José Francisco Rodrigues

An elliptic quasi-variational inequality with gradient constraints and some of its applications

We consider a class of quasi-variational inequalities for certain second-order elliptic operators, where the set of admissible functions is required to satisfy an implicit gradient bound which depends on the solutions itself. We give sufficient conditions for the existence of a solution, and we apply our results to stationary problems arising in superconductivity, in thermoplasticity, and in electrostatics with implicit ionization threshold. Copyright

Remarks on the Reynolds problem of elastohydrodynamic lubrication

The mathematical model of the flow of a viscous lubricant between elastic bearings leads to the study of a highly non-linear and non-local elliptic variational inequality. We discuss the existence of a solution by using an a priori L ∞ -estimate. This method allows us to solve a large class of problems, including those arising from the linear Hertzian theory, and yields new existence results for the cases of a pressure-dependent viscosity or the inclusion of a load constraint. For small data the uniqueness of the solution holds, and we show that in the cylindrical journal bearing problem with small eccentricity ratio, the free boundary is given by two disjoint differentiable arcs close to the free boundary of the first-order approximate solution.

The Stefan Problem Revisited

A large class of mathematical models for phase change problems in thermo-diffusion phenomena belong to the family of the so-called Stefan problems, after the Austrian mathematical-physicist Joseph Stefan, who, exactly one century ago, published a series of papers on some of these problems [St]. However, the first work on this type of problems seems to be due to Lame and Clapeyron [LC] and since then they have interested a large number of mathematicians.

A NONLINEAR PARABOLIC SYSTEM ARISING IN THERMOMECHANICS AND IN THERMOMAGNETISM

We consider a system of two parabolic equations modeling the thermo-convection of a Newtonian fluid, with temperature dependent viscosity of energy dissipation, as well as the thermal effects of the eddy currents, induced by a slowly varying magnetic field, in cylinders with arbitrary cross-section. We show the existence of a weak solution of the corresponding initial-boundary value problem and, under additional assumptions, we consider the question of the uniqueness and regularity of the solution.

The N-membranes problem for quasilinear degenerate systems

We study the regularity of the solution of the variational inequality for the problem of N -membranes in equilibrium with a degenerate operator of p-Laplacian type, 1 < p < 1, for which we obtain the corresponding Lewy‐Stampacchia inequalities. By considering the problem as a system coupled through the characteristic functions of the sets where at least two membranes are in contact, we analyze the stability of the coincidence sets.

A steady-state Boussinesq-Stefan problem with continuous extraction

SummaryOne establishes an existence result for the weak solution to a steady-state strongly coupled system between a nonlinear two phases heat equation with convection and the Navier-Stokes equation in the liquid phase. The two phases Rayleigh-Bénard problem is included as the particular case corresponding to a zero extraction velocity.

A class of stationary nonlinear Maxwell systems

We study a new class of electromagnetostatic problems in the variational framework of the subspace of W1,p(Ω) of vector functions with zero divergence and zero normal trace, for , in smooth, bounded and simply connected domains Ω of ℝ3. We prove a Poincare–Friedrichs type inequality and we obtain the existence of steady-state solutions for an electromagnetic induction heating problem and for a quasi-variational inequality modelling a critical state generalized problem for type-II superconductors.

On the stationary Boussinesq-Stefan problem with constitutive power-laws

Abstract We discuss the existence of weak solutions to a steady-state coupled system between a two-phase Stefan problem, with convection and non-Fourier heat diffusion, and an elliptic variational inequality traducing the non-Newtonian flow only in the liquid phase. In the Stefan problem for the p -Laplacian equation the main restriction comes from the requirement that the liquid zone is at least an open subset, a fact that leads us to search for a continuous temperature field. Through the heat convection coupling term, this depends on the q -integrability of the velocity gradient and the imbedding theorems of Sobolev. We show that the appropriate condition for the continuity to hold, combining these two powers, is pq > n . This remarkably simple condition, together with q > 3n (n + 2) , that assures the compactness of the convection term, is sufficient to obtain weak solvability results for the interesting space dimension cases n = 2 and n = 3.

On the two obstacles problem in Orlicz–Sobolev spaces and applications

We prove the Lewy–Stampacchia inequalities for the two obstacles problem in abstract form for T-monotone operators. As a consequence for a general class of quasi-linear elliptic operators of Ladyzhenskaya–Uraltseva type, including p(x)-Laplacian type operators, we derive new results of C 1,α regularity for the solution. We also apply those inequalities to obtain new results to the N-membranes problem and the regularity and monotonicity properties to obtain the existence of a solution to a quasi-variational problem in (generalized) Orlicz–Sobolev spaces.

Some remarks on the homogenization of the dam problem

The homogenization of the dam problem is discussed in the general weak formulation by means of the technique of correctors. In the two special cases of horizontal layers and of the isotropic coefficients with separated variables the problem is solved. However, in general, the unsaturated region is not stable for the homogenization, as a counterexample by Alt shows.