Keywords
- Weak Solution
- Variational Inequality
- Free Boundary
- Free Boundary Problem
- Stefan Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
[B] Baiocchi, C.—Sur un problème à frontière libre traduisant le filtrage de liquides à travers des milieux poreux, C.R. Acad. Sci. Paris, 273 (1971), 1215–1217.
[BC] Baiocchi, C. & Capelo, A.—Disequazioni variazionali e quasi variazionali. Applicazioni a problemi di frontiera libera, Vol. I, II, Quaderni dell’U. Mat. Ital., Pitagora, Bologna, 1978; English transl. J. Wiley, Chichester-New York, 1984.
[Ba1] Barbu, V.—Nonlinear semigroups and differential equations in Banach Spaces, Noordhoof Int. Publ., Leyden, 1976.
[Ba2] Barbu, V.—Optimal Control of Variational Inequalities, Research Notes in Math. No. 100, Pitman, Boston, London, 1984.
[BL] Bensoussan, A. & Lions J.L.—Applications des Inéquations Variationelles en Contrôle Stochastique, Dunod, Paris, 1978; English transl., North-Holland, Amsterdam, 1982.
[BDF] Bossavit, A., Damlamian, A. & Frémond, M.—Free Boundary problems: applications and theory, Vol. III, IV, Research Notes in Math. Nos. 120/121, Pitman, Boston, London, 1985.
[BFN] Brauner, C.M., Frémond, M. & Nicolaenko, B.—A new homographic approximation to multiphase Stefan problem, in: “Free Boundary Problems—Theory and Applications” (A. Fasano, M. Primicerio (Eds.), Vol. II, Pitman, Boston, 1983, 365–379.”
[B1] Brézis, H.—On some degenerate nonlinear parabolic equations, (in Browder F.E., ed.) Nonlinear functional analysis, Part I, Symp. Pure Math., 18 (1970), 28–38.
[B2] Brézis, H.—Problèmes unilatéraux, J. Math. Pures et Appl., 51 (1972), 1–168.
[C1] Caffarelli, L.A.—The Regularity of Free Boundaries in Higer Dimensionsm, Acta Math., 139 (1977), 155–184.
[C2] Caffarelli, L.A.—Some Aspects of the One-Phase Stefan Problem, Indiana Univ. Math. J., 27 (1978), 73–77.
[CE] Caffarelli, L. & Evans, L.—Continuity of the temperature in the two-phase Stefan problems, Arch. Rational Mech. Anal., 81, 3 (1983), 199–220.
[CF] Caffarelli, L.A. & Friedman, A.—Continuity of the Temperature in the Stefan Problem, Indiana Univ. Math. J., 28 (1979), 53–70.
[CD] Cannon, J.R. & DiBenedetto, E.—On the existence of weak solutions to an n-dimensional Stefan problem with nonlinear boundary conditions, SIAM J. Math. Anal., 11 (1980), 632–645.
[CR] Chipot, M. & Rodrigues, J.F.—On the Steady-State Continuous Casting Stefan Problem with Non-Linear Cooling, Quart. Appl. Math., 40 (1983), 476–491.
[Cr] Crank, J.—Free and moving boundary problems, Oxford, Univ. Press, Oxford, 1984.
[D1] Damlamian, A.—Résolution de certaines inéquations variationnelles stationnaires et d’évolution, Thése Doct., Univ., Paris VI, 1976.
[D2] Damlamian, A.—Some results on the multiphase Stefan problem, Comm. in P.D.E., 2(10) (1977), 1017–1044.
[D3] Damlamian, A.—On the Stefan problem: the variational approach and some applications, in “Math. Models Meth. in Mech.”, Banach Cent. Publ. (Warsaw), Vol. 15 (1985), 253–275.
[DK1] Damlamian A. & Kenmochi, N.—Le problème de Stefan avec conditions latérales variables, Hiroshima Math. J., 10 (1980), 271–293.
[DK2] Damlamian, A. & Kenmochi, N.—Asymptotic behaviour of the solution to a multiphase Stefan problem, Japan J. Appl. Math., 3 (1986), 15–36.
[DK3] Damlamian, A. & Kenmochi, N.—Periodicity and almost-periodicity of solutions to a multiphase Stefan problem in several space variables. Nonlinear analysis, Th. Math. and Appl., 12 (1988), 921–934.
[DK4] Damlamian, A. & Kenmochi, N.—Uniqueness of the solution of a Stefan problem with variable lateral boundary conditions, Adv. Math. Sci. Appl., 1 (1992), 175–194.
[Dan] Danilyuk, I.I.—On the Stefan problem, Russian Math. Surveys, 40:5 (1985), 157–223.
[Db] DiBenedetto, E.—Continuity of weak solutions to certain singular parabolic equations, Ann. Mat. Pura Appl., (4), 130 (1982), 131–176.
[DF] DiBenedetto, E. & Friedman, A.—Periodic Behaviour for the evolutionary dam problem and related free boundary problems, Comm. P.D.E., 11 (1986), 1297–1377.
[DO] DiBenedetto, E. & O’Leary, M.—3-D conduction-convection with change of phase, Arch. Rational Mech. Anal. (in press).
[Du1] Duvaut, G.—Résolution d’un problème de Stefan, C. R. Ac. Sci. Paris, 276-A (1973), 1461–1463.
[Du2] Duvaut, G.—The solution of a two-phase Stefan problem by a variational inequality, in: J.R. Ockendon, A.R. Hodgkins Eds., “Moving Boundary Problems in Heat Flow and Diffusion”, Clarendon Press, Oxford, 1975, 173–181.
[DL] Duvaut, G. & Lions, J.L.—Les inéquations en mécanique et en physique, Dunod, Paris, 1972; English transl. Springer, Berlin, 1976.
[EO] Elliott, C.M. & Ockendon, J.R.—Weak and Variational Methods for Moving Boundary Problems, Research Notes in Maths. 59, Pitman, Boston, London, 1982.
[FP] Fasano, A. & Primicerio, M. (eds.)—Free Boundary Problems: Th. & Appl., Research Notes in Math., 78/79, Pitman, Boston, 1983.
[Fr] Frémond, M.—Viriational formulation of the Stefan problem—Coupled Stefan problem—Frost propagation in porous media, in: “Computational Methods in Nonlinear Mechanics” (J.T. Oden, Ed.), The University of Texas, Austin, 1974.
[F1] Friedman, A.—The Stefan problem in several space variables, Trans. Amer. Math. Soc., 133 (1968), 51–87.
[F2] Friedman, A.—Variational Principles and Free-Boundary Problems, Wiley, New York, 1982.
[FH] Friedman, A. & Hoffmann, K.-H.—Control of free boundary problems with hysteresis, SIAM J. Control and Optim., 26 (1988), 42–55.
[FHY] Friedman, A., Huang, S. & Yong, J.—Optimal periodic control for the two-phase Stefan problem, SIAM J. Control and Optim., 26 (1988), 23–41.
[FK] Friedman, A. & Kinderlehrer, D.—A one phase Stefan problem, Indiana Univ. Math. J., 24 (1975), 1005–1035.
[GHM] Götz, I.G., Hoffmann, H.-H. & Meirmanov, A.M.—Periodic solutions of the Stefan problem with hysteresis-type boundary conditions, Rep. 399, T.U. München, 1992.
[GZ] Götz, I.G. & Zaltzman, B.B.—Nonincrease of mushy region in a nonhomogeneous Stefan problem, Quart. Appl. Math., 49 (1991), 741–746
[H] Hilpert, M.—On uniqueness for evolution problems with hysteresis, Int. Ser. Numer. Math., 88 (1989), 377–388.
[HN] Hoffmann, K.H. & Niezgodka, M.—Control of Parabolic Systems Involving Free Boundaries, in [FP], Vol. II (1983), 431–453.
[HS] Hoffmann, K.H. & Sprekels, J.—Free Boundary Problems, Theory and Applications, Pitman Res. Notes in Math., 185/186, Longman, 1990.
[Ka] Kamenomostkaya, S.—On the Stefan problem (in Russian), Naučnye Dokl. Vysšei Školy, 1 (1958), 60–62; Mat. Sb., 53(95) (1961), 489–514.
[KS] Kinderlehrer, D. & Stampacchia, G.—An introduction to variational inequalities and their applications, Acad. Press, New York, 1980.
[LSU] Ladyzenskaya, O.A., Solonnikov, V.A. & Ural’eva, N.N.—Linear and Quasilinear Equations of Parabolic Type, A.M.S. Transl. Monog. 23, Providence, 1968.
[LC] Lamé, G. & Clapeyron, B.P.—Mémoire sur la solidification par refroidissement d’un globe solid, Ann. Chem. Phys., 47 (1831), 250–256.
[L1] Lions, J.L.—Sur le contrôle optimale de systèmes governés par des équations aux dérivées partielles, Dunod, Paris, 1968.
[L2] Lions, J.L.—Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.
[L3] Lions, J.L.—Sur quelques questions d’Analyse, de Mécanique et de Contrôle Optimal, Presses Univ. Montreal, 1976.
[LR] Louro, B. & Rodrigues, J.F.—Remarks on the quasi-steady one phase Stefan problem, Proc. Royal Soc. Edinburg, 102-A (1986), 263–275.
[M1] Magenes, E. (ed.)—Free Boundary Problems, Proc. Pavia Seminar Sept.–Oct. 1979, Instituto Nazionale di Alta Matematica Francesco Severi, Vol. I and II, Roma, 1980.
[M2] Magenes, E.—Problemi di Stefan Bifase in più Variabili Spaziali, Le Matematiche, (Catania) 36 (1983), 65–108.
[MVV1] Magenes, E., Verdi, C. & Visintin, A.—Semigroup approach to the Stefan problem with nonlinear flux, Rend. Arad. Naz. Lincei, 75(2) (1983), 24–33.
[MVV2] Magenes, E., Verdi, C. & Visintin, A.—Theoretical and numerical results on the two-phase Stefan Problem, SIAM J. Numer. Anal., 26 (1989), 1425–1438.
[Me1] Meirmanov, A.M.—On the Classical Solution of the Multidimensional Stefan Problem for Quasilinear Parabolic Equations, Mat. Sbornik, 112 (1980), 170–192; Math. USSR-Sb., 40, 2 (1981), 157–179).
[Me2] Meirmanov, A.M.—The Stefan Problem, Nauka, Novosibirsk, 1986; English transl., De Gruyter, Berlin, 1992.
[Na1] Nagase, H.—On an application of Rothe’s method to nonlinear parabolic variational inequalities, Funkcialaj Ekvacioj, 32 (1989), 273–299.
[Na2] Nagase, H.—On an asymptotic behaviour of solutions of nonlinear parabolic variational inequalities, Japan. J. Math., 15 (1989), 169–189.
[N] Niezgodka, M.—Stefan-like problems, in [FP] (1983), 321–347.
[NP] Niezgodka, N. & Pawlow, I.—A generalized Stefan problem in several space variables, Appl. Math. Optim., 9 (1983), 193–224.
[No] Nochetto, R.—A class of non-degenerate two-phase Stefan problems in several space variables, Comm. P.D.E., 12 (1987), 21–45.
[O] Oleinik, O.A.—A method of solution of the general Stefan problem, Soviet Math. Dokl., 1 (1960), 1350–1354.
[OPR] Oleinik, O.A., Primicerio, M. & Radkevich, E.V.—Stefan-like problems, Meccanica, 28 (1993), 129–143.
[P1] Pawlow, I.—A variational inequality approach to generalized two-phase Stefan problem in several space variables, Annali Mat. Pura Appl., 131 (1982), 333–373.
[P2] Pawlow, I.—Analysis and control of evolution multiphase problems with free boundaries, Polska Akad. Nauk., Warszawa, 1987.
[Pr] Primicerio, M.—Problemi di diffusione a frontiera libera, Boll. U.M.I., (5) 18-A (1981), 11–68.
[PR] Primicerio, M. & Rodrigues, J.F.—The Hele-Shaw Problem with nonlocal injection condition, Gakuto Int. Ser., Math. Sci. Appl., Vol. 2 (1993), 375–390.
[R1] Rodrigues, J.F.—Some remarks on the asymptotic behaviour of strong solutions to monotone parabolic variational inequalities, Rendiconti di Mat., 3 (1984), 457–470.
[R2] Rodrigues, J.F.—On the variational inequality approach to the one-phase Stefan problem, Acta Appl. Math., 8 (1987), 1–35.
[R3] Rodrigues, J.F.—Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987.
[R4] Rodrigues, J.F.—The Stefan problem revisited, Int. Ser. Num. Math., 88 (1989), 129–190.
[R5] Rodrigues, J.F. (Editor)—Mathematical models for phase change problems, ISNM no. 88, Birkhäuser, Basel, 1989.
[R6] Rodrigues, J.F.—Weak solutions for thermoconvective flows of Boussinesq-Stefan type, Pitman Res. Notes in Math. Ser., 279 (1992), 93–116.
[RY] Rodrigues, J.F. & Yi Fahuai—On a two-phase continuous casting Stefan problem with nonlinear flux, Euro. J. Appl. Math., 1 (1990), 259–278.
[RZ] Rodrigues, J.F. & Zaltzman, B.—Free boundary optimal control in the multidimensional Stefan problem, in: Intern. Conf. on “Free Boundary Problems: Theory and Applications”, Toledo, June, 1993 (to appear).
[R] Rubinstein, L.I.—The Stefan problem, Amer. Math. Soc. Transl. Monogr. 27, Providence, 1971.
[Ru] Rulla, J.—Weak Solutions to Stefan Problems with Prescribed Convection, SIAM J. Math. Anal., 18 (1987), 1784–1800.
[S1] Saguez, C.—Contrôle optimal d’inéquations variationnelles avec observations de domains, Rapport no. 286, I.R.I.A. (1978); see also C. R. Acad. Sc. Paris, 287-A (1978), 957–959.
[S2] Saguez, C.—Contrôle optimal de systèmes à frontière libre, Thèse d’État, Université Technologie de Compiègne, 1980.
[Si] Simon, L.—Compact sets in the space L p(0,T;B), Annali Mat. pura ed Appl., 146 (1987), 65–96.
[Sh] Shinoda, J.—On a continuous casting problem with periodicity in time, Gakuto Int. Ser., Math. Sci. Appl., Vol. 2 (1993), 655–669.
[St] Stefan, J.—Über einige Probleme der Theorie der Wärmeleitung, Sitzungsber, Wien, Akad. Mat. Natur., 98 (1889), 473–484; see also pp. 614–634; 965–983; 1418–1442.
[T1] Tarzia, D.A.—Sur le problème de Stefan à deux phase, C. R. Acad. Sci. Paris, 288 (1980), 941–944.
[T2] Tarzia, D.A.—Etude de l’inéquation variationnelle proposée par Duvaut pour le problème de Stefan à deux phases, I, Boll. UMI, 1-B (1982), 865–883; II, Boll. UMI, 2-B (1983), 589–603.
[T3] Tarzia, D.A.—A bibliography on moving-free boundary problems for the heat diffusion equations, Rep. Math. Dept. Univ. Firenze, 1988.
[V1] Visintin, A.—Sur le problème de Stefan avec flux non linéaire, Boll. UMI C., 18 (1981), 63–86.
[V2] Visintin, A.—General free boundary evolution problems in several space dimensions, J. Math. Anal. Appl., 95 (1983), 117–143.
[V3] Visintin, A.—A phase transition problem with delay, Control and Cybernetics, 11 (1982), 5–18.
[V4] Visintin, A.—A model for hysteresis of distributed systems, Annal. Mat. Pura ed Appl., 131 (1982), 203–231.
[V5] Visintin, A.—Mathematical Models of Hysteresis, in: “Topics in Nonsmooth Mechanics” (J.J. Moreau, P.D. Panagiotopoulos, G. Strang, Eds.), Birkhäuser, 1988, 295–326.
[Y] Yi Fahuai—An evolutionary continuous casting problem of two-phase and its periodic behaviour, J. Part. Diff. Eq., 2(3) (1989), 7–22.
[YQ] Yi, F. & Qin, Y.—On the Stefan problem with prescribed convection (to appear).
[Za] Zaltzman, B.—Multidimensional two-phase quasistationary Stefan problem, Manuscripta Math., 78 (1993), 287–301.
[Z] Zeidler—Nonlinear functional analysis and its applications, Vol. II/B, Nonlinear monotone operators, Springer Verlag, New York, 1990.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag
About this chapter
Cite this chapter
Rodrigues, JF. (1994). Variational methods in the stefan problem. In: Visintin, A. (eds) Phase Transitions and Hysteresis. Lecture Notes in Mathematics, vol 1584. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073397
Download citation
DOI: https://doi.org/10.1007/BFb0073397
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58386-8
Online ISBN: 978-3-540-48678-7
eBook Packages: Springer Book Archive
