A shape-optimization technique for the capillary surface problem.
A sharp isoperimetric inequality in the plane
We show that among all the convex bounded domain in having an assigned Fraenkel asymmetry index, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We also show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domain.
A sharpening of a theorem of Bouligand. With an application to harmonic maps.
A simple proof in Monge-Kantorovich duality theory
A simple proof is given of a Monge-Kantorovich duality theorem for a lower bounded lower semicontinuous cost function on the product of two completely regular spaces. The proof uses only the Hahn-Banach theorem and some properties of Radon measures, and allows the case of a bounded continuous cost function on a product of completely regular spaces to be treated directly, without the need to consider intermediate cases. Duality for a semicontinuous cost function is then deduced via the use of an...
A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity
The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2the functional is I ϵ ( u ) = 1 2 ∫ Ω ϵ -1 1 − Du 2 2 + ϵ D 2 u 2 d z whereubelongs to the subset of functions in W02,2(Ω) whose gradient (in the sense of trace) satisfiesDu(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl....
A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity
The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functional is where u belongs to the subset of functions in whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal ...
A simple proof of the minimax-theorem
A simple proof of the Rademacher theorem
A simple solution to the finite-horizon LQ problem with zero terminal state
This short paper deals with the classical finite-horizon linear-quadratic regulator problem with the terminal state constrained to be zero, for both continuous and discrete-time systems. Closed-form expressions for the optimal state and costate trajectories of the Hamiltonian system, as well as the corresponding control law, are derived through the solutions of two infinite- horizon LQ problems, thus avoiding the use of the Riccati differential equation. The computation of the optimal value of the...
A simple trolley-like model in the presence of a nonlinear friction and a bounded fuel expenditure
We consider a problem of maximization of the distance traveled by a material point in the presence of a nonlinear friction under a bounded thrust and fuel expenditure. Using the maximum principle we obtain the form of optimal control and establish conditions under which it contains a singular subarc. This problem seems to be the simplest one having a mechanical sense in which singular subarcs appear in a nontrivial way.
A singular control problem with an expected and a pathwise ergodic performance criterion.
A singular perturbation problem in exact controllability of the Maxwell system
This paper studies the exact controllability of the Maxwell system in a bounded domain, controlled by a current flowing tangentially in the boundary of the region, as well as the exact controllability the same problem but perturbed by a dissipative term multiplied by a small parameter in the boundary condition. This boundary perturbation improves the regularity of the problem and is therefore a singular perturbation of the original problem. The purpose of the paper is to examine the connection,...
A Singular Perturbation Problem in Exact Controllability of the Maxwell System
This paper studies the exact controllability of the Maxwell system in a bounded domain, controlled by a current flowing tangentially in the boundary of the region, as well as the exact controllability the same problem but perturbed by a dissipative term multiplied by a small parameter in the boundary condition. This boundary perturbation improves the regularity of the problem and is therefore a singular perturbation of the original problem. The purpose of the paper is to examine the connection, for...
A Solution of the 'Converse Canonical Problem' for Multiple Integrals in the Calculus of Variations (Short Communication)
A Solution of the 'Converse Canonical Problem' for Multiple Integrals in the Calculus of Variations
A stability theorem in nonlinear bilevel programming.
In this short paper, we are concerned with the stability of nonlinear bilevel programs. A stability problem is proven and an example is given to illustrate this theorem.
A stochastic impulse control problem with quadratic growth Hamiltonian and the corresponding quasi variational inequality.
A stochastic inventory model with stock dependent demand items.
A strong minimax property of nondegenerate minimal submanifolds.
A strongly nonlinear problem arising in glaciology
The computation of glacier movements leads to a system of nonlinear partial differential equations. The existence and uniqueness of a weak solution is established by using the calculus of variations. A discretization by the finite element method is done. The solution of the discrete problem is proved to be convergent to the exact solution. A first simple numerical algorithm is proposed and its convergence numerically studied.