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We study a two-dimensional model for micromagnetics, which consists in an energy functional over S2-valued vector fields. Bounded-energy configurations tend to be planar, except in small regions which can be described as vortices (Bloch lines in physics). As the characteristic “exchange-length” tends to 0, they converge to planar divergence-free unit norm vector fields which jump along line singularities. We derive lower bounds for the energy, which are explicit functions of the jumps of the limit....
We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in -norm and -norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important...
We consider the operator on a complex Hilbert space, where is positive self-adjoint and is self-adjoint, and where, moreover, « is comparable to , », in a technical sense. Two applications are given.
We consider the lower semicontinuous functional of the form where satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar’s -convexity condition for the integrand extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex...
We consider the lower semicontinuous functional of the form
where u satisfies a given
conservation law defined by differential operator of degree one
with constant coefficients. We show that under certain constraints
the well known Murat and Tartar's Λ-convexity condition
for the integrand f extends to the new geometric conditions
satisfied on four dimensional symplexes. Similar conditions on
three dimensional symplexes were recently obtained by the second
author. New conditions apply...
In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence...
We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “Lipschitz-type” and center-“Lipschitz-type” instead of just “Lipschitz-type” conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.
In 1959, Nikaidô established a remarkable coincidence theorem in a compact Hausdorff topological space, to generalize and to give a unified treatment to the results of Gale regarding the existence of economic equilibrium and the theorems in game problems. The main purpose of the present paper is to deduce several generalized key results based on this very powerful result, together with some KKM property. Indeed, we shall simplify and reformulate a few coincidence theorems on acyclic multifunctions,...
We compute numerically the minimizers of the Dirichlet energyamong maps from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned...
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