Fractional Gaussian estimates and holomorphy of semigroups

Keyantuo, Valentin; Seoanes, Fabian; Warma, Mahamadi

doi:10.1007/s00013-019-01381-y

Cited by 2 publications

(2 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2.10), the semigroup (e −t(−∆) s D ) t≥0 can be extended to bounded analytic semigroups on L p (Ω), for every p ∈ (1, ∞). By [24], the semigroup is even analytic on L 1 (Ω).…”

Section: Preliminary Resultsmentioning

confidence: 98%

Exponential Turnpike property for fractional parabolic equations with non-zero exterior data

Warma

Zamorano

2021

ESAIM: COCV

Self Cite

View full text Add to dashboard Cite

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question arises: Do time averages of optimal controls and trajectories converge to the stationary optimal controls and states as the time-horizon goes to infinity? This question is very closely related to the so-called turnpike property that shows that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In the present paper we deal with heat equations with non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the fractional Laplace operator $(-\Delta)^s$ ( $0<s<1$ ). We prove the turnpike property for the nonlocal Robin optimal control problem and the exponential turnpike property for both Dirichlet and nonlocal Robin optimal control problems.

show abstract

Section: Preliminary Resultsmentioning

confidence: 98%

Exponential Turnpike property for fractional parabolic equations with non-zero exterior data

Warma

Zamorano

2021

ESAIM: COCV

Self Cite

View full text Add to dashboard Cite

show abstract

“…By Theorem 2.10 again the semigroups T D , T R and T N are analytic on L p (Ω) for every 1 < p < ∞. Very recently, using (5.1), it has been shown in [28] that the semigroup T D is also analytic of angle π 2 on L 1 (Ω).…”

Section: As We Have Already Mentioned In Section 3 It Is Then a Natur...mentioning

confidence: 89%

Realization of the fractional Laplacian with nonlocal exterior conditions via forms method

Claus

Warma

2020

J. Evol. Equ.

Self Cite

View full text Add to dashboard Cite

Let Ω ⊂ R n (n ≥ 1) be a bounded open set with a Lipschitz continuous boundary. In the first part of the paper, using the method of bilinear forms we give a characterization of the realization in L 2 (Ω) of the fractional Laplace operator (−∆) s (0 < s < 1) with the nonlocal Neumann and Robin exterior conditions. Contrarily to the classical local case s = 1, it turns out that the nonlocal (Robin and Neumann) exterior conditions can be incorporated in the form domain. We show that each of the above operators generates a strongly continuous submarkovian semigroup which is also ultracontractive. In the second part, we prove that the semigroup corresponding to the nonlocal Robin exterior condition is always sandwiched between the fractional Dirichlet semigroup and the fractional Neumann semigroup.2010 Mathematics Subject Classification. 35R11, 47D07, 47D06, 34B10.

show abstract

Fractional Gaussian estimates and holomorphy of semigroups

Cited by 2 publications

References 35 publications

Exponential Turnpike property for fractional parabolic equations with non-zero exterior data

Exponential Turnpike property for fractional parabolic equations with non-zero exterior data

Realization of the fractional Laplacian with nonlocal exterior conditions via forms method

Contact Info

Product

Resources

About