Gaussian lower bound for the Neumann Green function of a general parabolic operator

Mourad Choulli (1) , Laurent Kayser (1)

1 IECL - Institut Élie Cartan de Lorraine

Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a single-layer potential, we establish gaussian two-sided bounds for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for the gaussian two-sided bounds for the fundamental solution. We also indicate how our method can be adapted to get a gaussian lower bound for the Neumann heat kernel of a compact Riemannian manifold with boundary having non negative Ricci curvature.

Mots clés: Parabolic operator; fundamental solution; Gaussian lower bound; parametrix; Neumann Green function; heat kernel.

Domaines: Equations aux dérivées partielles [math.AP]

Mourad Choulli, Laurent Kayser. Gaussian lower bound for the Neumann Green function of a general parabolic operator. Positivity, 2015, 19 (3), pp.625-646. ⟨10.1007/s11117-014-0319-z⟩. ⟨hal-00773492v4⟩