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

By Alberto Farina, Berardino Sciunzi, Enrico Valdinoci. 2008

167 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/7d7cb0f619aaf6e080b188fa3ad6d748c528ad45

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Citations in the corpus

1D symmetry for solutions of semilinear and quasilinear elliptic equations (2011, methodology): “Without loss of generality, we suppose that the latter situation happens. Hence, from Lemma 4.1 of [ FSV07 ],”
Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces (2012, methodology): “2 in [FSV1] it follows that u (and therefore u) is one dimensional.”
Flattening Results for Elliptic PDEs in Unbounded Domains with Applications to Overdetermined Problems (2010, methodology): “25) which does not use the results of [FSV08]: when (C2) holds, we have ∆u∞ ≤ ∆u∞ + f(u∞) = 0 in R (2), so u∞ is constant by Liouville Theorem (see, for instance, Theorem 3.”
De Giorgi type results for elliptic systems (2012, methodology): “Later on, Farina, Sciunzi, and Valdinoci [10] used it to reprove the De Giorgi’s conjecture in dimension two, and Cabré used it (see Proposition 2.”
Monotonicity and one-dimensional symmetry for solutions of −Δpu = f(u) in half-spaces (2011, methodology): “1 we will use some recent results in [19] (see also [20]) to show that the solution u must have one-dimensional symmetry, that is, u must depend only on the variable xN .”
On Phase-Separation Models: Asymptotics and Qualitative Properties (2013, unknown): “We use the method in [31].”
Liouville results for m-Laplace equations of Lane―Emden―Fowler type (2009, background): “Further details about monotonicity and stabilities are given in Section 7 of [16].”
Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result (2008, methodology): “In particular, [SZ98a, SZ98b] obtained a similar inequality for stable solutions of the standard Allen-Cahn equation, and symmetry results for possibly singular or degenerate models have been obtained in [Far02, FSV07 ].”