Lisa Santos

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 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.

Remarks on the two and three membranes problem

We consider the problem of finding the equilibrium position of two or three membranes constrained not to pass through each other. For general linear second order elliptic operators with measurable coefficients we prove the Lewy-Stampacchia type inequalities and we establish sufficient conditions on the external forces to obtain the stability of the coincidence sets of the membranes, in analogy with the obstacle problem.

Free Boundary Problems

This book gathers a collection of refereed articles containing original results reporting the recent contributions of the lectures and communications presented at the Free Boundary Problems Conference that took place at the University of Coimbra, Portugal, from June 7 to 12, 2005 (FBP2005). They deal with the mathematics of a broad class of models and problems involving nonlinear partial differential equations arising in physics, engineering, biology and finance. Among the main topics, the talks considered free boundary problems in biomedicine, in porous media, in thermodynamic modeling, in fluid mechanics, in image processing, in financial mathematics or in computations for inter-scale problems. The mathematical analysis and fine properties of solutions and interfaces in free boundary problems have been an active subject in the last three decades and their mathematical understanding continues to be an important interdisciplinary tool for the scientific applications, on one hand, and an intrinsic aspect of the current development of several important mathematical disciplines.

Stationary Quasivariational Inequalities with Gradient Constraint and Nonhomogeneous Boundary Conditions

We study existence of solution of stationary quasivariational inequalities with gradient constraint and nonhomogeneous boundary condition of Neumann or Dirichlet type. Through two different approaches, one making use of a fixed point theorem and the other using a process of regularization and penalization, we obtain different sufficient conditions for the existence of solution.

The N -membranes Problem with Neumann Type 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 the conditions for stability in measure of the interior contact zones of the membranes.

Variational Limit of Compressible to Incompressible Fluid

We consider the flow with (or without) wake of a stationnary, irrotational, compressible fluid, with non-constant density, in a channel (or in the whole plane), with given velocity at infinity and at the wake.

Solutions for linear conservation laws with gradient constraint

We consider variational inequality solutions with prescribed gradient constraints for first order linear boundary value problems. For operators with coefficients only in

Some free boundary problems in theorical glaciology

L^2

A Diffusion Problem with Gradient Constraint

, we show the existence and uniqueness of the solution by using a combination of parabolic regularization with a penalization in the nonlinear diffusion coefficient. We also prove the continuous dependence of the solution with respect to the data, as well as, in a coercive case, the asymptotic stabilization as time