Sunto
Le equazioni paraboliche singolari (1.1)nella introduzione, si presentano come modelli di una classe generale di fenomeni di diffusione con cambio di fase. Le soluzioni deboli sono trovate in senso globale come classi di equivalenza in certi spazi di Sobolev. In questo lavoro si dimostra che le soluzioni deboli ammettono delle rappresentanti continue nell'interno del dominio di definizione. Si danno anche delle condizioni di continuità fino alla frontiera.
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References
H.Brèzis,On some degenerate non-linear parabolic equation, Proceedings AMS, Vol. I, XVII (1968).
L.Caffarelli - L. C.Evans,Continuity of the temperature in the two-phase Stefan problem, (to appear).
L.Caffarelli - A.Friedman,Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc. (to appear).
L.Caffarelli - A.Friedman,Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J.,29, no. 3 (1980).
J. R.Cannon - E.Di Benedetto,On the existence of solution of boundary value problems in Fast Chemical reactions, Bollettino U.M.I., (5),15-B (1978).
J. R.Cannon - E.Di Benedetto,On the existence of weak solutions to an n-dimensional Stefan problem with non-linear boundary conditions, SIAM J. on Math. Analysis,11, no. 4 (1980).
J. R.Cannon - E.Di Benedetto - G. H.Knightly,The Stefan problem with convection: the non-steady state case, (in preparation).
J. R.Cannon - C. D.Hill,On the movement of a chemical reaction interface, Indiana Math. J.,20 (1970).
J. R.Cannon - A.Fasano,Boundary value multidimensional problems in fast chemical reactions, Archive for Rat. Mech. and Analysis,53, no. 1 (1973).
J. R.Cannon - D. B.Henry - D. B.Kotlow,Classical solutions of the one-dimensional two-phase Stefan problem, Annali di Mat. Pura e Appl., (IV),107 (1976).
E.De Giorgi,Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sc. Fis. Mat. Nat., (3),3 (1957).
E.Di Benedetto,Regularity properties of the solution of an n-dimensional two-phase Stefan problem, Bollettino U.M.I. (to appear).
E.Di Benedetto - R. E.Showalter,Implicit degenerate evolution equations and applications, SIAM J. on Math. Anal., Vol.12, no. 5 (1981).
A. Fasano -M. Primicerio,Partially saturated porous media, J. Inst. Maths. Applics.,23 (1979), pp. 503–517.
A.Fasano - M.Primicerio - S.Kamin,Regularity of weak solutions of one-dimensional two-phase Stefan problem, Annali di Mat. Pura e Appl., (IV),115 (1977).
A.Friedman,The Stefan problem in several space variables, Trans. Amer. Mat. Soc.,132 (1968).
A. Friedman,Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J. (1964).
O. A.Ladyzenskaja - V. A.Solonnikov - N. N.Ural'ceva,Linear and Quasi-linear Equations of Parabolic Type, Amer. Math. Soc. Transl. Math. Mono,23, Providence, R.I. (1968).
O. A. Ladyzenskaja -N. N. Ural'ceva,Linear and Quasi-linear Elliptic Equations, Academic Press, New York (1968).
J. L. Lions,Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod Gauthier-Villars, Paris (1969).
J. L. Lions -E. Magenes,Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin, London and New York (1972).
S. N.Kruzkov,A priori estimates for generalized solutions of second-order elliptic and parabolic equations, Soviet Math. (1963).
S. N.Kružkov,A priori estimates of solutions of linear parabolic equations and of boundary value problems for a certain class of quasi-linear parabolic equations, Doklady Akad. Nauk,150 (1963).
S. N.Kružkov,Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications, Matematicheskie Zametki,6, no. 1 (1969).
S. N.Kružkov - S. M.Sukorjanskii,Boundary value problems of systems of equations of two-phase porous flow type: statement of the problems, questions of solvability, justification of approximate methods, Math. USSR Sbornik,33, no. 1 (1977).
J. Moser,A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math.,13 (1960), pp. 457–468.
O. A.Oleinik - A. S.Kalashnikov -Chzhou Yui-Lin,The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izvestija Akademii Nauk SSSR Ser. Mat.,22 (1958).
L.Rubinstein,The Stefan Problem, A.M.S. Translations of Mathematical Monographs, Vol.27, Providence, R.I. (1971).
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Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grant No. MCS78-09525 A01.
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Di Benedetto, E. Continuity of weak solutions to certain singular parabolic equations. Annali di Matematica pura ed applicata 130, 131–176 (1982). https://doi.org/10.1007/BF01761493
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DOI: https://doi.org/10.1007/BF01761493