Denk, Robert

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Convex monotone semigroups on lattices of continuous functions

2021, Denk, Robert, Kupper, Michael, Nendel, Max

We consider convex monotone C0-semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a σ-Dedekind complete Banach lattice. Typical examples include the space of all bounded uniformly continuous functions and the space of all continuous functions vanishing at infinity. We show that the domain of the classical generator of a convex semigroup is typically not invariant. Therefore, we propose alternative versions for the domain, such as the monotone domain and the Lipschitz set, for which we prove invariance under the semigroup. As a main result, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are illustrated with several examples related to Hamilton-Jacobi-Bellman equations, including nonlinear versions of the shift semigroup and the heat equation. In particular, we determine their symmetric Lipschitz sets, which are invariant and allow to understand the generators in a weak sense.

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Estimates for solutions of a parameter-elliptic multi-order system of differential equations

2009, Denk, Robert, Faierman, Melvin

This paper is concerned with a boundary value problem defined over a bounded region of Euclidean space, and in particular it is devoted to the establishment of a priori estimates for solutions of a parameter-elliptic multi-order system of differential equations under limited smoothness assumptions. In this endeavour we extend the results of Agranovich, Denk, and Faierman pertaining to a priori estimates for solutions associated with a parameter-elliptic scalar problem, as well as the results of various other authors who have extended the results of Agranovich et. al. from the scalar case to parameter-elliptic systems of operators which are either of homogeneous type or have the property that the diagonal operators are all of the same order. In addition, we extend some results of Kozhevnikov and Denk and Volevich who have also dealt with sytems of the kind under consideration here, in that one of the works of Kozhevnikov deals only with 2x2 systems, while the other, as well as the work of the last two authors, do not cover Dirichlet boundary conditions.

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Forward Simulation of Financial Problems via BSDEs

2005, Bender, Christian, Denk, Robert

We introduce a forward scheme to simulate backward SDEs and demonstrate the strength of the new algorithm by solving some financial problems numerically.

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Lp-resolvent estimates and time decay for generalized thermoelastic plate equations

2005, Denk, Robert, Racke, Reinhard

We consider the Cauchy problem for a coupled system generalizing the thermoelastic plate equations. First we prove resolvent estimates for the stationary operator and conclude the analyticity of the associated semigroup in Lp-spaces 1 < p <∞, for certain values of the parameters of the system; here the Newton polygon method is used.

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Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

2012, Barraza Martinez, Bienvenido, Denk, Robert, Hernandez Monzon, Jairo

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces Wpk(Rn,E). Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.

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Bounded H-infty-calculus for pseudodifferential Douglis-Nirenberg systems of mild regularity

2007, Denk, Robert, Saal, Jürgen, Seiler, Jörg

Parameter-ellipticity with respect to a closed subsector of the complex plane for pseudodifferential Douglis-Nirenberg systems is introduced and shown to imply the existence of a bounded H_\infty-calculus in suitable scales of Sobolev, Besov, and Hölder spaces. We also admit non pseudodifferential perturbations. Applications concern systems with coefficients of mild Hölder regularity and the generalized thermoelastic plate equations.

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Optimal Lp-Lq-regularity for parabolic problems with inhomogeneous boundary data

2005, Denk, Robert, Hieber, Matthias, Prüss, Jan

In this paper we investigate vector-valued parabolic initial boundary value problems (A(x,D), B1(x,D),..., Bm(x,D)) subject to general boundary conditions in a domain G with compact boundary. The top-order coefficients of the operator A are assumed to be continuous. We characterize optimal Lp-Lq-regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on A and the Lopatinskii-Shapiro condition on (A(x,D), B1(x,D),..., Bm(x,D)) are necessary for these Lp-Lq-estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.

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Parameter-dependent estimates for mixed-order boundary value problems

2010, Denk, Robert, Faierman, Melvin

In this paper we prove parameter-dependent a priori estimates for mixed-order boundary value problems of rather general structure. In particular, the diagonal operators are not assumed to be of the same order. Our assumptions on the structure of the boundary value problem covers the case of Dirichlet type boundary conditions.

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R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators

2006, Denk, Robert, Krainer, Thomas

It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with R-bounded symbols, yielding by an iteration argument the R-boundedness of the resolvent in a closed complex half-plane. To this end, elements of a symbolic and operator calculus of pseudodifferential operators with R-bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators with operator valued coefficients.

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A Forward Scheme for Backward SDEs

2005, Bender, Christian, Denk, Robert

We introduce a forward scheme to simulate backward SDEs. Compared to existing schemes, we avoid high order nestings of conditional expectations backwards in time. In this way the error, when approximating the conditional expectation, in dependence of the time partition is significantly reduced. Besides this generic result, we present an implementable algorithm and provide an error analysis for it. Finally, we demonstrate the strength of the new algorithm by solving some financial problems numerically.