### 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 Non-linear 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 mean-field games.

### Publications

__Papers published in scientific international journals with peer reviewing__

- R. Ferreira and D. Gomes

*Existence of Weak Solutions to Stationary Mean-Field Games through Variational Inequalities.*To appear in*SIAM J. Math. Anal.*, 2018.[Preprint]Here, we consider stationary monotone mean-field 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.TBA - D. Evangelista, R. Ferreira, D. Gomes, L. Nurbekyan, V. Voskanyan

*First-order, stationary mean-field games with congestion.*To appear in*Nonlinear Analysis*, 2018.[Preprint]Mean-field 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 high-density areas. Here, we study stationary MFGs with congestion with quadratic or power-like 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.TBA -
R. Ferreira, I. Fonseca, and M.L. Mascarenhas

*A chromaticity-brightness model for color images denoising in a Meyer's*Calc. Var. Partial Differential Equations 56, no. 5, Art. 140, 53 pp (2017).*"u + v''*framework.[Preprint]A variational model for imaging segmentation and denoising color images is proposed. The model combines Meyer's "$u+v$" decomposition with a chromaticity-brightness 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.@article {FeFoMa17,

AUTHOR = {Ferreira, Rita and Fonseca, Irene and Mascarenhas, M. Lu\'\i sa},

TITLE = {A chromaticity-brightness model for color images denoising in a {M}eyer's ``u + v'' framework},

JOURNAL = {Calc. Var. Partial Differential Equations},

FJOURNAL = {Calculus of Variations and Partial Differential Equations},

VOLUME = {56},

YEAR = {2017},

NUMBER = {5},

PAGES = {Art. 140, 53},

ISSN = {0944-2669},

MRCLASS = {94A08 (26B30 49J45)},

MRNUMBER = {3695373},

DOI = {10.1007/s00526-017-1223-8},

URL = {https://doi.org/10.1007/s00526-017-1223-8}, } - N. Almulla, R. Ferreira, and D.
Gomes

*Two numerical approaches to stationary mean-field games.*Dyn. Games Appl. 7, no. 4, 657--682 (2017).[Preprint] [Article]Here, we consider numerical methods for stationary mean-field games (MFG) and investigate two classes of algorithms. The first one is a gradient-flow 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 one-dimensional periodic MFG, congestion problems, and higher-dimensional models.TBA - 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. Velasco-Perez, V. Voskanyan, and X. Yang

*Existence of positive solutions for an approximation of stationary mean-field games.*Involve 10, no. 3, 473-493 (2017).[Preprint] [Article] [BibTex]Here, we consider a regularized mean-field game model that features a low-order 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 high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.TBA - R. Ferreira, M.L. Mascarenhas, and A. Piatnitski

*Spectral analysis in thin tubes with axial heterogeneities.*Port. Math. 72, no. 2, 247-266 (2015).[Preprint] [Article] [BibTex]In this paper, we present the 3D-1D asymptotic analysis of the Dirichlet spectral problem associated with an elliptic operator with axial periodic heterogeneities. We extend to the 3D-1D case previous 3D-2D 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), 427-451.

[17] L. J. Lauhon, M. S. Gudiksen, and C. M. Lieber, Semiconductor nanowire heterostructures. Philos. Trans. R. Soc. Lond. Ser. A-Math. Phys. Eng. Sci. 362, 1819 (2004), 1247-1260.TBA - R. Ferreira, C. Kreisbeck, and A.M. Ribeiro

*Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients.*Nonlinear Anal. 112, 199-214 (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 $k$th-order 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. 693--708] to the higher-order case, and extend the work by Borghol [Asymptotic Anal. 51 (2007), pp. 303--318] to a more general setting.TBA - R. Ferreira and D. Gomes

*On the convergence of finite state mean-field games through*J. Math. Anal. Appl. 418, no. 1, 211-230 (2014).Γ -convergence.[Preprint] [Article] [BibTex]In this study, we consider the long-term convergence (trend toward an equilibrium) of finite state mean-field games using $\Gamma $-convergence. Our techniques are based on the observation that an important class of mean-field games can be viewed as the Euler-Lagrange equation of a suitable functional. Therefore, using a scaling argument, one can convert a long-term convergence problem into a $\Gamma $-convergence problem. Our results generalize previous results related to long-term convergence for finite state problems.TBA - R. Ferreira and I. Fonseca

*Reiterated Homogenization in*. SIAM J. Math. Anal.*BV*via Multiscale Convergence**44**, no. 3, 2053-2098 (2012).[Preprint] [Article] [BibTex]Multiple-scale homogenization problems are treated in the space $BV$ of functions of bounded variation, using the notion of multiple-scale convergence developed in [J. Convex Anal. 19 (2012), no. 2, pp. 403--452] by the authors. In the case of one microscale, Amar's result [Asymptot. Anal. 16 (1998), pp. 65--84] is recovered under more general conditions; for two or more microscales, new results are obtained.TBA - R. Ferreira and I. Fonseca

*Characterization of the Multiscale Limit Associated with Bounded Sequences in*. J. Convex Anal.*BV***19**, no. 2, 403-452 (2012).[Preprint] [Article] [BibTex]The notion of two-scale 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. 2053--2098].TBA - 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, 427--451 (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.TBA - R. Ferreira and M.L. Mascarenhas

*Waves in a thin and periodically oscillating medium*. C. R. Acad. Sci. Paris**346**, no. 9-10, 579-584 (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.TBA

__Papers submitted to scientific international journals with peer reviewing__

- 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 Orlicz--Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces. - R. Ferreira and E. Zappale

*Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity. Submitted, 2017.*[Preprint]Here, we address a dimension-reduction problem in the context of nonlinear elasticity where the applied external surface forces induce bending-torsion moments. The underlying body is a multi-structure in ${\mathbb{R}}^{3}$ consisting of a thin tube-shaped domain placed upon a thin plate-shaped domain. The problem involves two small parameters, the radius of the cross-section of the tube-shaped domain and the thickness of the plate-shaped domain. We characterize the different limit models, including the limit junction condition, in the membrane-string regime according to the ratio between these two parameters as they converge to zero.

__Theses__

- 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 Gamma-Convergence.*Master Thesis Dissertation, Faculdade de Ciências da Universidade de Lisboa (College of Science of the University of Lisbon), Lisbon, 2006.