Papers by José Francisco Rodrigues
Mathematics in Engineering
In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $ \Ome... more In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $ \Omega\subset\mathbb{R}^d $ with time-dependent Dirichlet boundary condition for the temperature $ \vartheta = \vartheta(x, t) $, $ \vartheta = g $ on $ \Omega^c\times]0, T[$, and initial condition $ \eta_0 $ for the enthalpy $ \eta = \eta(x, t) $, given in $ \Omega\times]0, T[$ by \begin{document}$ \frac{\partial \eta}{\partial t} +\mathcal{L}_A^s \vartheta = f\quad\text{ with }\eta\in \beta(\vartheta), $\end{document} where $ \mathcal{L}_A^s $ is an anisotropic fractional operator defined in the distributional sense by \begin{document}$ \langle\mathcal{L}_A^su, v\rangle = \int_{\mathbb{R}^d}AD^su\cdot D^sv\, dx, $\end{document} $ \beta $ is a maximal monotone graph, $ A(x) $ is a symmetric, strictly elliptic and uniformly bounded matrix, and $ D^s $ is the distributional Riesz fractional gradient for $ 0 < s < 1 $. We show the existence of a unique weak solution with its correspondin...
Journal of Elliptic and Parabolic Equations
In this paper we prove the existence and uniqueness of the solution to the one and the two obstac... more In this paper we prove the existence and uniqueness of the solution to the one and the two obstacles problems associated with a linear elliptic operator, which is non coercive due to the presence of a convection term. We show that the operator is weakly T-monotone and, as a consequence, we establish the Lewy–Stampacchia dual estimates and we study the comparison and the continuous dependence of the solutions as the obstacles vary. As an application, we prove also the existence of solutions for a class of non coercive implicit obstacle problems.
We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the... more We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the type |D σ u| ≤ g, involving the distributional fractional Riesz gradient D σ , 0 < σ < 1, extending previous results on the existence of solutions and Lagrange multipliers of these nonlocal problems. We also prove their convergence as σ 1 towards their local counterparts with the gradient constraint |Du| ≤ g.
Boletim da Sociedade Portuguesa de Matemática, 2016
Nonlinear Analysis, 2022
We formulate and study two mathematical models of a thermoforming process involving a membrane an... more We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.
Visualization has always been an essential aid in the communication of mathematics. It is an impo... more Visualization has always been an essential aid in the communication of mathematics. It is an important way to concretize concepts, to develop abstraction skills, and to motivate learning, for example in topology and geometry, and in the application of numerical methods to simulations of the real world. Video has proven to be one of the most adequate ways to communicate visualization results, allowing to present in a rich cultural context a large quantity and diversity of information in a brief period of time. However, by itself, video has a limited capability to support learning. The structure and interaction introduced by hypervideo allow providing the user with greater control and autonomy, exploring links among the information conveyed by the video and complemented by other materials, augmenting its capabilities as a cognitive artifact. This paper develops these ideas, presenting The Story of Pi hypervideo as a case study.
Luso-Chinese Symposium on Nonlinear Evolution Equations and Their Applications : Macau, 7-9 October 1998
This book discusses recent trends and developments in the area of nonlinear evolution equations. ... more This book discusses recent trends and developments in the area of nonlinear evolution equations. It is a collection of invited lectures on the following topics: nonlinear parabolic equations (systems); nonlinear hyperbolic systems; free boundary problems; conservation laws and shock waves; travelling and solitary waves; regularity, stability and singularity, etc.
In a joint initiative of the Centro Internacional de Matemática (CIM) and the Instituto de Cienci... more In a joint initiative of the Centro Internacional de Matemática (CIM) and the Instituto de Ciencias Matemáticas (ICMAT) a simbolic mathematical celebration of the Periodic Table took place at the Academy of Sciences of Lisbon, the 21st November 2019. It consisted of four talks, by two mathematicians and two chemists: Some mathematical aspects of the periodic table, by José Francisco Rodrigues (CIM and FCiências/ULisboa), The power of systematisation. The importance of precision, by Manuel Yáñez and Otilia Mo (Universidad Autónoma de Madrid), The periodic table: Are atoms the bricks of molecules? by Adelino Galvão (ISTécnico/ULisboa) and Counting lattice points and atomic energies oscillations, by Antonio Córdoba (ICMAT and UAMadrid). They were streamed online and their record can be found at http://www.cim.pt/agenda/event/208. The UNESCO decided to celebrate the year 2019 as
In this work, we consider the nonlocal obstacle problem with a given obstacle ψ in a bounded Lips... more In this work, we consider the nonlocal obstacle problem with a given obstacle ψ in a bounded Lipschitz domain Ω in R, such that Kψ = {v ∈ H 0(Ω) : v ≥ ψ a.e. in Ω} 6= ∅, given by u ∈ Kψ : 〈Lau, v − u〉 ≥ 〈F, v − u〉 ∀v ∈ Kψ, for F in H−s(Ω), the dual space of the fractional Sobolev space H 0(Ω), 0 < s < 1. The nonlocal operator La : H 0(Ω)→ H−s(Ω) is defined with a measurable, bounded, strictly positive singular kernel a(x, y) : R × R → [0,∞), by the bilinear form 〈Lau, v〉 = P.V. ˆ
We consider some properties of the solutions of free boundary problems of obstacletype with two p... more We consider some properties of the solutions of free boundary problems of obstacletype with two phases for a class of heterogeneous quasilinear elliptic operators, including the p-Laplacian operator with 1 < p < ∞. Under a natural non-degeneracy assumption on the interface, when the level set of the change of phase has null Lebesgue measure, we prove a continuous dependence result for the characteristic functions of each phase and we establish sharp estimates on the variation of its Lebesgue measure with respect to the L1-variation of the data, in a rather general framework. For elliptic quasilinear equations which heterogeneities have appropriate integrable derivatives, we show that the characteristic functions of both phases are of bounded variation for general data with bounded variation. This extends recent results for the obstacle problem and is a first result on the regularity of the free boundary of the heterogeneous two phases problem, which is therefore an interface l...
Applied Mathematics & Optimization, 2021
Revista de Ciência Elementar, 2017
Este artigo é de acesso livre, distribuído sob licença Creative Commons com a designação CC-BY-NC... more Este artigo é de acesso livre, distribuído sob licença Creative Commons com a designação CC-BY-NC-SA 4.0, que permite
Applied Mathematics & Optimization, 2019
We extend classical results on variational inequalities with convex sets with gradient constraint... more We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the σ-gradient (0 < σ < 1). We establish continuous dependence results with respect to the data, including the threshold of the fractional σ-gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with fractional gradient constraint via compactness and via contraction arguments. Using the approximation of the solutions with a family of quasilinear penalisation problems we show the existence of generalised Lagrange multipliers for the σ-gradient constrained problem, extending previous results for the classical gradient case, i.e., with σ = 1.
Portugaliae Mathematica, 2019
We study the homogenization of obstacle problems in Orlicz-Sobolev spaces for a wide class of mon... more We study the homogenization of obstacle problems in Orlicz-Sobolev spaces for a wide class of monotone operators (possibly degenerate or singular) of the p(•)-Laplacian type. Our approach is based on the Lewy-Stampacchia inequalities, which then give access to a compactness argument. We also prove the convergence of the coincidence sets under non-degeneracy conditions.
Advances in Nonlinear Analysis, 2018
This paper considers a general framework for the study of the existence of quasi-variational and ... more This paper considers a general framework for the study of the existence of quasi-variational and variational solutions to a class of nonlinear evolution systems in convex sets of Banach spaces describing constraints on a linear combination of partial derivatives of the solutions. The quasi-linear operators are of monotone type, but are not required to be coercive for the existence of weak solutions, which is obtained by a double penalization/regularization for the approximation of the solutions. In the case of time-dependent convex sets that are independent of the solution, we show also the uniqueness and the continuous dependence of the strong solutions of the variational inequalities, extending previous results to a more general framework.
Nonlinear Evolution Equations and Their Applications
Nonlinear Evolution Equations and Their Applications, 1999
Revista de Ciência Elementar, 2017
Este artigo é de acesso livre, distribuído sob licença Creative Commons com a designação CC-BY-NC... more Este artigo é de acesso livre, distribuído sob licença Creative Commons com a designação CC-BY-NC-SA 4.0, que permite a utilização e a partilha para fins não comerciais, desde que citado o autor e a fonte original do artigo.
Quarterly of Applied Mathematics, 1983
A steady-state one-phase Stefan problem corresponding to the solidification process of an ingot o... more A steady-state one-phase Stefan problem corresponding to the solidification process of an ingot of pure metal by continuous casting with nonlinear lateral cooling is considered via the weak formulation introduced in [5] for the dam problem. Two existence results are obtained, for a general nonlinear flux and for a maximal monotone flux. Comparison results and the regularity of the free boundary are discussed. An uniqueness theorem is given for the monotone case.
Homogenization of the dam problem with layer structure
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Papers by José Francisco Rodrigues