Regularity of minimizers for three elliptic problems: minimal cones, harmonic maps, and semilinear equations

Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutions

We will present several results on the classification of stable solutions to some nonlinear elliptic equations. These results are a crucial step within the regularity theory of minimizers to such problems. We will mainly center our attention to three different (but connected) equations. Some techniques and ideas in the three settings are quite similar. The first one is the celebrated result of ...

Regularity of Minimizers of Semilinear Elliptic Problems up to Dimension Four

Abstract. We consider the class of semi-stable solutions to semilinear equations −∆u = f(u) in a bounded smooth domain Ω of R (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an priori L∞ bound which holds for every semi-stable solution and every nonlinearity f . This estimate leads to the boundedness o...

The Nehari Manifold for a Class of Indefinite Weight Semilinear Elliptic Equations

ar X iv : m at h / 06 04 63 5 v 1 [ m at h . A P ] 2 8 A pr 2 00 6 PARTIAL REGULARITY FOR HARMONIC MAPS , AND RELATED PROBLEMS

Via gauge theory, we give a new proof of partial regularity for harmonic maps in dimensions m ≥ 3 into arbitrary targets. This proof avoids the use of adapted frames and permits to consider targets of ”minimal” C regularity. The proof we present moreover extends to a large class of elliptic systems of quadratic growth.

Nonexistence of Nonconstant Global Minimizers with Limit at ∞ of Semilinear Elliptic Equations in All of Rn

We prove nonexistence of nonconstant global minimizers with limit at infinity of the semilinear elliptic equation −∆u = f(u) in the whole R , where f ∈ C(R) is a general nonlinearity and N ≥ 1 is any dimension. As a corollary of this result, we establish nonexistence of nonconstant bounded radial global minimizers of the previous equation.