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One-dimensional Maxwell's equations are considered in a ferromagnetic body surrounded by vacuum. Existence and uniqueness of solution for the resulting system of P.D.E.s with hysteresis on the whole real line is proved under suitable constitutive hypotheses.

We prove regularity results for minimizers of functionals F(u, Ω) := ∫Ω f(x, u, Du) dx in the class K := {u ∈ W1,p(x)(Ω, R) : u ≥ ψ}, where ψ : Ω → R is a fixed function and f is quasiconvex and fulfills a growth condition of the type L−1|z|p(x) ≤ f(x, ξ, z) ≤ L(1 + |z|p(x)), with growth exponent p : Ω → (1, ∞).

In this paper we deal with a class of parabolic partial differential equations containing a continuous hysteresis operator. We get an existence result by means of a technique based on an implicit time discretization scheme and we also analyse the dependence of the solution on the data. This model equation appears in the context of magnetohydrodynamics.

A partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero.

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The aim of this thesis is to present some results concerning some new classes of P.D.E.s containing a continuous hysteresis operator. Even if in the most frequent applications the operators involved in our treatment will be also rate independent, our results turn to be valid for the more general class of memory operators. Throughout the text, we focus our attention on the well-posedness of our model problems, dealing, when possible, with several kind of boundary conditions; in the last chapter, we present instead a result of asymptotic behaviour.

We consider an evolutionary P.D.E. motivated by models for electromagnetic processes in ferromagnetic materials. Magnetic hysteresis is represented by means of a hysteresis operator. Under suitable assumptions, an existence and uniqueness theorem is obtained, together with the Lipschitz continuous dependence on the data and some further regularity results. The discussion of the behaviour of the solution in dependence on physical parameters of the problem is also outlined.

We prove regularity results for minimizers of the integral functional f(x,u,Du)dx under non-standard growth condotions of p(x)-type, i.e. L-1|x| p(x) ≤ f(x,s,z) ≤ L (1+|x| p(x)) under sharp assumptions on the continuuus function p(x)>1