Damião Júnio Araújo
Associate Professor
Department of Mathematics
Universidade Federal da Paraíba · UFPB
araujo@mat.ufpb.br
Interests
I am currently based in João Pessoa - Paraíba - Brazil, in the Department of Mathematics at UFPB. I am conducting research into geometric and analytical techniques for partial differential equations and free boundary problems. Our aim is to provide precise regularity (smoothness) estimates of solutions to these problems. Regularity is crucial for numerical computations, and a lack of smoothness can describe how wild the behavior of models are. Of particular interest are my recent works on problems involving degenerate PDEs, in which the diffusion properties of governing operators degenerate along free boundaries - a priori unknown sets. One notable example is the phenomenon of ice melting into water, where the free boundary refers to the interface between water and ice, namely, the solid-liquid interface. Such mathematical scenarios are also related to jet and cavitational flows, gas and liquid flows in porous media, and financial mathematics, among others. Image: NASA/Kathryn Hansen
Published and Accepted papers
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Sharp regularity for singular obstacle problems.
(with R. Teymurazyan and V. Voskanyan)
Math. Ann., to appear.
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Improved boundary and global regularity for degenerate fully nonlinear elliptic equations.
(with B. Sirakov)
J. Math. Pures Appl. (9) 169 (2023), 138-154.
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Infinity Laplacian equations with singular absorptions.
(with Ginaldo Sá)
Calc. Var. 61, 132 (2022).
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An optimal Liouville theorem for the porous medium equation. (with R. Teymurazyan)
Arch. Math. 118 (2022), 427-433.
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On a two-phase free boundary problem ruled by the infinity Laplacian. (with E. Teixeira and J.M. Urbano)
Isr. J. Math. 245, 773–785 (2021).
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Higher regularity estimates for the porous medium equation near the heat equation.
Rev. Mat. Iberoam. 37 (2021), 1747-1760.
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Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian.
(with L. Mari
and L. Pessoa) J. Differential Equations, Volume 281, 25 April 2021, Pages 550-587.
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Optimal C^{1,\alpha} estimates for a class of elliptic quasilinear equations. (with L. Zhang)
Commun. Contemp. Math. 22 (2020), no. 5, 1950014, 19 pp.
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Sharp regularity for the inhomogeneous porous medium equation.
(with A. Maia and J.M. Urbano)
Journal d'Analyse Mathématique, J. Anal. Math. 140 (2020), 395-407.
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Towards the C^p'-regularity conjecture in higher dimensions. (with E. Teixeira and J.M. Urbano)
Int Math Res Notices, Volume 2018, Issue 20, 2018, P. 6481-6495.
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A proof of the C^p'-regularity conjecture in the plane. (with E. Teixeira and J.M. Urbano)
Adv. Math. 316, 541–553, (2017).
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Singularly pertubed equations of degenerate type. (with E. Teixeira and G. Ricarte)
Ann. Inst. H. Poincaré Anal. Non linéaire 34, no. 3, 655-678,(2017).
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Infinity Laplacian equation with strong absorptions. (with E. Teixeira and R. Leitão)
J. Funct. Analysis 270, pp.2249-2267, (2016).
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Geometric gradient estimates for solutions to degenerate elliptic equations. (with E. Teixeira and G. Ricarte)
Cal. of Var. and PDEs, Vol.53, Issue 3-4, pp 605-625, (2015).
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Geometric approach nonvariational singular elliptic equations. (with E. Teixeira)
Arc. for Rat. Mech. and Anal., Vol.209, Issue 3, pp 1019-1054, (2013).
Submitted papers
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Fully nonlinear dead-core systems.
(with R. Teymurazyan)
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Sharp regularity for a singular fully nonlinear parabolic free boundary problem.
(with G. Sá and J.M. Urbano)
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Regularity in diffusion models with gradient activation.
(with A. Sobral and E. Teixeira)
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On free boundary problems with varying singularities.
(with A. Sobral, E. Teixeira and J.M. Urbano)
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On fractional quasilinear equations with elliptic degeneracy.
(with D. dos Prazeres and E. Topp)
CV and others 
Teamwork
Aelson O. Sobral, Ph.D. student
Diego dos Santos, M.S. student
Renato de Andrade, Undergrad student
Felipe de Andrade, Undergrad student
Lucas Pontes, Undergrad student
Gustavo Gabriel Cunha, Highschool student
We warmly invite students of all levels and disciplines related to mathematics to reach out to us regarding research projects centered around introductory topics in the analysis of differential equations, functional analysis, and measure theory, whether you are an undergraduate, master's, or Ph.D. We strongly encourage girls to join our research group, as we are committed to promoting inclusivity and gender equality in scientific research.
Past students
Ginaldo S. Sá, Ph.D. student | 2023
Co-advised by José Miguel Urbano (KAUST - Saudi Arabia)
Thesis Publications: Calc. Var. Partial Differential Equations
Current Position: Post-Doctorate Fellow at the University of Central Florida
Harish Shrivastava, Post-Doctorate | 2019