We consider, for any odd positive integer k, the degenerate Partial Differential Equationu_t = u_xx + x^k u_yand we prove a Harnack inequality which is expressed in terms of the integral curves of the vector fields that occur in the PDE. The novelty of our result is in that, as k>1, we cannot assume the existence of any Lie group in R^3 such that the vector fields are invariant. As an application of the Harnack inequality we prove a lower bond of the fundamental solution.
Harnack inequalities and Lifting Procedure for Evolution Hypoelliptic Equations / C., Cinti; Polidoro, Sergio. - STAMPA. - VII:(2008), pp. 93-105. (Intervento presentato al convegno Geometric Methods in PDE's: a conference on the occasion of the 65th birthday of Ermanno Lanconelli tenutosi a Bologna nel 27-30 maggio 2008).
Harnack inequalities and Lifting Procedure for Evolution Hypoelliptic Equations
POLIDORO, Sergio
2008
Abstract
We consider, for any odd positive integer k, the degenerate Partial Differential Equationu_t = u_xx + x^k u_yand we prove a Harnack inequality which is expressed in terms of the integral curves of the vector fields that occur in the PDE. The novelty of our result is in that, as k>1, we cannot assume the existence of any Lie group in R^3 such that the vector fields are invariant. As an application of the Harnack inequality we prove a lower bond of the fundamental solution.
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