Advances in Analysis

5

Authors: Ricardo Estrada

We prove that the product p (x) ∇2mδ (x) of a homogeneous polynomial of degree j in n variables, p, and the iterated Laplacian of the Dirac delta function is of the form p (∇) q (∇) δ (x) for some polynomial q if and only if p (x) = |x| 2l pk (x) for some harmonic homogeneous polynomial of some degree k, j = 2l + k. We also show that the product p (x) ∇2mδ (x), where p is homogeneous of degree j, vanishes for all m > j if and only if p is a harmonic polynomial.

10

Authors: Attou A. Miloua

Let Ω be a region in R 2 and f be a positive C 1 function satisfying lim u→0+ f (u) = ∞. We consider the quasi-linear elliptic equations of the form div (a (u) ∇u) = a 0 (u) 2 |∇u| 2 + f (u) where a is a positive C 1 function. Motivated by the thin film equations, a solution u is said to be a point rupture solution if for some p ∈ Ω, u (p) = 0 and u (p) > 0 in Ω\ {p}. Our main result is a sufficient condition on a and f for the existence of radial point rupture solutions.

14

Authors: Veronika Chrastinová

In the common theory of the inverse problem, a system of differential equations is given and we ask whether this system is identical with the Lagrange system of an appropriate variational integral. In this article, only a small part of the Euler–Lagrange system may be prescribed in advance.The lack of information does not affect the results. The classical Helmholz solvability conditions and the Tonti resolving formula are adapted for this incomplete problem. Elementary and self–contained algorithmical approach is applied.