We consider the Brascamp–Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse th...

We consider the parabolic regularization of a scalar conservation law in which the Laplacian operator has been replaced by a fractional power of itself. Using a splitting method, we prove the existence of a solution to the problem and, thanks to the Banach fixed point theorem, its uniqueness and regularity. We also show that, as the regularization vanishes, the solution converge to the entropy ...

We consider the null controllability problem for wave equation, and analyse a stabilized finite element method formulated on global, unstructured spacetime mesh. prove error estimates approximate control given by computational method. The proofs are based regularity properties of Hilbert Uniqueness Method, together with stability numerical scheme. Numerical experiments illustrate results.

In 1928 Emanuel Sperner presented a simple. Cohen, On the Sperner lemma, J. Theory 2 1967.Abstract. This paper presents elementary combinatorial proofs of Sperners. Advanced topics like degree theory or homology. Before beginning, it.ple combinatorial result known as Sperners lemma. As motivation, we examine a special case of Sperners lemma. Vick, Homology Theory.extremal two-part Sperner famil...

We consider a quantum two-level system perturbed by classical noise. The noise is implemented as a stationary diffusion process in the off-diagonal matrix elements of the Hamiltonian, representing a transverse magnetic field. We determine the invariant measure of the system and prove its uniqueness. In the case of Ornstein–Uhlenbeck noise, we determine the speed of convergence to the invariant ...

Here ν is the viscosity, p is the pressure, and f1, f2 are the components of an external forcing which may be time-dependent. As our setting is periodic, the functions u1, u2, ∇p, f1, and f2 are all periodic in x. For simplicity, we take the period to be one. The first existence and uniqueness theorems for weak solutions of (1) were proven by Leray ([Ler34]) in whole plane R. Later these result...

In the recent paper [2], the authors obtained new proofs on the existence and uniqueness of the solution of the Volterra linear equation. Applying their results, in this paper we express the exact and approximate solution of the equation in the field of Mikusi´nski operators, F, which corresponds to an integro–differential equation.

The aim of this paper is to give two uniqueness results for the Dirichlet problem associated to the constant mean curvature equation. We study constant mean curvature graphs over strips of R. The proofs are based on height estimates and the study of the asymptotic behaviour of solutions to the Dirichlet problem. 2000 Mathematics Subject Classification. 53A10.