Andrew Acker

Remarks on quenching

On considere le probleme aux limites parabolique non lineaire suivant: u t −Δ u =f(u), t>0, x∈Ω, u=0, t>0, n∈∂Ω, u(0, x)=u 0 (x)≥0, x∈Ω. On demontre un phenomene de trempe pour des non linearites non convexes dans un espace de dimension arbitraire

On the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions

We prove the existence of convex classical solutions for a general multidimensional, multilayer free-boundary problem. The geometric context of this problem is a nested family of closed, convex surfaces. Except for the innermost and outermost surfaces, which are given, these surfaces are interpreted as unknown layer-interfaces, where the layers are the bounded annular domains between them. Each unknown interface is characterized by a quite general nonlinear equation, called a joining condition, which relates the first derivatives (along the interface) of the capacitary potentials in the two adjoining layers, as well as the spatial variables. A well-known special case of this problem involves several stationary, immiscible, two-dimensional flows of ideal fluid, related along their interfaces by Bernoulli’s law.

The multi-layer free boundary problem for the p-Laplacian in convex domains

The main result of this paper concerns existence of classical solutions to the multi-layer Bernoulli free boundary problem with nonlinear joining conditions and the p-Laplacian as governing operato ...

A trial-free-boundary method for computing Batchelor flows

The solutions of Batchelor flows in bounded domains are computed by a nonstandard trial-free-boundary method.

Convex Configurations for Solutions to Semilinear Elliptic Problems in Convex Rings

For a given convex ring and an L 1 function f:Ω × ℝ → ℝ+ we show, under suitable assumptions on f, that there exists a solution (in the weak sense) to with {x ∈ Ω: u(x) > s} ∪ Ω1 convex, for all s ∈ (0, M).

One-Layer Free Boundary Problems with Two Free Boundaries

We study the uniqueness and successive approximation of solutions of a class of two-dimensional steady-state fluid problems involving infinite periodic flows between two periodic free boundaries, each characterized by a flow-speed condition related to Bernoulli’s law.

Isoperimetric inequalities for the energy of uniform-vorticity flows in channels of uniform width

We prove several isoperimetric inequalities involving the kinetic energy of constant-vorticity flows through channels of uniform width.