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Weak differentiability of scalar hysteresis operators
1. | Fakultät für Mathematik, TU München, Boltzmannstr. 3, D 85747 Garching bei München |
2. | Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1 |
References:
[1] |
J.-J. Moreau, Problème d'evolution associé à un convexe mobile d'un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A791-A794. (In French.) |
[2] |
J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Eq., 26 (1977), 347-374.
doi: 10.1016/0022-0396(77)90085-7. |
[3] |
A. Mielke, Evolution of rate independent systems, in Evolutionary Equations, vol. II (eds. C.M. Dafermos and and E. Feireisl), Handbook of Differential Equations, Elsevier/North Holland, Amsterdam, II (2005), 461-559. |
[4] |
A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177.
doi: 10.1007/s002050200194. |
[5] |
M. A. Krasnosel'skiĭ, B. M. Darinskiĭ, I. V. Emelin, P. P. Zabrejko, E. A. Lifshits and A. V. Pokrovskiĭ, An operator-hysterant, Dokl. Akad. Nauk SSSR, 190 (1970), 34-37; Soviet Math. Dokl., 11 (1970), 29-33. |
[6] |
M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Nauka, Moscow, 1983. (In Russian.) |
[7] |
M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, Heidelberg, 1989.
doi: 10.1007/978-3-642-61302-9. |
[8] |
A. Visintin, Differential models of hysteresis, Springer, Berlin, 1994.
doi: 10.1007/978-3-662-11557-2. |
[9] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[10] |
P. Krejĭ, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996. |
[11] |
P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep., 35 (1991), 31-62.
doi: 10.1080/17442509108833688. |
[12] |
P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183. |
[13] |
P. Krejčí, Hysteresis in singularly perturbed problems, in Singular Perturbations and Hysteresis (eds. M.P. Mortell, R.E. O'Malley, A. Pokrovskiĭ, and V. Sobolev), SIAM, (2005), 73-100.
doi: 10.1137/1.9780898717860.ch3. |
[14] |
M. Brokate and P. Krejčí, Duality in the space of regulated functions and the play operator, Math. Z., 245 (2003), 667-688.
doi: 10.1007/s00209-003-0563-6. |
[15] |
R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 7 (2008), 97-169. |
[16] |
I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Springer, Berlin, 1972. |
[17] |
D. Fraňková, Regulated functions, Math. Bohem., 116 (1991), 20-59. |
[18] |
L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. f. angew. Math., 8 (1928), 85-106. (In German.) |
[19] |
A. Yu. Ishlinskiĭ, Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR Techn. Ser., 9 (1928), 583-590. (In Russian.) |
[20] |
F. Preisach, Über die magnetische Nachwirkung, Z. Phys., 94 (1935), 277-302. (In German.) |
[21] |
P. Krejčí, On Maxwell equations with the Preisach operator: The one-dimensional time-periodic case, Apl. Mat., 34 (1989), 364-374. |
[22] |
M. Brokate, Some BV properties of the Preisach hysteresis operator, Appl. Anal., 32 (1989), 229-252.
doi: 10.1080/00036818908839851. |
show all references
References:
[1] |
J.-J. Moreau, Problème d'evolution associé à un convexe mobile d'un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B, 276 (1973), A791-A794. (In French.) |
[2] |
J.-J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Eq., 26 (1977), 347-374.
doi: 10.1016/0022-0396(77)90085-7. |
[3] |
A. Mielke, Evolution of rate independent systems, in Evolutionary Equations, vol. II (eds. C.M. Dafermos and and E. Feireisl), Handbook of Differential Equations, Elsevier/North Holland, Amsterdam, II (2005), 461-559. |
[4] |
A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177.
doi: 10.1007/s002050200194. |
[5] |
M. A. Krasnosel'skiĭ, B. M. Darinskiĭ, I. V. Emelin, P. P. Zabrejko, E. A. Lifshits and A. V. Pokrovskiĭ, An operator-hysterant, Dokl. Akad. Nauk SSSR, 190 (1970), 34-37; Soviet Math. Dokl., 11 (1970), 29-33. |
[6] |
M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Nauka, Moscow, 1983. (In Russian.) |
[7] |
M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis, Springer, Heidelberg, 1989.
doi: 10.1007/978-3-642-61302-9. |
[8] |
A. Visintin, Differential models of hysteresis, Springer, Berlin, 1994.
doi: 10.1007/978-3-662-11557-2. |
[9] |
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[10] |
P. Krejĭ, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakkōtosho, Tokyo, 1996. |
[11] |
P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep., 35 (1991), 31-62.
doi: 10.1080/17442509108833688. |
[12] |
P. Krejčí and Ph. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183. |
[13] |
P. Krejčí, Hysteresis in singularly perturbed problems, in Singular Perturbations and Hysteresis (eds. M.P. Mortell, R.E. O'Malley, A. Pokrovskiĭ, and V. Sobolev), SIAM, (2005), 73-100.
doi: 10.1137/1.9780898717860.ch3. |
[14] |
M. Brokate and P. Krejčí, Duality in the space of regulated functions and the play operator, Math. Z., 245 (2003), 667-688.
doi: 10.1007/s00209-003-0563-6. |
[15] |
R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 7 (2008), 97-169. |
[16] |
I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, Springer, Berlin, 1972. |
[17] |
D. Fraňková, Regulated functions, Math. Bohem., 116 (1991), 20-59. |
[18] |
L. Prandtl, Ein Gedankenmodell zur kinetischen Theorie der festen Körper, Z. f. angew. Math., 8 (1928), 85-106. (In German.) |
[19] |
A. Yu. Ishlinskiĭ, Some applications of statistical methods to describing deformations of bodies, Izv. AN SSSR Techn. Ser., 9 (1928), 583-590. (In Russian.) |
[20] |
F. Preisach, Über die magnetische Nachwirkung, Z. Phys., 94 (1935), 277-302. (In German.) |
[21] |
P. Krejčí, On Maxwell equations with the Preisach operator: The one-dimensional time-periodic case, Apl. Mat., 34 (1989), 364-374. |
[22] |
M. Brokate, Some BV properties of the Preisach hysteresis operator, Appl. Anal., 32 (1989), 229-252.
doi: 10.1080/00036818908839851. |
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