On a long standing conjecture of E De Giorgi old and recent results

On a long - standing conjecture of E . De Giorgi : symmetry in 3 d for

On a Long-standing Conjecture of E. De Giorgi: Symmetry in 3D for General Nonlinearities and a Local Minimality Property

This paper studies a conjecture made by E. De Giorgi in 1978 concerning the onedimensional character (or symmetry) of bounded, monotone in one direction, solutions of semilinear elliptic equations ∆u = F (u) in all of Rn. We extend to all nonlinearities F ∈ C the symmetry result in dimension n = 3 previously established by the second and the third authors for a class of nonlinearities F which i...

On De Giorgi Conjecture in Dimension

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation ∆u+(1−u2)u = 0 in RN with ∂yN u > 0 must be such that its level sets {u = λ} are all hyperplanes, at least for dimension N ≤ 8. A counterexample for N ≥ 9 has long been believed to exist. Based on a minimal graph Γ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in RN , N ≥ 9, we prove t...

On De Giorgi Conjecture in Dimension N ≥ 9

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation ∆u+(1−u2)u = 0 in RN with ∂yN u > 0 must be such that its level sets {u = λ} are all hyperplanes, at least for dimension N ≤ 8. A counterexample for N ≥ 9 has long been believed to exist. Based on a minimal graph Γ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in RN , N ≥ 9, we prove t...

On Silverman's conjecture for a family of elliptic curves

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...