A Harnack inequality for a degenerate parabolic equation

ABSTRACT Since Moser’s fundamental work on Harnack estimates for second order parabolic partial differential equations with bounded and measurable coefficients ([M]) this kind of inequalities have become more and more important for a wide class of evolution equations. In the case of degenerate parabolic equations of p-laplacian type, Harnack estimates were known only in some particular instances and the theory was fragmented and incomplete. One of the main difficulties was in the information transport which characterizes Harnack inequalities. Actually, in the context of parabolic De Giorgi classes, naturally associated to second order parabolic partial differential equations with bounded and measurable coefficients, this transport is realized using proper estimates on the expansion of positivity of the solution. This method typically relies on the homogeneity of the classes and depends only on the data as we proved in [GV]. If we consider the equation $$u_t – Div(|Du|^{p-2}Du) = 0, p>2$$ the corresponding De Giorgi class does not show the homogeneity we had before and therefore new techniques are needed. In [DB] things are solved using a proper comparison with the Barenblatt fundamental solution. Unfortunately the method stops working as soon as an explicit expression for the fundamental solution is not at hand. However if we look more carefully at the method set forth in [DB], we quickly realize that we don’t need information on the value of u at each point, but just information on the measure of proper level sets of u; in fact De Giorgi Lemma transforms such measure theoretic estimates in point estimates. This is exactly the kind of approach we used in this paper to prove a Harnack inequality for the equation $$u_t – \sum_{i=1}^N $\frac{\partial}{\partial_{x_i}}(|D_iu|^{p-2}D_iu = 0, p>2$$ which was originally considered by Vishik in [V] and further studied by Lions in [L]. To get this result we used the Rayleigh quotient. This quantity was largely used to study the asymptotic behaviour of singular and degenerate parabolic evolution equations (see for example [DBKV], [MV], [SV]) and here we used in completely different situations. This new approach was the key point in [DBGV] to prove the Harnack estimates for general quasilinear degenerate operators. From Google Scholar database it results that this paper was quoted 9 times. REFERENCES: [DB] DiBenedetto,E.; Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Rat. Mech. Anal. 100 (1988), pp. 129-147. [DBGV] DiBenedetto, E.; Gianazza, U.; Vespri, V; Harnack estimates for quasi linear degenerate parabolic differential equations Acta Math 200 (2008), no. 2, 181-209. [DBKV] DiBenedetto, E., Kwong, Y. and Vespri, V., Local space-analiticity of Solutions of Certain Singular Parabolic Equations - Indiana Univ. Math. J. 40 (1991), 741–765. [GV] Gianazza, U.; Vespri, V.; Parabolic De Giorgi classes of order $p$ and the Harnack inequality. Calc. Var. Partial Differential Equations 26 (2006), no. 3, 379-399 [L] Lions, J. L., Quelques Methodes des Resolution des problemes aux limites non lineaires Dunod, Paris (1969). [MV] Manfredi, J. J. and Vespri, V., Large time behaviour of Solutions to a class of Doubly Nonlinear Parabolic Equations, Electronic J. Diff. Eq. 2 (1994), 1–17. [M] Moser, J., A Harnack Inequality for Parabolic Differential Equations, Comm. Pure Appl. Math. 17 (1964), 101–134. [SV] Savare', G. , Vespri, V., The Asymptotic Profile of Solutions of a Class of Doubly Nonlinear Parabolic Equations, Nonlinear Analysis, 22(12) (1994), 1553–1565. [V] Vishik, I. M., Sur la resolution des problemes aux limites pour des equations paraboliques quasi lineaires d’ordre quelconque, Mat. Sbornik, 59(101) (1962), 289–325.