Rita Ferreira
I am a Research Scientist in the Computer, Electrical and Mathematical Sciences & Engineering Division (CEMSE) of King Abdullah University of Science and Technology (KAUST). I am a member of Prof. Diogo Gomes's Group Nonlinear PDEs & Viscosity Solutions Research Group @ KAUST.
Research
My research interests lie in the areas of Partial Differential Equations, Continuum Mechanics, Calculus of Variations, and Homogenization. My research activities have been focused on asymptotic analysis using variational methods of dimension reduction and homogenization problems. Currently, I am also interested in the mathematical study of problems in imaging processing and in meanfield games.
Publications

R. Ferreira, I. Fonseca, and M.L. Mascarenhas
A chromaticitybrightness model for color images denoising in a Meyer's "u + v'' framework. To appear in Calculus of Variations and PDEs, 2017.[Preprint]A variational model for imaging segmentation and denoising color images is proposed. The model combines Meyer's "$u+v$" decomposition with a chromaticitybrightness framework and is expressed by a minimization of energy integral functionals depending on a small parameter $\epsilon >0$. The asymptotic behavior as $\mathrm{\epsilon \; \to}{0}^{+}$ is characterized, and convergence of infima, almost minimizers, and energies are established. In particular, an integral representation of the lower semicontinuous envelope, with respect to the ${L}^{1}$norm, of functionals with linear growth and defined for maps taking values on a certain compact manifold is provided. This study escapes the realm of previous results since the underlying manifold has boundary, and the integrand and its recession function fail to satisfy hypotheses commonly assumed in the literature. The main tools are $\Gamma $convergence and relaxation techniques.TBA  N. Almulla, R. Ferreira, and D.
Gomes
Two numerical approaches to stationary meanfield games. To appear in Dynamic Games and Applications, 2016.[Preprint] [Article]Here, we consider numerical methods for stationary meanfield games (MFG) and investigate two classes of algorithms. The first one is a gradientflow method based on the variational characterization of certain MFG. The second one uses monotonicity properties of MFG. We illustrate our methods with various examples, including onedimensional periodic MFG, congestion problems, and higherdimensional models.  N. Almayouf, E. Bachini, A. Chapouto, R. Ferreira, D. Gomes, D. Jordão, D. E. Junior, A. Karagulyan, J. Monasterio,
L. Nurbekyan, G. Pagliar, M. Piccirilli, S. Pratapsi, M. Prazeres, J. Reis, A. Rodrigues, O. Romero,
M. Sargsyan, T. Seneci, C. Song, K. Terai, R. Tomisaki, H. VelascoPerez, V. Voskanyan, and X. Yang
Existence of positive solutions for an approximation of stationary meanfield games. Involve 10, no. 3, 473493 (2017).[Preprint] [Article] [BibTex]Here, we consider a regularized meanfield game model that features a loworder regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with highorder regularizations, the loworder regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.  R. Ferreira, M.L. Mascarenhas, and A. Piatnitski
Spectral analysis in thin tubes with axial heterogeneities. Port. Math. 72, no. 2, 247266 (2015).[Preprint] [Article] [BibTex]In this paper, we present the 3D1D asymptotic analysis of the Dirichlet spectral problem associated with an elliptic operator with axial periodic heterogeneities. We extend to the 3D1D case previous 3D2D results (see [10]) and we analyze the special case where the scale of thickness is much smaller than the scale of the heterogeneities and the planar coefficient has a unique global minimum in the periodic cell. These results are of great relevance in the comprehension of the wave propagation in nanowires showing axial heterogeneities (see [17]).
[10] R. Ferreira, M. L. Mascarenhas, and A. Piatnitski, Spectral analysis in a thin domain with periodically oscillating characteristics. ESAIM Control Optim. Calc. Var. 18, 2 (2012), 427451.
[17] L. J. Lauhon, M. S. Gudiksen, and C. M. Lieber, Semiconductor nanowire heterostructures. Philos. Trans. R. Soc. Lond. Ser. AMath. Phys. Eng. Sci. 362, 1819 (2004), 12471260.  R. Ferreira, C. Kreisbeck, and A.M. Ribeiro
Characterization of polynomials and higherorder Sobolev spaces in terms of functionals involving difference quotients. Nonlinear Anal. 112, 199214 (2015).[Preprint] [Article] [BibTex]The aim of this paper, which deals with a class of singular functionals involving difference quotients, is twofold: deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a $k$thorder Sobolev space. One of the main theorems is a new characterization of ${W}^{k,p}(\mathrm{\Omega})$, $k\in \mathbb{N}$ and $p\in (1,+\mathrm{\infty})$, for arbitrary open sets $\mathrm{\Omega}\subset {\mathbb{R}}^{n}$. In particular, we provide natural generalizations of the results regarding Sobolev spaces summarized in Brézis' overview article [Russ. Math. Surv. 57 (2002), pp. 693708] to the higherorder case, and extend the work by Borghol [Asymptotic Anal. 51 (2007), pp. 303318] to a more general setting.  R. Ferreira and D. Gomes
On the convergence of finite state meanfield games through Gammaconvergence. J. Math. Anal. Appl. 418, no. 1, 211230 (2014).[Preprint] [Article] [BibTex]In this study, we consider the longterm convergence (trend toward an equilibrium) of finite state meanfield games using $\Gamma $convergence. Our techniques are based on the observation that an important class of meanfield games can be viewed as the EulerLagrange equation of a suitable functional. Therefore, using a scaling argument, one can convert a longterm convergence problem into a $\Gamma $convergence problem. Our results generalize previous results related to longterm convergence for finite state problems.  R. Ferreira and I. Fonseca
Reiterated Homogenization in BV via Multiscale Convergence, SIAM J. Math. Anal. 44 , no. 3, 20532098 (2012).[Preprint] [Article] [BibTex]Multiplescale homogenization problems are treated in the space $BV$ of functions of bounded variation, using the notion of multiplescale convergence developed in [J. Convex Anal. 19 (2012), no. 2, pp. 403452] by the authors. In the case of one microscale, Amar's result [Asymptot. Anal. 16 (1998), pp. 6584] is recovered under more general conditions; for two or more microscales, new results are obtained.  R. Ferreira and I. Fonseca
Characterization of the Multiscale Limit Associated with Bounded Sequences in BV , J. Convex Anal. 19, no. 2, 403452 (2012).[Preprint] [Article] [BibTex]The notion of twoscale convergence for sequences of Radon measures with finite total variation is generalized to the case of multiple periodic length scales of oscillations. The main result concerns the characterization of $(n+1)$scale limit pairs $(u,U)$ of sequences $\{({u}_{\epsilon}{{\mathcal{L}}^{N}}_{\lfloor \Omega},{D{u}_{\epsilon}}_{\lfloor \Omega}){\}}_{\epsilon >0}$ $\subset \mathcal{M}(\Omega ;{\mathbb{R}}^{d})\times \mathcal{M}(\Omega ;{\mathbb{R}}^{d\times N})$ whenever $\{{u}_{\epsilon}{\}}_{\epsilon >0}$ is a bounded sequence in $BV(\Omega ;{\mathbb{R}}^{d})$ . This characterization is useful in the study of the asymptotic behavior of periodically oscillating functionals with linear growth, defined in the space $BV$ of functions of bounded variation and described by $n\in \mathbb{N}$ microscales, undertaken in another paper of the authors [SIAM J. Math. Anal. 44 (2012), no. 3, pp. 20532098].  R. Ferreira, M.L. Mascarenhas, and A. Piatnitski
Spectral Analysis in a Thin Domain with Periodically Oscillating Characteristics, ESAIM Control Optim. Calc. Var. 18, no. 2, 427451 (2012).[Preprint] [Article] [BibTex]The paper deals with a Dirichlet spectral problem for an elliptic operator with $\epsilon $periodic coefficients in a 3D bounded domain of small thickness $\delta $. We study the asymptotic behavior of the spectrum as $\epsilon $ and $\delta $ tend to zero. This asymptotic behavior depends crucially on whether $\epsilon $ and $\delta $ are of the same order $(\delta \approx \epsilon )$, or $\epsilon $ is much less than $\delta $ $(\delta ={\epsilon}^{\tau}$, $\tau <1$), or $\epsilon $ is much greater than $\delta $ ($\delta ={\epsilon}^{\tau}$, $\tau >1$). We consider all three cases.  R. Ferreira and M.L. Mascarenhas
Waves in a thin and periodically oscillating medium, C. R. Acad. Sci. Paris 346, no. 910, 579584 (2008).[Preprint] [Article] [BibTex]We study the asymptotic behavior of the spectrum of an elliptic operator with periodically oscillating coefficients, in a thin domain, with vanishing Dirichlet conditions. Two cases are treated: the case where the periodicity of the oscillations and the thickness of the domain have the same order of magnitude and the case where the oscillations have a frequency much greater than the thickness of the domain. A physical motivation can be to understand the behavior of the probability density associated to the wave function of a particle confined to a very thin domain, with periodically varying characteristics.
 R. Ferreira and D. Gomes
Existence of Weak Solutions to Stationary MeanField Games through Variational Inequalities. Submitted, 2016.[Preprint]Here, we consider stationary monotone meanfield games (MFGs) and study the existence of weak solutions. First, we introduce a regularized problem that preserves the monotonicity. Next, using variational inequality techniques, we prove the existence of solutions to the regularized problem. Then, using Minty's method, we establish the existence of solutions for the original MFG. Finally, we examine the properties of these weak solutions in several examples. Our methods provide a general framework to construct weak solutions to stationary MFGs with local, nonlocal, or congestion terms.  R. Ferreira, P. Hästö, and A. Ribeiro
Characterization of generalized Orlicz spaces. Submitted, 2016.[Preprint]The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized OrliczSobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.  D. Evangelista, R. Ferreira, D. Gomes, L. Nurbekyan, V. Voskanyan
Firstorder, stationary meanfield games with congestion . Submitted, 2017.[Preprint]Meanfield games (MFGs) are models for large populations of competing rational agents that seek to optimize a suitable functional. In the case of congestion, this functional takes into account the difficulty of moving in highdensity areas. Here, we study stationary MFGs with congestion with quadratic or powerlike Hamiltonians. First, using explicit examples, we illustrate two main difficulties: the lack of classical solutions and the existence of areas with vanishing density. Our main contribution is a new variational formulation for MFGs with congestion. This formulation was not previously known, and, thanks to it, we prove the existence and uniqueness of solutions. Finally, we consider applications to numerical methods.  R. Ferreira and E. Zappale
Bendingtorsion moments in thin multistructures in the context of nonlinear elasticity. Submitted, 2017.[Preprint]Here, we address a dimensionreduction problem in the context of nonlinear elasticity where the applied external surface forces induce bendingtorsion moments. The underlying body is a multistructure in ${\mathbb{R}}^{3}$ consisting of a thin tubeshaped domain placed upon a thin plateshaped domain. The problem involves two small parameters, the radius of the crosssection of the tubeshaped domain and the thickness of the plateshaped domain. We characterize the different limit models, including the limit junction condition, in the membranestring regime according to the ratio between these two parameters as they converge to zero.
 R. Ferreira
Spectral and Homogenization Problems. Ph.D. Thesis Dissertation, Carnegie Mellon University and Universidade Nova de Lisboa, Pittsburgh and Lisbon, 2011.  R. Ferreira
Dimension Reduction in Nonlinear Elasticity via GammaConvergence. Master Thesis Dissertation, Faculdade de Ciências da Universidade de Lisboa (College of Science of the University of Lisbon), Lisbon, 2006.