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Polyhedral convex set optimization problems are the simplest optimization problems with set-valued objective function. Their role in set optimization is comparable to the role of linear programs in scalar optimization. Vector linear…
Convergence results are stated for the variational iteration method applied to solve an initial value problem for a system of ordinary differential equations.
We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a $\bS^1$-symmetry group.…
We prove a result on the convex dependence of solutions of ordinary differential equations on an ordered finite-dimensional real vector space with respect to the initial data.
We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of…
Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution…
We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to…
This paper deals with elliptic equations in the plane with degeneracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed.…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
Solutions of a variational inequality are found by giving conditions for the monotone convergence with respect to a cone of the Picard iteration corresponding to its natural map. One of these conditions is the isotonicity of the projection…
We carry on the approach used in [Sch] to provide a solution for the inverse problem of the calculus of variations for Maxwell equations in vacuum and we provide an abstract theory including all implicit differential equations that can be…
We introduce "chain by chain" method for constructing the constraint structure of a system possessing both first and second class constraints. We show that the whole constraints can be classified into completely irreducible first or second…
Given a collection of N asymptotically Euclidean ends with zero scalar curvature, we construct a Riemannian manifold with zero scalar curvature and one asymptotically Euclidean end, whose boundary has a neighborhood isometric to the…
We use ODEs and symmetry arguments to construct closed constant curvature space curves, first on cylinders, next on tori, at last with the Frenet-Serret equations.
Using a recent result of C. De Lellis and L. Sz\'{e}kelyhidi Jr. we show that, in the case of periodic boundary conditions and for dimension greater or equal 2, there exist infinitely many global weak solutions to the incompressible Euler…
This is the second part of a two parts work on the analysis of heat-type equations on manifolds with fibered boundary equipped with a $\Phi$-metric. This setting generalizes the asymptotically conical (scattering) spaces and includes…
In this paper, we consider an initial boundary value problem for Maxwell's equations. For this hyperbolic type problem, we derive guaranteed and computable upper bounds for the difference between the exact solution and any pair of vector…
We construct high-frequency initial data for the Einstein vacuum equations in dimension 3+1 by solving the constraint equations on $\mathbb{R}^3$. Our family of solutions $(\bar{g}_\lambda,K_\lambda)_{\lambda\in(0,1]}$ is defined through a…
We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. The analytical nature of these initial data…
In Part I, we construct a class of examples of initial velocities for which the unique solution to the Euler equations in the plane has an associated flow map that lies in no Holder space of positive exponent for any positive time. In Part…