Degenerate parabolic equations and Harnack inequality

SuntoViene 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.

[1]  F. Rühs,et al.  J. L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites. IX + 292 S. Berlin/Göttingen/Heidelberg 1961. Springer-Verlag. Preis geb. 64,— , 1962 .

[2]  B. Muckenhoupt,et al.  Weighted norm inequalities for the Hardy maximal function , 1972 .

[3]  G. Stampacchia,et al.  Inverse Problem for a Curved Quantum Guide , 2012, Int. J. Math. Math. Sci..

[4]  N. Trudinger.  On the regularity of generalized solutions of linear, non-uniformly elliptic equations , 1971 .

[5]  Properties of solutions of linear and quasilinear second-order equations with measurable coefficients which are neither strictly nor uniformly parabolic , 1978 .

[6]  The boundary-value problem for linear parabolic equations of the divergence type with measurable coefficients , 1978 .

[7]  N. Trudinger.  Linear elliptic operators with measurable coe cients , 1973 .

[8]  Generalized solutions of quasilinear, differential inequalities. I. Elliptic operators , 1971 .

[9]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[10]  J. Lions.  Equations Differentielles Operationnelles , 1961 .

[11]  Francesco Nicolosi.  Soluzioni deboli dei problemi al contorno per operatori parabolici che possono degenerare , 1980 .

[12]  G. Stampacchia,et al.  Boundary value problems for some degenerate-elliptic operators , 1968 .

[13]  B. Muckenhoupt,et al.  Weighted norm inequalities for fractional integrals , 1974 .

[14]  M. Gurtin,et al.  On the semi-discrete Galerkin method for hyperbolic problems and its application to problems in elastodynamics , 1972 .

[15]  F. Trèves.  Basic Linear Partial Differential Equations , 1975 .

[16]  Carlos E. Kenig,et al.  The local regularity of solutions of degenerate elliptic equations , 1982 .

[17]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[18]  Ronald R. Coifman,et al.  Weighted norm inequalities for maximal functions and singular integrals , 1974 .