Denk, Robert

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Robert
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An elliptic boundary problem acting on generalized Sobolev spaces

2017-10-05T10:47:25Z, Denk, Robert, Faierman, Melvin

We consider an elliptic boundary problem over a bounded region Ω in Rn and acting on the generalized Sobolev space W0p(Ω) for 1n or a closed manifold acting on W02(Ω), called H\"{o}rmander space, have been the subject of investigation by various authors. Then in this paper we will, under the assumption of parameter-ellipticity, establish results pertaining to the existence and uniqueness of solutions of the boundary problem. Furthermore, under the further assumption that the boundary conditions are null, we will establish results pertaining to the spectral properties of the Banach space operator induced by the boundary problem, and in particular, to the angular and asymptotic distribution of its eigenvalues.

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Maximal Lp-regularity of non-local boundary value problems

2013, Denk, Robert, Seiler, Jörg

We investigate the R-boundedness of operator families belonging to the Boutet de Monvel calculus. In particular, we show that weakly and strongly parameter-dependent Green operators of nonpositive order are R-bounded. Such operators appear as resolvents of non-local (pseudodifferential) boundary value problems. As a consequence, we obtain maximal Lp-regularity for such boundary value problems. An example is given by the reduced Stokes equation in waveguides.

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Pseudodifferential operators with non-regular operator-valued symbols

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

In this paper, we consider pseudodifferential operators with operator-valued symbols and their mapping properties, without assumptions on the underlying Banach space. We show that, under suitable parabolicity assumptions, the realization of the operator in the standard Sobolev space generates an analytic semigroup. An application to non-autonomous pseudodifferential Cauchy problems gives the existence and uniqueness of a classical solution. Our approach is based on oscillatory integrals and kernel estimates for them.

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Resolvent Estimates for Elliptic Systems in Function Spaces of Higher Regularity

2010, Denk, Robert, Dreher, Michael

We consider parameter-elliptic boundary value problems and uniform a priori estimates in Lp-Sobolev spaces of Bessel potential and Besov type. The problems considered are systems of uniform order and mixed-order systems (Douglis-Nirenberg systems). It is shown that compatibility conditions on the data are necessary for such estimates to hold. In particular, we consider the realization of the boundary value problem as an unbounded operator with the ground space being a closed subspace of a Sobolev space and give necessary and sufficient conditions for the realization to generate an analytic semigroup.

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Maximal regularity for the thermoelastic plate equations with free boundary conditions

2016, Denk, Robert, Shibata, Yoshihiro

We consider the linear thermoelastic plate equations with free boundary conditions in the Lp in time and Lq in space setting. We obtain unique solvability with optimal regularity for the inhomogeneous problem in a uniform C4-domain, which includes the cases of a bounded domain and of an exterior domain with C4-boundary. Moreover, we prove uniform a priori-estimates for the solution. The proof is based on the existence of R-bounded solution operators of the corresponding generalized resolvent problem which is shown with the help of an operator-valued Fourier multiplier theorem due to Weis.

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Inhomogeneous boundary value problems in spaces of higher regularity

2013, Denk, Robert, Seger, Tim

Uniform a priori estimates for parameter-elliptic boundary value problems are well-known if the underlying basic space equals Lp(Ω). However, much less is known for the Wsp(Ω)-realization, s>0, of a parameter-elliptic boundary value problem. We discuss a priori estimates and the generation of analytic semigroups for these realizations in various cases. The Banach scale method can be applied for homogeneous boundary conditions if the right-hand side satis es certain compatibility conditions, while for the general case parameter-dependent norms are used. In particular, we obtain a resolvent estimate for the General situation where no analytic semigroup is generated.

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Necessity of parameter-ellipticity for multi-order systems of differential equations

2011, Denk, Robert, Faierman, Melvin

In this paper we investigate parameter-ellipticity conditions for multi-order systems of differential equations on a bounded domain. Under suitable assumptions on smoothness and on the order structure of the system, it is shown that parameter-dependent a priori-estimates imply the conditions of parameter-ellipticity, i.e., interior ellipticity, conditions of Shapiro-Lopatinskii type, and conditions of Vishik-Lyusternik type. The mixed-order systems considered here are of general form; in particular, it is not assumed that the diagonal operators are of the same order. This paper is a continuation of an article by the same authors where the sufficiency was shown, i.e., a priori-estimates for the solutions of parameter-elliptic multi-order systems were established.

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A structurally damped plate equation with Dirichlet-Neumann boundary conditions

2014, Denk, Robert, Schnaubelt, Roland

We investigate sectoriality and maximal regularity in Lp-Lq-Sobolev spaces for the structurally damped plate equation with Dirichlet-Neumann (clamped) boundary conditions. We obtain unique solutions with optimal regularity for the inhomogeneous problem in the whole space, in the half-space, and in bounded domains of class C4.



It turns out that the first-order system related to the scalar equation on Rn is sectorial only after a shift in the operator. On the half-space one has to include zero boundary conditions in the underlying function space in order to obtain sectoriality of the shifted operator and maximal regularity for the case of homogeneous boundary conditions. We further show that the semigroup solving the problem on bounded domains is exponentially stable.

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Lp-estimates for a transmission problem of mixed elliptic-parabolic type

2013, Denk, Robert, Seger, Tim

We consider the situation when an elliptic problem in a subdomain Ω1 of an n-dimensional bounded domain Ω is coupled via inhomogeneous canonical transmission conditions to a parabolic problem in Ω\Ω1. In particular, we can treat elliptic-parabolic equations in bounded domains with discontinuous coefficients. Using Fourier multiplier techniques, we prove an a priori estimate for strong solutions to the equations in Lp-Sobolev spaces.

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Discrete Fourier multipliers and cylindrical boundary value problems

2011, Denk, Robert, Nau, Tobias

We consider operator-valued boundary value problems in (0;2π)n with periodic or, more generally, v-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary value problem. As an application, we study vector-valued parabolic initial boundary value problems in cylindrical domains (0;2π)n x V with v-periodic boundary conditions in the cylindrical directions. We show that under suitable assumptions on the coefficients, we obtain maximal Lq-regularity for such problems.