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Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
A variational approach to strongly damped wave equations
 Functional Analysis and Evolution Equations – The Günter Lumer Volume
, 2008
"... Abstract. We discuss a Hilbert space method that allows to prove analytical wellposedness of a class of linear strongly damped wave equations. The main technical tool is a perturbation lemma for sesquilinear forms, which seems to be new. In most common linear cases we can furthermore apply a recent ..."
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Cited by 6 (6 self)
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Abstract. We discuss a Hilbert space method that allows to prove analytical wellposedness of a class of linear strongly damped wave equations. The main technical tool is a perturbation lemma for sesquilinear forms, which seems to be new. In most common linear cases we can furthermore apply a recent result due to Crouzeix–Haase, thus extending several known results and obtaining optimal analyticity angle.
Wellposedness and symmetries of strongly coupled network equations
 J. Phys. A
"... Abstract. We consider a class of evolution equations taking place on the edges of a finite network and allow for feedback effects between different, possibly nonadjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the bo ..."
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Cited by 5 (5 self)
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Abstract. We consider a class of evolution equations taking place on the edges of a finite network and allow for feedback effects between different, possibly nonadjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i.e., in the nodes of the network. We discuss wellposedness of the associated initial value problem as well as contractivity and positivity properties of its solutions. Finally, we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space, i.e., of symmetries of the associated physical system. Applications to a neurobiological model as well as to a system of linear Schrödinger equations on a quantum graph are discussed. 1.
Convex domains and Kspectral sets
 Math. Z
"... Let Ω be an open convex domain of C. We study constants K such that Ω is Kspectral or complete Kspectral for each continuous linear Hilbert space operator with numerical range included in Ω. Several approaches are discussed. ..."
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Cited by 5 (2 self)
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Let Ω be an open convex domain of C. We study constants K such that Ω is Kspectral or complete Kspectral for each continuous linear Hilbert space operator with numerical range included in Ω. Several approaches are discussed.
Qualitative properties of coupled parabolic systems of evolution equations. Ulmer Seminare über Differentialgleichungen und Funktionalanalysis
, 2006
"... Abstract. We apply functional analytical and variational methods in order to study wellposedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic prob ..."
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Cited by 4 (4 self)
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Abstract. We apply functional analytical and variational methods in order to study wellposedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equation, a heat equation with dynamic boundary conditions, and a general semilinear Hodgkin–Huxley sytem. 1.
A parabolic free boundary problem modeling electrostatic
"... ABSTRACT. A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrowgap model is given by showing that ..."
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Cited by 3 (3 self)
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ABSTRACT. A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrowgap model is given by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrowgap model when the aspect ratio of the device tends to zero.
MAXIMAL PARABOLIC REGULARITY FOR DIVERGENCE OPERATORS INCLUDING MIXED BOUNDARY CONDITIONS
, 903
"... Abstract. We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly nonsmooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to q ..."
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Cited by 2 (1 self)
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Abstract. We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly nonsmooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with nonsmooth data are presented. 1.
CONVERGENCE OF SECTORIAL OPERATORS ON VARYING HILBERT SPACES
"... Abstract. Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on varying spaces is natural. However, it seems that the first results in this direction have been obtained only recen ..."
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Cited by 1 (0 self)
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Abstract. Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on varying spaces is natural. However, it seems that the first results in this direction have been obtained only recently, to the best of our knowledge. Here we consider sectorial operators on scales of Hilbert spaces. We define a notion of convergence that generalises convergence of the resolvents in operator norm to the case when the operators act on different spaces and show that this kind of convergence is compatible with the functional calculus of the operator and moreover implies convergence of the spectrum. Finally, we present examples for which this convergence can be checked, including convergence of coefficients of parabolic problems. Convergence of a manifold (roughly speaking consisting of thin tubes) towards the manifold’s skeleton graph plays a prominent role, being our main application. 1.
Maximal Regularity for Evolution Equations Governed by NonAutonomous Forms
"... We consider a nonautonomous evolutionary problem ..."
A FOURTHORDER MODEL FOR MEMS WITH CLAMPED BOUNDARY CONDITIONS
"... ABSTRACT. The dynamical and stationary behaviors of a fourthorder equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameterλ. It is ..."
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Cited by 1 (1 self)
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ABSTRACT. The dynamical and stationary behaviors of a fourthorder equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameterλ. It is shown that there is a threshold valueλ ∗> 0 of the voltage parameter such that no radially symmetric stationary solution exists forλ> λ∗, while at least two such solutions exist forλ ∈ (0,λ∗). Local and global wellposedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities whenλ> λ∗. hal00809296, version 1 8 Apr 2013 1.