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One-dimensional symmetry for solutions of quasilinear equations in R 2

Alberto Farina — 2003

Bollettino dell'Unione Matematica Italiana

In this paper we consider two-dimensional quasilinear equations of the form div a u u + f u = 0 and study the properties of the solutions u with bounded and non-vanishing gradient. Under a weak assumption involving the growth of the argument of u (notice that arg u is a well-defined real function since u > 0 on R 2 ) we prove that u is one-dimensional, i.e., u = u ν x for some unit vector ν . As a consequence of our result we obtain that any solution u having one positive derivative is one-dimensional. This result provides a proof of...

Spazi BV e di Nikolskii e applicazioni al problema di Stefan

Alberto Farina — 1995

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Questa Nota è dedicata a mettere in evidenza alcune proprietà degli spazi B V Ω = N 1 Ω delle funzioni a variazione limitata e degli spazi di Nikolskii N 1 λ Ω = N λ Ω ed N λ , 0 Ω , ( λ 0 , 1 ), che non mi risulta siano già state esposte nella forma generale qui enunciata, quali la non separabilità, l'essere il duale di uno spazio di Banach separabile, la convergenza e la compattezza debole * in L W * 0 , T ; N λ Ω e le loro applicazioni al classico problema di Stefan bifase.

Simmetria delle soluzioni di equazioni ellittiche semilineari in R N

Alberto Farina — 1999

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Nella prima parte di questa Nota si dimostrano dei risultati di simmetria unidimensionale e radiale per le soluzioni di Δ u + f u = 0 in R N . Questi risultati sono legati a due congetture (De Giorgi, 1978 e Gibbons, 1994) riguardanti la classificazione delle soluzioni dell’equazione Δ u + u 1 - u 2 = 0 in R N . Si dimostra, in particolare, la seguente generalizzazione della congettura di Gibbons: se N > 1 e se l’insieme degli zeri di u è limitato nella direzione ν , allora u x = u 0 ν x , ovvero, u è unidimensionale. Nella seconda parte si considerano...

Stable solutions of Δ u = f ( u ) in N

Louis DupaigneAlberto Farina — 2010

Journal of the European Mathematical Society

Several Liouville-type theorems are presented for stable solutions of the equation - Δ u = f ( u ) in N , where f > 0 is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.

Bernstein and De Giorgi type problems: new results via a geometric approach

Alberto FarinaBerardino SciunziEnrico Valdinoci — 2008

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form div a ( | u ( x ) | ) u ( x ) + f ( u ( x ) ) = 0 . Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in  2 and  3 and of the Bernstein problem on the flatness of minimal area graphs in  3 . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our...

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