Results 1  10
of
32
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 ..."
Abstract

Cited by 23 (9 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
(Show Context)
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.
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. ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 6 (6 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
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 regularity of discrete second order Cauchy problems in Banach spaces
 J. Differ. Equ. Appl
"... Abstract. We characterize the discrete maximal regularity for second order difference equations by means of spectral and Rboundedness properties of the resolvent set. 1. ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We characterize the discrete maximal regularity for second order difference equations by means of spectral and Rboundedness properties of the resolvent set. 1.
Parabolic systems with coupled boundary conditions
, 2009
"... Abstract. We consider elliptic operators with operatorvalued coefficients and discuss the associated parabolic problems. The unknowns are functions with values in a Hilbert space W. The system is equipped with a general class of coupled boundary conditions of the form f ∂Ω ∈ Y and ∂f ∂ν ∈ Y ⊥ , wh ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We consider elliptic operators with operatorvalued coefficients and discuss the associated parabolic problems. The unknowns are functions with values in a Hilbert space W. The system is equipped with a general class of coupled boundary conditions of the form f ∂Ω ∈ Y and ∂f ∂ν ∈ Y ⊥ , where Y is a closed subspace of L 2 (∂Ω; W). We discuss wellposedness and further qualitative properties, systematically reducing features of the parabolic system to operatortheoretical properties of the orthogonal projection onto Y. 1.
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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.
Neerven, Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces, submitted
"... Dedicated to Professor Alan M c Intosh on the occasion of his 65th birthday Abstract. Let (E, H, µ) be an abstract Wiener space and let DV: = V D, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space H. Given a bounded operator B o ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Dedicated to Professor Alan M c Intosh on the occasion of his 65th birthday Abstract. Let (E, H, µ) be an abstract Wiener space and let DV: = V D, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space H. Given a bounded operator B on H, coercive on the range R(V), we consider the operators A: = V ∗ BV in H and A: = V V ∗ B in H, as well as the realisations of the operators L: = D ∗ V BDV and L: = DV D ∗ V B in Lp (E, µ) and L p (E, µ; H) respectively, where 1 < p < ∞. Our main result asserts that the following four assertions are equivalent: (1) D ( √ L) = D(DV) with ‖ √ Lf‖p � ‖DV f‖p for f ∈ D ( √ L); (2) L admits a bounded H∞functional calculus on R(DV); (3) D ( √ A) = D(V) with ‖ √ Ah ‖ � ‖V h ‖ for h ∈ D ( √ A); (4) A admits a bounded H∞functional calculus on R(V). Moreover, if these conditions are satisfied, then D(L) = D(D2 V) ∩ D(DA). The equivalence (1)–(4) is a nonsymmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where H = H, V = I, B = 1 2I). A onesided version of (1)–(4), giving Lpboundedness of the Riesz transform DV / √ L in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C0contraction semigroup on a Hilbert space H and let −L be the Lprealisation of the generator of its second quantisation. Our results imply that twosided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. 1.