Consistent operator semigroups and their interpolation
Under a mild regularity condition we prove that the generator of the interpolation of two C0-semigroups is the interpolation of the two generators.
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arXiv subjects
A. F. M. ter Elst
Publications and source records attributed to A. F. M. ter Elst.
Construction of dynamical semigroups by a functional regularisation à la Kato
A functional version of the Kato one-parametric regularisation for the construction of a dynamical semigroup generator of a relative bound one perturbation is introduced. It does not require that the minus generator of the unperturbed semigroup is a positivity preserving operator. The regularisation is illustrated by an example of a boson-number cut-off regularisation.
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Weak and strong approximation of semigroups on Hilbert spaces
For a sequence of uniformly bounded, degenerate semigroups on a Hilbert space, we compare various types of convergences to a limit semigroup. Among others, we show that convergence of the semigroups, or of the resolvents of the generators, in the weak operator topology, in the strong operator topology or in certain integral norms are equivalent under certain natural assumptions which are frequently met in applications.
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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.
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On maximal parabolic regularity for non-autonomous parabolic operators
We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents $r\neq 2$. This allows us to prove maximal parabolic $L^r$-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.
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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.
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On one-parameter Koopman groups
We characterize Koopman one-parameter $C_0$-groups in the class of all unitary one-parameter $C_0$-groups on $L_2(X)$ as those that preserve $L_\infty(X)$ and for which the infinitesimal generator is a derivation on the bounded functions in its domain.
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Hölder estimates for parabolic operators on domains with rough boundary
We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the domain, including a very weak compatibility condition between the Dirichlet boundary part and its complement, we prove Hölder continuity of the solution in space and time.
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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.
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Dirichlet-to-Neumann maps on bounded Lipschitz domains
The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of the Dirichlet-to-Neumann map and its inverse, the Neumann-to-Dirichlet map, in the framework of linear relations in Hilbert spaces. Our treatment is inspired by abstract methods from extension theory of symmetric operators, utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded Lipschitz domains.
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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.
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On series of sectorial forms
We prove a convergence theorem for partial sums of sectorial forms with vertex zero and a common semi-angle. As an example we prove an absorption theorem for sectorial forms.
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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.
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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.
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Analysis of the heat kernel of the Dirichlet-to-Neumann operator
We prove Poisson upper bounds for the kernel $K$ of the semigroup generated by the Dirichlet-to-Neumann operator if the underlying domain is bounded and has a $C^\infty$-boundary. We also prove Poisson bounds for $K_z$ for all $z$ in the right half-plane and for all its derivatives.
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Parabolic equations with dynamical boundary conditions and source terms on interfaces
We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable $L^p$-setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.
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Square roots of perturbed subelliptic operators on Lie groups
We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower order terms. In this general setting we deduce inhomogeneous estimates. In case the group is nilpotent and the subelliptic operator is pure second order, then we prove stronger homogeneous estimates. Furthermore, we prove Lipschitz stability of the estimates under small perturbations of the coefficients.
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Partial spectral multipliers and partial Riesz transforms for degenerate operators
We consider degenerate differential operators $A = \displaystyle{\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)}$ on $L^2(\mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given a function $χ\in C_b^\infty(\mathbb{R}^d)$ (respectively, $Ω$ a bounded Lipschitz domain) and suppose that $(a_{kj}) \ge μ> 0$ a.e.\ on $ \supp χ$ (resp., a.e.\ on $Ω$). We prove a spectral multiplier type result: if $F\colon [0, \infty) \to \mathbb{C}$ is such that $\sup_{t > 0} \| φ(.) F(t .) \|_{C^s} < \infty$ for some non-trivial function $φ\in C_c^\infty(0,\infty)$ and some $s > d/2$ then $M_χF(I+A) M_χ$ is weak type $(1,1)$ (resp.\ $P_ΩF(I+A) P_Ω$ is weak type $(1,1)$). We also prove boundedness on $L^p$ for all $p \in (1,2]$ of the partial Riesz transforms $M_χ\nabla (I + A)^{-1/2}M_ χ$. The proofs are based on a criterion for a singular integral operator to be weak type $(1,1)$.
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