The initial temperature of a heat conductor with non-compact boundary is zero and its boundary temperature is kept equal to one at each time. The conductor contains a stationary isothermic surface, that is, a time-invariant spatial level surface of the temperature. We prove that a kind of Liouville theorem: the unbounded boundary of the conductor, if it satisfies certain sufficient global assumptions, must be either a hyperplane or the union of two parallel hyperplanes. We present two different proofs (based on different assumptions): one is based on a short-time asymptotic formula for the heat content of a ball touching the boundary from inside and on a Bernstein-type theorem; the other one relies on a sliding method due to Berestycky, Caffarelli and Nirenberg.
Stationary isothermic surfaces for unbounded domains / R. MAGNANINI; S. SAKAGUCHI. - In: INDIANA UNIVERSITY MATHEMATICS JOURNAL. - ISSN 0022-2518. - STAMPA. - 56:(2007), pp. 2723-2738.
Stationary isothermic surfaces for unbounded domains
MAGNANINI, ROLANDO;
2007
Abstract
The initial temperature of a heat conductor with non-compact boundary is zero and its boundary temperature is kept equal to one at each time. The conductor contains a stationary isothermic surface, that is, a time-invariant spatial level surface of the temperature. We prove that a kind of Liouville theorem: the unbounded boundary of the conductor, if it satisfies certain sufficient global assumptions, must be either a hyperplane or the union of two parallel hyperplanes. We present two different proofs (based on different assumptions): one is based on a short-time asymptotic formula for the heat content of a ball touching the boundary from inside and on a Bernstein-type theorem; the other one relies on a sliding method due to Berestycky, Caffarelli and Nirenberg.File | Dimensione | Formato | |
---|---|---|---|
iumj-56-2007.pdf
Accesso chiuso
Descrizione: ReprintIUMJ
Tipologia:
Pdf editoriale (Version of record)
Licenza:
Tutti i diritti riservati
Dimensione
249.88 kB
Formato
Adobe PDF
|
249.88 kB | Adobe PDF | Richiedi una copia |
I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.