Volume 3, Issue 1

This article studies the problem of minimizing ∫ Ω F ( Du ( x )) dx over the functions u ∈ W 1,1 (Ω) that assume given boundary values φ on Γ ≔ ∂Ω. The Lagrangian F and the domain Ω are assumed convex but not necessarily strictly conxex. When φ is continuous and F superlinear, we prove the existence of a minimum which is continuous on the closure of Ω. We also consider the case when F is not superlinear.

We show that an initial and boundary value problem related to the parabolic p -Laplace equation is stable with respect to p if the complement of the cylindrical domain satisfies a uniform capacity density condition. This condition is essentially optimal for our stability results.

We prove Liouville type theorems for biharmonic maps from complete manifolds and from Euclidean balls.

We continue to study ℍ-regular graphs, a class of intrinsic regular hypersurfaces in the Heisenberg group endowed with a left-invariant metric d ∞ equivalent to its Carnot–Carathéodory metric. Here we investigate their relationships with suitable weak solutions of non-linear first-order PDEs. As a consequence this implies some of their geometric properties: a uniqueness result for ℍ-regular graphs of prescribed horizontal normal as well as their (Euclidean) regularity as long as there is regularity on the horizontal normal.

We present a new proof for a pointwise upper bound in terms of Wolff potential for -superharmonic functions, which are the pointwise defined solutions to elliptic equations involving nonnegative measure data.