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Hölder-estimates for non-autonomous parabolic problems with rough data
A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications
1. | Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States |
References:
[1] |
C. Baiocchi, Un teorema di interpolazione: Applicazioni ai problemi ai limiti per le equazioni a derivate parziali, Ann. Mat. Pura Appl., 73 (1966), 233-251.
doi: 10.1007/BF02415089. |
[2] |
A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, $2^{nd}$ edition, Birkhauser, 2007, 575 pages.
doi: 10.1007/978-0-8176-4581-6. |
[3] |
S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case $\alpha = 1/2$, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234-256. Proceedings of Seminar on Approximation and Optimization, University of Havana, Cuba (January 1987).
doi: 10.1007/BFb0089601. |
[4] |
S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$), Pacific J. Math., 136 (1989), 15-55.
doi: 10.2140/pjm.1989.136.15. |
[5] |
S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Diff. Eqns., 88 (1990), 279-293.
doi: 10.1016/0022-0396(90)90100-4. |
[6] |
L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rendiconti del Seminario Matematico della Universita di Padova, 34 (1964), 205-223. |
[7] |
D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 (1967), 82-86.
doi: 10.3792/pja/1195521686. |
[8] |
P. Grisvard, Characterization de qualques espaces d' interpolation, Arch. Pat. Mech. Anal., 25 (1967), 40-63.
doi: 10.1007/BF00281421. |
[9] |
T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan., 13 (1961), 246-274.
doi: 10.2969/jmsj/01330246. |
[10] |
I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach, Appl. Math. & Optimiz., 6 (1980), 287-333.
doi: 10.1007/BF01442900. |
[11] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. |
[12] |
I. Lasiecka and R. Triggiani, Domains of fractional powers of matrix-valued Operators: A general approach, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Operator Theory Advances and Applications, W.Arendt, R.Chill and Y.Tomilov, Editors, 250 (2015), 297-309.
doi: 10.1007/978-3-319-18494-4_20. |
[13] |
I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, Communications on Pure & Applied Analysis, to appear. |
[14] |
C. Lebiedzik and R. Triggaini, The optimal interior regularity for the critical case of a clamped thermoelastic system with point control revisited, Modern Aspects of the Theory of PDEs. Vol. 216 of Operator Theory: Advances and Applications, 243-259, Birkhäuser/Springer, Basel, 2011. M. Ruzhansky and J. Wirth, eds.
doi: 10.1007/978-3-0348-0069-3_14. |
[15] |
J. L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs, J. Math Soc., 14 (1962), 233-241.
doi: 10.2969/jmsj/01420233. |
[16] |
J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I, Springer-Verlag (1972), 357 pp. |
[17] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[18] |
R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A.V.Balakrishnan, to appear. |
show all references
References:
[1] |
C. Baiocchi, Un teorema di interpolazione: Applicazioni ai problemi ai limiti per le equazioni a derivate parziali, Ann. Mat. Pura Appl., 73 (1966), 233-251.
doi: 10.1007/BF02415089. |
[2] |
A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, $2^{nd}$ edition, Birkhauser, 2007, 575 pages.
doi: 10.1007/978-0-8176-4581-6. |
[3] |
S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case $\alpha = 1/2$, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234-256. Proceedings of Seminar on Approximation and Optimization, University of Havana, Cuba (January 1987).
doi: 10.1007/BFb0089601. |
[4] |
S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$), Pacific J. Math., 136 (1989), 15-55.
doi: 10.2140/pjm.1989.136.15. |
[5] |
S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Diff. Eqns., 88 (1990), 279-293.
doi: 10.1016/0022-0396(90)90100-4. |
[6] |
L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rendiconti del Seminario Matematico della Universita di Padova, 34 (1964), 205-223. |
[7] |
D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 (1967), 82-86.
doi: 10.3792/pja/1195521686. |
[8] |
P. Grisvard, Characterization de qualques espaces d' interpolation, Arch. Pat. Mech. Anal., 25 (1967), 40-63.
doi: 10.1007/BF00281421. |
[9] |
T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan., 13 (1961), 246-274.
doi: 10.2969/jmsj/01330246. |
[10] |
I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach, Appl. Math. & Optimiz., 6 (1980), 287-333.
doi: 10.1007/BF01442900. |
[11] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. |
[12] |
I. Lasiecka and R. Triggiani, Domains of fractional powers of matrix-valued Operators: A general approach, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Operator Theory Advances and Applications, W.Arendt, R.Chill and Y.Tomilov, Editors, 250 (2015), 297-309.
doi: 10.1007/978-3-319-18494-4_20. |
[13] |
I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, Communications on Pure & Applied Analysis, to appear. |
[14] |
C. Lebiedzik and R. Triggaini, The optimal interior regularity for the critical case of a clamped thermoelastic system with point control revisited, Modern Aspects of the Theory of PDEs. Vol. 216 of Operator Theory: Advances and Applications, 243-259, Birkhäuser/Springer, Basel, 2011. M. Ruzhansky and J. Wirth, eds.
doi: 10.1007/978-3-0348-0069-3_14. |
[15] |
J. L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs, J. Math Soc., 14 (1962), 233-241.
doi: 10.2969/jmsj/01420233. |
[16] |
J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I, Springer-Verlag (1972), 357 pp. |
[17] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[18] |
R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A.V.Balakrishnan, to appear. |
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