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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.
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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
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