Extrapolation of $L^p$ maximal regularity for second order Cauchy problems

Chill, Ralph; Król, Sebastian

doi:10.4064/bc112-0-3

Cited by 14 publications

(27 citation statements)

References 26 publications

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“…Note that arg α ± = ±ϑ(ρ) and ϑ(ρ) π 2 for ρ 0. By the theory of quadratic operator pencils and second-order Cauchy problems, we can invert the operator V (λ) and show maximal L p -regularity, see Theorem 3.4 of [16] and Theorem 4.1 of [7], as well as [4] and [28]. However, a more detailed investigation of the related first-order system will be useful for the analysis of the half-space.…”

Section: The Full Space Casementioning

confidence: 99%

“…[4,7,28]. However, in this context the available existence results are restricted to the very special case that the damping operator is a multiple of the square root B 1/2 of the elastic operator B (which we call the square root case).…”

Section: Introductionmentioning

confidence: 97%

“…We thus present detailed proofs although some of the results could also be deduced from e.g. [7] and [16]. In Section 3 we derive the crucial solution formula for the parameter-dependent elliptic boundary value problem (3.1) corresponding to (1.1) on R n + and establish the core estimates on the operators appearing there, see Theorem 3.5 and Corollary 3.6.…”

Section: Introductionmentioning

confidence: 98%

“…In the square root case one can easily compute the resolvent of the associated generator in terms of the given operators and show its sectoriality, see [16] and the references therein, as well as [4,7,8,15,28] for more recent contributions. Moreover, Theorem 4.1 of [7] shows maximal regularity in the square root case if the elastic operator B has an 'R-bounded H ∞ -calculus' (which can be applied to our case if G = R n ). In these papers, inhomogeneous boundary data have not been considered.…”

Section: Introductionmentioning

confidence: 99%

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A structurally damped plate equation with Dirichlet–Neumann boundary conditions

Denk

Schnaubelt

2015

Journal of Differential Equations

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We investigate sectoriality and maximal regularity in L p -L q -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 C 4 . It turns out that the first-order system related to the scalar equation on R n 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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Section: The Full Space Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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A structurally damped plate equation with Dirichlet–Neumann boundary conditions

Denk

Schnaubelt

2015

Journal of Differential Equations

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“…The theory of maximal L p -regularity of the Equation 3 was been studied in the last years (see, for instance, other studies 15,16 ). However, in our definition, we remove the condition "u ∈ L p " to avoid triviality, since this fact, in discrete context, implies in exponential decay of solutions (see Section 4.1 for similar property).…”

Section: Introductionmentioning

confidence: 99%

A note on temporal regularity of second order difference equations

Dantas

2019

Math Methods in App Sciences

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This work deals with the study of maximal ℓp‐regularity of a pair (A,B) of bounded linear operators on a complex Banach space double-struckX associated to the second order difference equation un + 2 = Bun + 1 + Aun + fn, where f is a given sequence on double-struckX. We obtain results of characterization based on spectral analysis of the discrete sine family, which is the resolvent family of this equation.

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