This work is concerned with the numerical computation of null controls for the heat equation. The goal is to compute an approximation of controls that drives the solution from a prescribed initial state at t=0 to zero at t=T. In spite of the diffusion of the heat equation, recent developments indicate that this issue is difficult and still largely open. Most of the existing literature, concerned with controls of minimal L2-norm, make use of dual convex arguments and introduce backward adjoint system. In practice, the null control problem is then reduced to the minimization of a dual conjugate function with respect to the final condition of the adjoint state. As a consequence of the highly regularizing property of the heat kernel, this final condition – which may be seen as the Lagrange multiplier for the null controllability condition – does not belong to L2, but to a much larger space than can hardly be approximated by finite (discrete) dimensional basis. This phenomenon, unavoidable whatever be the numerical approximation used, strongly deteriorates the efficiency of minimization algorithms. In this work, we do not use duality arguments and in particular do not introduce any backward heat equation. For the boundary case, the approach consists first in introducing a class of functions satisfying a priori the boundary conditions in space and time, in particular the null controllability condition at time T, and then finding among this class one element satisfying the heat equation. This second step is done by minimizing a convex functional among the admissible corrector functions of the heat equation. The inner case is performed in a similar way. We present the (variational) approach, discuss the main features of it and then describe some numerical experiments highlighting the interest of the method. The method holds in any dimension but, for the sake of simplicity, we provide details in the one-space dimensional case.

We present a two-parameter family $(G_{m,k})_{m, k \in \mathbb{N}_{\geq 2}}$, of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of Gm,k is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erdős–Rényi random graph. As well as being of independent interest, our groups would, if our conjecture is true, provide a large family of counterexamples to the conjecture of Iranmanesh and Jafarzadeh that the commuting graph of a finite group, if connected, must have a bounded diameter. Simulations of our model yielded explicit examples of groups whose commuting graphs have all diameters from 2 up to 10.

We present the first macroscopical model for charge transport in compound semiconductors to make use of analytic ellipsoidal approximations for the energy dispersion relationships in the neighbours of the lowest minima of the conduction bands. The model considers the main scattering mechanisms charges undergo in polar semiconductors, that is the acoustic, polar optical, intervalley non-polar optical phonon interactions and the ionized impurity scattering. Simulations are shown for the cases of bulk 4H and 6H-SiC.

We study the local and global well-posedness of a full system of magnetohydrodynamic equations. The system is a coupling of the incompressible Navier—Stokes equations with the Maxwell equations through the Lorentz force and Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale-invariant spaces classically used for Navier—Stokes. These solutions are global if the initial data are small enough. Our results not only simplify and unify the proofs for the space dimensions 2 and 3, but also refine those in [8]. The main simplification comes from an a prioriLt2 (Lx∞) estimate for solutions of the forced Navier—Stokes equations.

In this work we study the multiplicity results for a class of critical elliptic systems related to the Brézis–Nirenberg problem with the Neumann boundary condition on a ball. Our approach relies on a minimization argument for an auxiliary problem with a mixed boundary condition and on suitable estimates of the critical level for the system case.

We study the radial symmetry of large solutions of the semilinear elliptic problem Δu + ∇h · ∇u = f (∣x∣, u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of growth of the solution at infinity.

This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region Ω ⊂ ℝN. The inequalities bound (semi-)norms of the boundary trace by certain norms of the function, its gradient on the region and by two specific constants κρ and κΩ associated with the domain and a weight function, respectively. These inequalities are sharp in that there exist functions for which equality holds. Explicit inequalities in some special cases when the region is a ball, or the region between two balls, are evaluated.

The aim of this paper is to show that the p-local homotopy type of the gauge group of a principal bundle over an even-dimensional sphere is completely determined by the divisibility of the classifying map by p. In particular, for gauge groups of principal SU(n)-bundles over S2d for 2 ≤ d ≤ p − 1 and n ≤ 2p − 1, we give a concrete classification of their p-local homotopy types.