Sunto
Viene risolto il problema di Cauchy Dirichlet relativo all'operatore parabolico degenere ∂u/∂t−∂/∂xi(aij(x, t) ∂u/∂xj), in opportune ipotesi di integrabilità per gli autovalori di aij(x, t). Vengono inoltre forniti controesempi circa l'impossibilità di risultati di regolarità per le soluzioni deboli mostrando in tal modo che operatori parabolici degeneri hanno un comportamento radicalmente differente da quello dei corrispondenti operatori ellittici degeneri.
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References
R. Coifman -C. Fefferman,Weighted norm inequalities for maximal functions and singular integrals, Studia Math., T.LI (1974), pp. 241–250.
E.Fabes - C.Kenig - R.Serapioni,The local regularity of solutions of degenerate elliptic equations, Comm. Partial Diff. Equations,7 (1) (1982).
A. V. Ivanov,Smoothness of generalized solutions of degenerate parabolic equations of second order, Proc. Steklov Inst. Math.,116 (1971), pp. 52–67.
A. V. Ivanov,The boundary-value problem for linear parabolic equations of the divergence type with measurable coefficients, J. Soviet Math.,9 (1978), pp. 651–680.
A. V. Ivanov,Properties of solutions of linear and quasilinear second order equations with measurable coefficients which are neither strictly nor uniformly parabolic, J. Soviet Math.,10 (1978), pp. 29–43.
S.Kruzhkov - I.Kolodii,A priori estimates and Harnack's inequality for generalized solutions of degenerate quasi linear second order parabolic equations, Soviet Math. Dokl., Vol.13, N∘3 (1972).
L. J.Lions,Equations différentielles opérationnelles et problèmes aux limites, Springer Verlag, 1961.
O. Ladyzenskaja -V. Solonnikov -N. Ural'ceva,Linear and quasilinear equations of parabolic type, Amer. Math. Soc., Providence, 1968.
B. Muckenhoupt,Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc.,165 (1972), pp. 207–226.
M. K. V. Murthy -G. Stampacchia,Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl., (4)20 (1968), pp. 1–122.
M. K. V. Murthy -G. Stampacchia,Errata Corrige, ibidem,90 (1971), pp. 413–414.
B. Muckenhoupt -R. Wheeden,Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc.,198 (1974), pp. 361–274.
F. Nicolosi,Soluzioni deboli dei problemi al eontorno per operatori parabolici che possono degenerare, Ann. Mat. Pura Appl., (4)185 (1980), pp. 135–155.
F.Nicolosi,Sulla limitatezza delle soluzioni deboli dei problemi al contorno per operatori parabolici degeneri, to appear in Rendiconti Circ. Mat. Palermo.
G.Stampacchia,Equations elliptiques du second ordre à coefficients discontinus, Montreal, 1966.
E.Stein,Singular integrals and differentiability properties of functions, Princeton University Press, 1970.
F. Treves,Basic linear partial differential equations, Academic Press, New York, 1975.
N. Trudinger,On the regularity of generalized solutions of linear, non-uniformly elliptic equations, Arch. Rat. Mech. Anal.,48 (1971), pp. 51–62.
N. Trudinger,Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Super. Pisa,87 (1973), pp. 265–308.
N. Trudinger,Generalized solutions of quasilinear differential inequalities. I. Elliptic operators, Bull. Amer. Math. Soc.,77 (1971), pp. 576–579.
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Both the authors were supported in part by a grant of the italian C.N.R.
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Chiarenza, F., Serapioni, R. Degenerate parabolic equations and Harnack inequality. Annali di Matematica pura ed applicata 137, 139–162 (1984). https://doi.org/10.1007/BF01789392
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DOI: https://doi.org/10.1007/BF01789392