Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains
Abstract
:1. Introduction
2. Preliminaries
2.1. Geometry
2.2. Sobolev Spaces
- (i)
- is a linear and continuous operator from to ;
- (ii)
- there exists a linear and continuous operator Ext from to such that is the identity operator in .
2.3. Besov Spaces
- (i)
- is a linear and continuous operator from to ;
- (ii)
- there exists a linear and continuous operator Ext from to such that is the identity operator in .
2.4. Convergence of Hilbert Spaces
2.5. Fractional-in-Time Derivatives
- (i)
- The Riemann–Liouville fractional derivative of order is defined as follows:for a.e. .
- (ii)
- The Caputo-type fractional derivative of order is defined as follows:for a.e. .
- (i)
- (The case ) The function is such that , for all , and . Moreover, the equation is satisfied on .
- (ii)
- (The case ) The function is such that , for , and . Moreover, the equation is satisfied on .
3. The Energy Forms
3.1. The Fractal Energy Form
3.2. The Pre-Fractal Energy Forms
3.3. Resolvents and Associated Semigroups
4. Existence and Uniqueness Results
4.1. The Abstract Cauchy Problems
- (i)
- (The case )for some ;
- (ii)
- (The case there exists such thatfor some .
- (i)
- (The case )for some ;
- (ii)
- (The case there exists such thatfor some .
4.2. The Venttsel’ Boundary Value Problems
5. Convergence Results
5.1. Convergence of Spaces and M-Convergence of the Energy Forms
- (i)
- for every weakly converging to in
- (ii)
- for every there exists a sequence , with strongly converging to u in , such that
5.2. Convergence of the Solutions of the Abstract Cauchy Problems
- (i)
- converges to in for every fixed ;
- (ii)
- converges to u in .
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Capitanelli, R.; Creo, S.; Lancia, M.R. Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains. Fractal Fract. 2023, 7, 479. https://doi.org/10.3390/fractalfract7060479
Capitanelli R, Creo S, Lancia MR. Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains. Fractal and Fractional. 2023; 7(6):479. https://doi.org/10.3390/fractalfract7060479
Chicago/Turabian StyleCapitanelli, Raffaela, Simone Creo, and Maria Rosaria Lancia. 2023. "Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains" Fractal and Fractional 7, no. 6: 479. https://doi.org/10.3390/fractalfract7060479
APA StyleCapitanelli, R., Creo, S., & Lancia, M. R. (2023). Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains. Fractal and Fractional, 7(6), 479. https://doi.org/10.3390/fractalfract7060479