Optimal existence results for the 2d elastic contact problem with Coulomb friction

In this article, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini-Coulomb problem) is unraveled and optimal existence results are proved for the most general bidimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray-Lions operators, so that a result of Br\'ezis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray-Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the bidimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini-Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient

A sensitivity analysis of a class of semi-coercive variational inequalities using recession tools

International audienceUsing the recession analysis we study necessary and sufficient conditions for the existence and the stability of a finite semi-coercive variational inequality with respect to data perturbation. Some applications of the abstract results in mechanics and in electronic circuits involving devices like ideal diode and practical diode are discussed

Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis

We derive a thermodynamically consistent general continuum-mechanical model describing mutually coupled martensitic and ferro/paramagnetic phase transformations in electrically-conductive magnetostrictive materials such as NiMnGa. We use small-strain and eddy-current approximations, yet large velocities and electric current injected through the boundary are allowed. Fully nonlinear coupling of magneto-mechanical and thermal effects is considered. The existence of energy-preserving weak solutions is proved by showing convergence of time-discrete approximations constructed by a carefully designed semi-implicit regularized scheme. The research that led to the present paper was partially supported by a grant of the group GNFM of INdA

Strong solutions to nonlocal 2D Cahn--Hilliard--Navier--Stokes systems with nonconstant viscosity, degenerate mobility and singular potential

We consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We suppose that the viscosity depends smoothly on the order parameter as well as the mobility. Moreover, we assume that the mobility is degenerate at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with no-slip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak-strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world

Homogeneización y diferenciación de formas de ecuaciones elípticas cuasilineales

Tesis de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático y Matemática Aplicada, leída el 19-12-2017Esta tesis se ha divido en dos partes de tamaños desiguales. La primera parte es la componentecentral del trabajo del candidato. Se encarga de la optimización de reactores químicosde lecho fijo, y el estudio de su efectividad, como se expondrá en los siguientes párrafos. Lasegunda parte es el resultado de la visita del candidato al Prof. Häim Brezis en el InstitutoTecnológico de Israel (Technion) en Haifa, Israel. Se entra en una pregunta concreta sobrebases óptimas en L2, que es de importancia en Tratamiento de Imágenes, y que fue formuladopor el Prof. Brezis.La primera parte de la tesis, que estudio reactores químicos, se ha dividido en 4 capítulos.Estudia un modelo establecido que tiene aplicaciones directas en Ingeniería Química, y lanoción de efectividad. Una de las mayores dificultades con la que nos enfrentamos es elhecho que, por las aplicaciones en Ingeniería Química, estamos interesados en reacciones deorden menor que uni (de tipo raíz).El primer capítulo se centra en la modelización: obtener un modelo macroscópico (homogéneo)a partir de un comportamiento microscópico prescrito. A este método se le conocecomo homogeneización. La idea es considerar partículas periódicamente repetidas, de formafija G0, a una distancia ε, y que han sido reescaladas por un factor aε . La expresión habitualde este factor es aε = C0εα, donde α ≥ 1 y C0 es una constante positiva. El objetivo esestudiar los diferentes comportamientos cuando ε →0, y ya no se consideran las partículas.Primero, los casos de partículas grandes y partículas pequeños se tratan de formas distintas.Este segundo, que ha sido el central en esta tesis, se divide en subcrítico, crítico y supercrítico.En términos generales, existe un valor α∗ tal que los comportamientos de los casos α = 1(partículas grandes), 1 α∗ (partículas supercríticos) son significativamente distintos...This thesis has been divided into two parts of different proportions. The first part is the mainwork of the candidate. It deals with the optimization of chemical reactors, and the study ofthe effectiveness, as it will explained in the next paragraphs. The second part is the result ofthe visit of the candidate to Prof. Häim Brezis at the Israel Institute of Technology (Technion)in Haifa, Israel. It deals with a particular question about optimal basis in L2 of relevance inImage Proccesing, which was raised by Prof. Brezis.The first part of the thesis, which deals with chemical reactors, has been divided intofour chapters. It studies well-established models which have direct applications in ChemicalEngineering, and the notion of “effectiveness of a chemical reactor”. One of the maindifficulties we faced is the fact that, due to the Chemical Engineering applications, we wereinterested in dealing with root-type nonlinearities. The first chapter focuses on modeling: obtaining a macroscopic (homogeneous) modelfrom a prescribed microscopic behaviour. This method is known as homogenization. Theidea is to consider periodically repeated particles of a fixed shape G0, at a distance ε, whichhave been rescaled by a factor aε . This factor is usually of the form aε =C0εα, where α ≥ 1and C0 is a positive constant. The aim is to study the different behaviours as ε →0, whenthe particles are no longer considered. It was known that depending of this factor there areusually different behaviours as ε →0. First, the case of big particles and small particles aretreated differently. The latter, which have been the main focus of this chapter, are dividedinto subcritical, critical and supercritical holes. Roughly speaking, there is a critical valueα∗ such that the behaviours α = 1 (big particles), 1 α∗ (supercritical particles) are significantly different...Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEunpu

Cahn-Hilliard-Brinkman models for tumour growth: Modelling, analysis and optimal control

Phase field models recently gained a lot of interest in the context of tumour growth models. In this work we study several diffuse interface models for tumour growth in a bounded domain with sufficiently smooth boundary. The basic model consists of a Cahn–Hilliard type equation for the concentration of tumour cells coupled to a convection-reaction-diffusion-type equation for an unknown species acting as a nutrient and a Brinkman-type equation for the velocity. The system is equipped with Neumann boundary conditions for the phase field and the chemical potential, a Robin-type boundary condition for the nutrient and a “no-friction” boundary condition for the velocity which allows us to consider solution dependent source terms. We derive the model from basic thermodynamic principles, conservation laws for mass and momentum and constitutive assumptions. Using the method of formal matched asymptotics, we relate our diffuse interface model with free boundary problems for tumour growth that have been studied earlier. For the basic model, we show the existence of weak solutions under suitable assumptions on the source terms and the potential by using a Galerkin method, energy estimates and compactness arguments. If the velocity satisfies a no-slip boundary condition and is divergence free, we can establish the existence of weak solutions for degenerate mobilities and singular potentials. From a modelling point of view, it seems to be more appropriate to describe the nutrient evolution by a so-called quasi-static equation of reaction-diffusion type. For this model we establish existence of both weak and strong solutions for regular potentials and a continuous dependence result yields the uniqueness of weak solutions and thus the model is well-posed. These results build the basis to study an optimal control problem where the control acts as a cytotoxic drug. Moreover, we rigorously prove the zero viscosity limit in two and three space dimensions which allows us to relate the Cahn–Hilliard–Brinkman model with Cahn–Hilliard–Darcy models which have been studied earlier. Finally, we also analyse the model with quasi-static nutrients and classical singular potentials like the logarithmic and double-obstacle potential which enforce the phase field to stay in the physical relevant range. Under suitable assumptions on the source terms, we can establish the existence of weak solutions for these kinds of potentials