H ∞ Functional Calculus and Characterization of Domains of Fractional Powers

Yagi, Akira

doi:10.1007/978-3-7643-8893-5_15

Cited by 9 publications

(13 citation statements)

References 32 publications

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“…However, there is an interesting result by Yagi that is relevant for the present purposes. He considers an operator

A = - \sum_{j, k = 1, \dots, n} \partial_{j} a_{j k} (x) \partial_{k} + c (x), with \sum_{j, k = 1, \dots, n} a_{j k} (x) ξ_{j} ξ_{j} \geq c_{0} {| ξ |}^{2},

a_{j k} = a_{k j}

real in

C^{1} (\bar{Ω})

c (x)

real bounded ⩾0 and

c_{0} > 0

, on a bounded C 2 ‐domain

Ω \subset R^{n}

.…”

Section: Further Developmentsmentioning

confidence: 91%

“…In Section , we first briefly discuss extensions to more general scales of function spaces. Next, for generalizations to nonsmooth domains, we show how a recent result of Denk, Dore, Hieber, Prüss and Venni , on the existence of

H^{\infty}

‐calculi for boundary problems, can be combined with more recent results of Yagi , , to extend the regularity properties of Sections and to suitable nonsmooth situations for small s , leading to new results.…”

Section: Introductionmentioning

confidence: 92%

“…Combined with the existence of an

H^{\infty}

‐calculus, Theorem of then shows: Theorem Let

1 < p < \infty

. For

0 \leq a \leq 1

, the fractional powers

{(A_{γ})}^{a}

L_{p} (Ω)

have domains

D_{p} ({(A_{γ})}^{a}) = \{\begin{matrix} right H_{p}^{2 a} (Ω) & right if 0 \leq 2 a < 0false \frac{1}{p}, \\ right H_{p, γ}^{2 a} (Ω) & right if 0false \frac{1}{p} < 2 a \leq 2, 2 a \neq 1 + 0false \frac{1}{p} . \end{matrix}

…”

Section: Further Developmentsmentioning

confidence: 99%

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Regularity of spectral fractional Dirichlet and Neumann problems

Grubb

2015

Mathematische Nachrichten

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Received XXXX, revised XXXX, accepted XXXX Published online XXXX MSC (2000) 35P99, 35S15, 47G30Consider the fractional powers (ADir) a and (ANeu) a of the Dirichlet and Neumann realizations of a secondorder strongly elliptic differential operator A on a smooth bounded subset Ω of R n . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of H s p -Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on H ∞ -calculus. We also include an overview of the various Dirichlet-and Neumann-type boundary problems associated with the fractional Laplacian.

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“…However, there is an interesting result by Yagi that is relevant for the present purposes. He considers an operator

A = - \sum_{j, k = 1, \dots, n} \partial_{j} a_{j k} (x) \partial_{k} + c (x), with \sum_{j, k = 1, \dots, n} a_{j k} (x) ξ_{j} ξ_{j} \geq c_{0} {| ξ |}^{2},

a_{j k} = a_{k j}

real in

C^{1} (\bar{Ω})

c (x)

real bounded ⩾0 and

c_{0} > 0

, on a bounded C 2 ‐domain

Ω \subset R^{n}

.…”

Section: Further Developmentsmentioning

confidence: 91%

H^{\infty}

Section: Introductionmentioning

confidence: 92%

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Regularity of spectral fractional Dirichlet and Neumann problems

Grubb

2015

Mathematische Nachrichten

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“…For example, we consider the regularity of functions in X α if Ω is bounded and convex. Then, it is well known that the function space X α is equivalent to the fractional Sobolev space H 2α (Ω) for 0 ≤ α < 1/4 [17]. Note that all elements u ∈ X α satisfy u = 0 on ∂Ω in the trace sense for 1/4 < α ≤ 1 and α = 3/4 [17].…”

Section: Remarkmentioning

confidence: 99%

“…Then, it is well known that the function space X α is equivalent to the fractional Sobolev space H 2α (Ω) for 0 ≤ α < 1/4 [17]. Note that all elements u ∈ X α satisfy u = 0 on ∂Ω in the trace sense for 1/4 < α ≤ 1 and α = 3/4 [17]. Moreover, if Ω ⊂ R 2 is a convex polygon, it is proved that all elements u ∈ X α satisfy u = 0 on ∂Ω…”

Section: Remarkmentioning

confidence: 99%

On the embedding constant of the Sobolev type inequality for fractional derivatives

Mizuguchi

Takayasu

Kubo

et al. 2016

NOLTA

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Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

2020

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We consider the numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise. For the spatial approximation we consider a standard finite element method and for the temporal approximation, a rational approximation of the exponential function. We first show strong convergence of this approximation in both positive and negative order norms. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.1991 Mathematics Subject Classification. 60H15, 65M12, 60H35, 65C30, 65M60, 60H07. Key words and phrases. Stochastic partial differential equations, stochastic wave equations, stochastic hyperbolic equations, weak convergence, finite element methods, Galerkin methods, rational approximations of semigroups, Crank-Nicolson method, Malliavin calculus.

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H ∞ Functional Calculus and Characterization of Domains of Fractional Powers

Cited by 9 publications

References 32 publications

Regularity of spectral fractional Dirichlet and Neumann problems

Regularity of spectral fractional Dirichlet and Neumann problems

On the embedding constant of the Sobolev type inequality for fractional derivatives

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

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