We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution. We exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnack inequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer Holder continuity, an elliptic Harnack inequality and a Liouville theorem.

PARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION / Ciani, S; Mosconi, S; Vespri, V. - In: JOURNAL D'ANALYSE MATHEMATIQUE. - ISSN 0021-7670. - STAMPA. - 149:(2023), pp. 611-642. [10.1007/s11854-022-0261-0]

PARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION

Vespri, V
2023

Abstract

We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution. We exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnack inequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer Holder continuity, an elliptic Harnack inequality and a Liouville theorem.
2023
149
611
642
Goal 4: Quality education
Ciani, S; Mosconi, S; Vespri, V
File in questo prodotto:
File Dimensione Formato  
s11854-022-0261-0.pdf

accesso aperto

Tipologia: Pdf editoriale (Version of record)
Licenza: Open Access
Dimensione 277.18 kB
Formato Adobe PDF
277.18 kB Adobe PDF

I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1347365
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 4
social impact