and symmetric operators

Given a densely defined skew-symmetric operators A 0 on a real or complex Hilbert space V , we parametrize all m-dissipative extensions in terms of contractions $Φ$ : H-$\rightarrow$ H + , where Hand H + are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary C 0-group if and only if $Φ$ is a unitary operator. As corollary we obtain the parametrization of all selfadjoint extensions of a symmetric operator by unitary operators from Hto H +. Our results extend the theory of boundary triples initiated by von Neumann and developed by V. I. and M. L. Gorbachuk, J. Behrndt and M. Langer, S. A. Wegner and many others, in the sense that a boundary quadruple always exists (even if the defect indices are different in the symmetric case).

math.NA↗

arXiv2022

Lions' representation theorem and applications

The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram Theorem where completeness of one of the spaces is not complete. In this paper, RTL is deduced from an operator-theoretical version on normed space. The main point of the paper is a theory of derivations, based on RTL, for which well-posedness is proved. One application concerns non-autonomous evolution equations with a new initial-value and a periodic boundary condition for the time variable.

math.FA↗

arXiv2022

Semilinear elliptic equations on rough domains

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form.

math.AP↗

arXiv2020

Positive irreducible semigroups and their long-time behaviour

The notion \emph{Perron-Frobenius theory} usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as $t \to \infty$. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large $t$, while being absent for smaller times.

math.FA↗

arXiv2020

Strict positivity for the principal eigenfunction of elliptic operators with various boundary conditions

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain $Ω\subset \mathbb{R}^d$ and show that the first eigenfunction $v$ satisfies $v(x) \ge δ> 0$ for all $x \in \overlineΩ$, even if the boundary $\partial Ω$ is only Lipschitz continuous. Under such weak regularity assumptions the Hopf-Ole\uınik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map $L_p(Ω)$ into $C(\overlineΩ)$. The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.

math.AP↗

arXiv2020

Galerkin approximation of linear problems in Banach and Hilbert spaces

In this paper we study the conforming Galerkin approximation of the problem: find u $\in$ U such that a(u, v) = for all v $\in$ V, where U and V are Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L $\in$ V' a given data. The approximate solution is sought in a finite dimensional subspace of U, and test functions are taken in a finite dimensional subspace of V. We provide a necessary and sufficient condition on the form a for convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint problem. We also characterize the fact that U has a finite dimensional Schauder decomposition in terms of properties related to the Galerkin approximation. In the case of Hilbert spaces, we prove that the only bilinear or sesquilinear forms for which any Galerkin approximation converges (this property is called the universal Galerkin property) are the essentially coercive forms. In this case, a generalization of the Aubin-Nitsche Theorem leads to optimal a priori estimates in terms of regularity properties of the right-hand side L, as shown by several applications. Finally, a section entitled "Supplement" provides some consequences of our results for the approximation of saddle point problems.

math.NA↗

arXiv2020

Disjointness-preserving operators and isospectral Laplacians

All the known counterexamples to Kac' famous question "can one hear the shape of a drum", i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent, consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts as a sum of overlapping "local" isometries mapping the copies to each other. We prove and explore a complementary positive statement: if an operator intertwining two appropriate realisations of the Laplacian on a pair of domains preserves disjoint supports, then under additional assumptions on it generally far weaker than unitarity, the domains are congruent. We show this in particular for the Dirichlet, Neumann and Robin Laplacians on spaces of continuous functions and on $L^2$-spaces.

math.SP↗

arXiv2019

In Koenigs' footsteps: Diagonalization of composition operators

Let $φ:\mathbb{D} \to \mathbb{D}$ be a holomorphic map with a fixed point $α\in\mathbb{D}$ such that $0\leq |φ'(α)|<1$. We show that the spectrum of the composition operator $C_φ$ on the Fréchet space $ \textrm{Hol}(\mathbb{D})$ is $\{0\}\cup \{ φ'(α)^n:n=0,1,\cdots\}$ and its essential spectrum is reduced to $\{0\}$. This contrasts the situation where a restriction of $C_φ$ to Banach spaces such as $H^2(\mathbb{D})$ is considered. Our proofs are based on explicit formulae for the spectral projections associated with the point spectrum found by Koenigs. Finally, as a byproduct, we obtain information on the spectrum for bounded composition operators induced by a Schröder symbol on arbitrary Banach spaces of holomorphic functions.

math.SP↗

arXiv2019

Elliptic problems and holomorphic functions in Banach spaces

In the first part we show that a vector-valued almost separably valued function $f$ is holomorphic (harmonic) if and only if it is dominated by an $L^1_\mathrm{loc}$ function and there exists a separating set $W\subset X'$ such that $\langle f,x'\rangle$ is holomorphic (harmonic) for all $x'\in W$. This improves a known result which requires $f$ to be locally bounded. In the second part we consider classical results in the $L^p$ theory for elliptic differential operators of second order. In the vector-valued setting these results are shown to be equivalent to the UMD property.

math.FA↗

arXiv2019

Maximal $L^2$-regularity in nonlinear gradient systems and perturbations of sublinear growth

The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function $φ$ has a smoothing effect, discovered by H. Brézis, which implies maximal regularity for the evolution equation. We use this and Schaefer's fixed point theorem to solve the evolution equation perturbed by a Nemytskii-operator of sublinear growth. For this, we need that the sublevel sets of $φ$ are not only closed but even compact. We apply our results to the $p$-Laplacian and also to the Dirichlet-to-Neumann operator with respect to $p$-harmonic functions.

math.AP↗

arXiv2017

Diffusion with nonlocal Robin boundary conditions

We investigate a second order elliptic differential operator $A_{β, μ}$ on a bounded, open set $Ω\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq β\in L^{\infty}(\partialΩ)$ and $μ\colon \partial Ω\to \mathscr{M}(\overlineΩ)$, and boundary conditions of the form \[ \partial_ν^{\mathscr{A}}u(z)+β(z)u(z)=\int_{\overlineΩ}u(x)μ(z)(dx),\ z\in\partialΩ, \] where $\partial_ν^{\mathscr{A}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $μ$ we show that $A_{β, μ}$ generates a holomorphic semigroup $T_{β,μ}$ on $L^{\infty}(Ω)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overlineΩ)$. Its restriction to $C(\overlineΩ)$ is strongly continuous and holomorphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.

math.FA↗

arXiv2016

J. L. Lions' Problem on Maximal Regularity

This is a survey on recent progress concerning maximal regularity of non-autonomous equations governed by time-dependent forms on a Hilbert space. It also contains two new results showing the limits of the theory.

math.FA↗