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On a p-curl system arising in electromagnetism

We prove existence of solution of a $p$-curl type evolutionary system arising in electromagnetism with a power nonlinearity of order $p$, $1<p<\infty$, assuming natural tangential boundary conditions. We consider also the asymptotic behaviour in the power obtaining, when $p$ tends to infinity, a variational inequality with a curl constraint. We also discuss the existence, uniqueness and continuous dependence o...

Quasivariational solutions for first order quasilinear equations with gradient ...

We prove the existence of solutions for an evolution quasi-variational inequality with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regu...

On a constrained reaction-diffusion system related to a multiphase problem

We solve and characterize the Lagrange multipliers of a reaction- -diffusion system in the Gibbs simplex of $\R^{N+1}$ by considering strong solutions of a system of parabolic variational inequalities in $\R^N$. Exploring properties of the two obstacles evolution problem, we obtain and approximate a $N$-system involving the characteristic functions of the saturated and/or degenerated phases in the nonlinear re...

The nonlinear N-membranes evolution problem

The parabolic N-membranes problem for the p-Laplacian and the complete order constraint on the components of the solution is studied in what concerns the approximation, the regularity and the stability of the variational solutions. We extend to the evolutionary case the characterization of the Lagrange multipliers associated with the ordering constraint in terms of the characteristic functions of the coincidenc...

A class os stationary nonlinear Maxwell systems

We study a new class of electromagnetostatic problems in the variational framework of the subspace of $W^{1,p}(\Omega)^3$ of vector functions with zero divergence and zero normal trace, for $p>6/5$, in smooth, bounded and simply connected domains $\Omega$ of $\mathbb R^3$. We prove a Poincaré-Friedrichs type inequality and we obtain the existence of steady-state solutions for an electromagnetic induction heati...

The nonlinear N-membranes evolution problem

The parabolic N-membranes problem for the p-Laplacian and the complete order constraint on the components of the solution is studied in what concerns the approximation, the regularity and the stability of the variational solutions. We extend to the evolutionary case the characterization of the Lagrange multipliers associated with the ordering constraint in terms of the characteristic functions of the coincidenc...

A unilateral multiphase problem with Neumann type boundary condition

On non-Newtonian incompressible fluids with phase transitions

A modified model for a binary fluid is analysed mathematically. The governing equations of the motion consists of a Cahn-Hilliard equation coupled with a system describing a class of non-Newtonian incompressible fluid with p-structure. The existence of weak solutions for the evolution problems is shown for the space dimension d=2 with p≥ 2 and for d=3 with p≥ 11/5. The existence of measure-valued solutions is o...

The obstacle problem for nonlinear elliptic equations with variable growth and ...

The aim of this paper is twofold: to prove, for L1-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy–Stampacchia inequalities to the general framework of L1

The N-membranes problem with neumann boundary condition

We consider the problem of finding the equilibrium position of N membranes constrained not to pass through each other, under prescribed volumic forces and boundary tensions. This model corresponds to solve variationally a N-system for linear second order elliptic equations with sequential constraints. We obtain interior and boundary Lewy-Stampacchia type inequalities for the respective solution and we establish...