We consider optimal control problems for the bidomain equations of cardiacelectrophysiology together with two-variable ionic models, e.g. theRogers–McCulloch model. After ensuring the existence of global minimizers, we provide arigorous proof for the system of first-order necessary optimality conditions. The proof isbased on a stability estimate for the primal equations and an existence theorem for weaksolutions of the adjoint system.

In this paper we investigate the equivalence of the sequentialweak lower semicontinuity of the total energy functional and the quasiconvexity of thestored energy function of the nonlinear micropolar elasticity. Based on techniques of Acerbi and Fusco [Arch. Rational Mech. Anal.86 (1984) 125–145] we extend the result from Tambača and Velčić [ESAIM: COCV (2008) DOI: 10.1051/cocv:2008065] for energies thatsatisfy the growth of order p≥ 1. This result is the mainstep towards the general existence theorem for the nonlinear micropolarelasticity.

In this paper, we first prove an existence theorem for the integrodifferential equation (*)where f,k,x are functions with values in a Banach space E and the integral is taken in the sense of Henstock–Kurzweil–Pettis. In the second part of the paper we show that the set S of all solutions of the problem (*) is compact and connected in (C(Id,E),ω), where .

In this paper we give an existence theorem for the equilibrium problem for nonlinear micropolar elastic body. We consider the problem in its minimization formulation and apply the direct methods of the calculus of variations. As the main step towards the existence theorem, under some conditions, we prove the equivalence of the sequential weak lower semicontinuity of the total energy and the quasiconvexity, in some variables, of the stored energy function.

The class field theory for the fraction field of a two-dimensional complete normal local ring with finite residue field is established by S. Saito. In this paper, we investigate the index of the norm group in the K2-idele class group for a finite Abelian extension of such fields and deduce that the existence theorem does not hold for almost fields in this case.