No abstract

Models of phase transitions / Augusto Visintin. p.cm. --(Progress in nonlinear differential equations and their applications: v. 28)Includes bibliographical references and indes.

No abstract

According to the classical theory of WeiB, on a microscopic scale a ferromagnetic body is magnetically saturated (i.e. [MI =~r162 constant) and is composed by uniformly magnetized regions separated by thin transition layers. At equilibrium this corresponds to the minimization of the magnetic energy functionat 8mag(M ) under the above constraint; this problem has at least one solution M ofclass H 1, which in general is not unique. The evolution is governed by Landau-Lifshitz' equations dM -21MxHe-22Mx(MxHe) (21, ~2: constant; 22>0 ) t3t t3e==8(M) He---(e=ag: density of magnetie energy); dM these are coupled with Maxwell's equations. An existence result based on the energy estimate and some complementary properties are proved for the corresponding variational formulation. Finally the magnetostrictive effect is included.

Abstract. We define and characterize weak and strong two-scale convergence in L p , C 0 and other spaces via a transformation of variable, extending Nguetseng's definition. We derive several properties, including weak and strong two-scale compactness; in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions. We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey.Mathematics Subject Classification.