Showing 1–20 of 20 results for author: Andrade, P

Calderón-Zygmund estimates for parabolic systems with $p$-growth and non-divergence data

Authors: Pêdra Andrade, Kristian Moring

Abstract: We obtain Calderón--Zygmund estimates for weak solutions to nonlinear parabolic systems with polynomial $p$-growth and the right-hand side in non-divergence form. Our approach does not require differentiability of the vector field with respect to the gradient variable and provides a unified treatment of both the singular and degenerate regimes. In particular, we establish local higher-integrabilit… ▽ More We obtain Calderón--Zygmund estimates for weak solutions to nonlinear parabolic systems with polynomial $p$-growth and the right-hand side in non-divergence form. Our approach does not require differentiability of the vector field with respect to the gradient variable and provides a unified treatment of both the singular and degenerate regimes. In particular, we establish local higher-integrability estimates for the gradient of the solution under appropriate dependency on the integrability of the right-hand side. △ Less

Submitted 20 August, 2026; originally announced August 2026.

Comments: 25 pages

MSC Class: 35K40; 35B65; 35K65; 35K67

Calderón-Zygmund estimates for parabolic $p$-Laplacian systems with non-divergence form right-hand sides

Authors: Pêdra Andrade, Verena Bögelein, Frank Duzaar, Kristian Moring

Abstract: We establish local Calderón-Zygmund type estimates for weak solutions to nonlinear parabolic systems with $p$-growth and VMO coefficients. In particular, we prove that if the right-hand side belongs locally to $L^{μs}$, where the exponent $μ$ depends explicitly on $p$, $N$, and a prescribed target exponent $s>p$, then the spatial gradient of the solution enjoys improved integrability… ▽ More We establish local Calderón-Zygmund type estimates for weak solutions to nonlinear parabolic systems with $p$-growth and VMO coefficients. In particular, we prove that if the right-hand side belongs locally to $L^{μs}$, where the exponent $μ$ depends explicitly on $p$, $N$, and a prescribed target exponent $s>p$, then the spatial gradient of the solution enjoys improved integrability $Du \in L^s_{\rm{loc}}$. The result provides a sharp transfer of integrability from the data to the gradient, consistent with the natural parabolic scaling, and recovers the optimal exponents in the linear case $p=2$. The proof combines intrinsic scaling techniques with a Calderón-Zygmund type iteration scheme. △ Less

Submitted 23 April, 2026; originally announced April 2026.

Boundary estimates for parabolic non-divergence equations in $C^1$ domains

Authors: Pêdra D. S. Andrade, Clara Torres-Latorre

Abstract: We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence form parabolic equations in parabolic $C^1$ domains, providing explicit moduli of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with $C^{1,\mathrm{Dini}}$ boundaries, while also recovering the known $C^{1-\varepsilon}$ regularity for parabolic Li… ▽ More We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence form parabolic equations in parabolic $C^1$ domains, providing explicit moduli of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with $C^{1,\mathrm{Dini}}$ boundaries, while also recovering the known $C^{1-\varepsilon}$ regularity for parabolic Lipschitz domains, unifying both regimes with a single proof. △ Less

Submitted 6 April, 2026; originally announced April 2026.

Comments: 29 pages

MSC Class: 35B65; 35K20

Optimal regularity for degenerate elliptic equations with Hamiltonian terms

Authors: Pêdra D. S. Andrade, Thialita M. Nascimento

Abstract: We establish optimal, quantitative Höder estimates for the gradient of solutions to a class of degenerate elliptic equations with Hamiltonian terms. The presence of such lower-order terms introduces additional challenges, particularly in regimes where the gradient is either very small or very large. Our approach adapts perturbative techniques to capture the interplay between the degeneracy rate an… ▽ More We establish optimal, quantitative Höder estimates for the gradient of solutions to a class of degenerate elliptic equations with Hamiltonian terms. The presence of such lower-order terms introduces additional challenges, particularly in regimes where the gradient is either very small or very large. Our approach adapts perturbative techniques to capture the interplay between the degeneracy rate and the Hamiltonian's growth. Our results naturally extend the regularity theory developed by Araujo-Ricarte-Teixeira (Calc. Var. 53:605-625, 2015) and Birindelli-Demengel (Nonlinear Differ. Equ. Appl. 23:41, 2016) to a more general setting. △ Less

Submitted 5 August, 2025; originally announced August 2025.

Comments: This paper was originally submited for publication on July 2024, and this is and updated version

Interior regularity estimates for fully nonlinear equations with arbitrary nonhomogeneous degeneracy laws

Authors: Pêdra D. S. Andrade, Thialita M. Nascimento

Abstract: In this paper, we study regularity estimates for a class of degenerate, fully nonlinear elliptic equations with arbitrary nonhomogeneous degeneracy laws. We establish that viscosity solutions are locally continuously differentiable under suitable conditions on the degeneracy laws. Our proof employs improvement of flatness techniques alongside an alternative recursive algorithm for renormalizing th… ▽ More In this paper, we study regularity estimates for a class of degenerate, fully nonlinear elliptic equations with arbitrary nonhomogeneous degeneracy laws. We establish that viscosity solutions are locally continuously differentiable under suitable conditions on the degeneracy laws. Our proof employs improvement of flatness techniques alongside an alternative recursive algorithm for renormalizing the approximating solutions, linking our model to the homogeneous, fully nonlinear, uniformly elliptic equation. △ Less

Submitted 6 January, 2025; originally announced January 2025.

Comments: 19 pages

MSC Class: 35B65; 35J70; 37J60

Spectral Stability of Periodic Traveling Wave Solutions for a Double Dispersion Equation

Authors: Fábio Natali, Thiago P. de Andrade

Abstract: In this paper, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method we find explict periodic waves and we also present a characterization for all positive and periodic solutions for the model using the monotonicity of the period map in terms of the energy levels. The monotonicity of the period map is also… ▽ More In this paper, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method we find explict periodic waves and we also present a characterization for all positive and periodic solutions for the model using the monotonicity of the period map in terms of the energy levels. The monotonicity of the period map is also useful to obtain the quantity and multiplicity of non-positive eigenvalues for the associated linearized operator and to do so, we use tools of the Floquet theory. Finally, we prove the spectral stability by analysing the difference between the number of negative eigenvalues of a convenient linear operator restricted to the space constituted by zero-mean periodic functions and the number of negative eigenvalues of the matrix formed by the tangent space associated to the low order conserved quantities of the evolution model. △ Less

Submitted 11 July, 2023; originally announced July 2023.

Comments: 18 pages and 2 pictures

MSC Class: 76B25; 35Q51; 35Q53

Two-phase free boundary problems for a class of fully nonlinear double-divergence systems

Authors: Pêdra D. S. Andrade, Julio C. Correa

Abstract: In this article, we study a class of fully nonlinear double-divergence systems with free boundaries associated with a minimization problem. The variational structure of Hessian-dependent functional plays a fundamental role in proving the existence of the minimizers and then the existence of the solutions for the system. In addition, we establish gains of the integrability for the double-divergence… ▽ More In this article, we study a class of fully nonlinear double-divergence systems with free boundaries associated with a minimization problem. The variational structure of Hessian-dependent functional plays a fundamental role in proving the existence of the minimizers and then the existence of the solutions for the system. In addition, we establish gains of the integrability for the double-divergence equation. Consequently, we improve the regularity for the fully nonlinear equation in Sobolev and Hölder spaces. △ Less

Submitted 19 August, 2025; v1 submitted 30 May, 2023; originally announced May 2023.

Comments: 18 pages, 1 figure

MSC Class: 35B65; 35J35; 35R35; 35A01

Spectral partition problems with volume and inclusion constraints

Authors: Pêdra D. S. Andrade, Ederson Moreira dos Santos, Makson S. Santos, Hugo Tavares

Abstract: In this paper, we discuss a class of spectral partition problems with a measure constraint, for partitions of a given bounded connected open set. We establish the existence of an optimal open partition, showing that the corresponding eigenfunctions are locally Lipschitz continuous, and obtain some qualitative properties for the partition. The proof uses an equivalent weak formulation that involves… ▽ More In this paper, we discuss a class of spectral partition problems with a measure constraint, for partitions of a given bounded connected open set. We establish the existence of an optimal open partition, showing that the corresponding eigenfunctions are locally Lipschitz continuous, and obtain some qualitative properties for the partition. The proof uses an equivalent weak formulation that involves a minimization problem of a penalized functional where the variables are functions rather than domains, suitable deformations, blowup techniques, and a monotonicity formula. △ Less

Submitted 21 June, 2023; v1 submitted 4 May, 2023; originally announced May 2023.

Comments: From v1 to v2 we have clarified, corrected and simplified some arguments in Section 3 (Continuity of Minimizers). We also added more references

MSC Class: 35B65; 49J30; 49J35

Sharp regularity estimates for a singular inhomogeneous (m, p)-Laplacian equation

Authors: Pêdra D. S. Andrade, João Vitor da Silva, Giane C. Rampasso, Makson S. Santos

Abstract: In this paper, we investigate a class of doubly nonlinear evolutions PDEs. We establish sharp regularity for the solutions in Hölder spaces. The proof is based on the geometric tangential method and intrinsic scaling technique. Our findings extend and recover the results in the context of the classical evolution PDEs with singular signature via a unified treatment in the slow, normal and fast diff… ▽ More In this paper, we investigate a class of doubly nonlinear evolutions PDEs. We establish sharp regularity for the solutions in Hölder spaces. The proof is based on the geometric tangential method and intrinsic scaling technique. Our findings extend and recover the results in the context of the classical evolution PDEs with singular signature via a unified treatment in the slow, normal and fast diffusion regimes. In addition, we provide some applications to certain nonlinear evolution models, which may have their own mathematical interest. △ Less

Submitted 4 May, 2023; originally announced May 2023.

Comments: 21 pages

MSC Class: 35B65; 35K55; 35K67; 35K92

Regularity estimates for fully nonlinear integro-differential equations with nonhomogeneous degeneracy

Authors: Pêdra D. S. Andrade, Disson S. dos Prazeres, Makson S. Santos

Abstract: We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class $C_{loc}^{1, α}$, for some constant $α\in (0, 1)$. In addition, under suitable conditions on $σ$, we prove regularity estimates in Hölder spaces for any viscosity solution. We also e… ▽ More We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class $C_{loc}^{1, α}$, for some constant $α\in (0, 1)$. In addition, under suitable conditions on $σ$, we prove regularity estimates in Hölder spaces for any viscosity solution. We also examine the singular setting and prove Hölder regularity estimates for the gradient of the solutions. △ Less

Submitted 1 February, 2023; originally announced February 2023.

MSC Class: 35B65; 35R11; 35R09; 35D40

$C^1$-regularity for degenerate diffusion equations

Authors: Pêdra Andrade, Daniel Pellegrino, Edgard A. Pimentel, Eduardo V. Teixeira

Abstract: We prove that any solution of a degenerate elliptic PDE is of class $C^1$, provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. $σ^{-1} \in L^1\left (\frac{1}λ {\bf d}λ\right )$. The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate $u(x)$, near $x_0$, by an error of order $\text{o}(|x-x_0|)$. Explicit… ▽ More We prove that any solution of a degenerate elliptic PDE is of class $C^1$, provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. $σ^{-1} \in L^1\left (\frac{1}λ {\bf d}λ\right )$. The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate $u(x)$, near $x_0$, by an error of order $\text{o}(|x-x_0|)$. Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the ${C}^1$--regularity of solutions. Among the main new ingredients required in the proof, we develop an alternative recursive algorithm for the renormalization of approximating solutions. This new method is based on a technique tailored to prevent the sequence of degeneracy laws constructed through the process from being, itself, degenerate. △ Less

Submitted 23 August, 2022; originally announced August 2022.

Comments: To appear in Advances in Mathematics

MSC Class: 35B65; 35J70; 35D40; 37J60

No-Regret Learning in Games is Turing Complete

Authors: Gabriel P. Andrade, Rafael Frongillo, Georgios Piliouras

Abstract: Games are natural models for multi-agent machine learning settings, such as generative adversarial networks (GANs). The desirable outcomes from algorithmic interactions in these games are encoded as game theoretic equilibrium concepts, e.g. Nash and coarse correlated equilibria. As directly computing an equilibrium is typically impractical, one often aims to design learning algorithms that iterati… ▽ More Games are natural models for multi-agent machine learning settings, such as generative adversarial networks (GANs). The desirable outcomes from algorithmic interactions in these games are encoded as game theoretic equilibrium concepts, e.g. Nash and coarse correlated equilibria. As directly computing an equilibrium is typically impractical, one often aims to design learning algorithms that iteratively converge to equilibria. A growing body of negative results casts doubt on this goal, from non-convergence to chaotic and even arbitrary behaviour. In this paper we add a strong negative result to this list: learning in games is Turing complete. Specifically, we prove Turing completeness of the replicator dynamic on matrix games, one of the simplest possible settings. Our results imply the undecicability of reachability problems for learning algorithms in games, a special case of which is determining equilibrium convergence. △ Less

Submitted 23 February, 2022; originally announced February 2022.

Comments: 18 pages, 1 figure

Homeostatic Mechanisms in Biological Systems

Authors: Pedro P. A. Cardoso de Andrade, João L. O. Madeira, Fernando Antoneli

Abstract: In this paper we investigate the homeostatic mechanism in two biologically motivated models: intracellular copper regulation and self immune recognition. The analysis is based on the notions of infinitesimal homeostasis and near-perfect homeostasis. We introduce a theoretical background that makes it possible to consider points of infinitesimal homeostasis that lie at the boundary of the domain of… ▽ More In this paper we investigate the homeostatic mechanism in two biologically motivated models: intracellular copper regulation and self immune recognition. The analysis is based on the notions of infinitesimal homeostasis and near-perfect homeostasis. We introduce a theoretical background that makes it possible to consider points of infinitesimal homeostasis that lie at the boundary of the domain of definition of the input-output function. We show that the two models display near-perfect homeostasis. Moreover, we show that, unlike the examples of [M. Reed, J. Best, M. Golubitsky, I. Stewart, and H. F. Nijhout. Analysis of homeostatic mechanisms in biochemical networks. Bull. Math. Biol., 79(11):2534-2557, 2017], the obstruction of occurrence of infinitesimal homeostasis in both of them is caused by the modeling assumptions that force the point of infinitesimal homeostasis to lie at the boundary of domain of definition of the respective input-output functions. △ Less

Submitted 22 February, 2022; originally announced February 2022.

Comments: 31 pages, 5 figures

Improved regularity for the parabolic normalized p-Laplace equation

Authors: Pêdra D. S. Andrade, Makson S. Santos

Abstract: We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity solutions is locally asymptotically Lipschitz continuous when $p$ is sufficiently close to 2. In addition, we establish regularity estimates in Sobolev spaces. We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity solutions is locally asymptotically Lipschitz continuous when $p$ is sufficiently close to 2. In addition, we establish regularity estimates in Sobolev spaces. △ Less

Submitted 18 August, 2021; originally announced August 2021.

Comments: 18 pages

MSC Class: 35B65; 35K55; 35Q91

Learning in Matrix Games can be Arbitrarily Complex

Authors: Gabriel P. Andrade, Rafael Frongillo, Georgios Piliouras

Abstract: A growing number of machine learning architectures, such as Generative Adversarial Networks, rely on the design of games which implement a desired functionality via a Nash equilibrium. In practice these games have an implicit complexity (e.g. from underlying datasets and the deep networks used) that makes directly computing a Nash equilibrium impractical or impossible. For this reason, numerous le… ▽ More A growing number of machine learning architectures, such as Generative Adversarial Networks, rely on the design of games which implement a desired functionality via a Nash equilibrium. In practice these games have an implicit complexity (e.g. from underlying datasets and the deep networks used) that makes directly computing a Nash equilibrium impractical or impossible. For this reason, numerous learning algorithms have been developed with the goal of iteratively converging to a Nash equilibrium. Unfortunately, the dynamics generated by the learning process can be very intricate and instances of training failure hard to interpret. In this paper we show that, in a strong sense, this dynamic complexity is inherent to games. Specifically, we prove that replicator dynamics, the continuous-time analogue of Multiplicative Weights Update, even when applied in a very restricted class of games -- known as finite matrix games -- is rich enough to be able to approximate arbitrary dynamical systems. Our results are positive in the sense that they show the nearly boundless dynamic modelling capabilities of current machine learning practices, but also negative in implying that these capabilities may come at the cost of interpretability. As a concrete example, we show how replicator dynamics can effectively reproduce the well-known strange attractor of Lonrenz dynamics (the "butterfly effect") while achieving no regret. △ Less

Submitted 4 March, 2021; originally announced March 2021.

Summability characterizations of positive sequences

Authors: Douglas Azevedo, Thiago P. Andrade

Abstract: In this paper, we propose extensions for the classical Kummer test, which is a very far-reaching criterion that provides sufficient and necessary conditions for convergence and divergence of series of positive terms. Furthermore, we present and discuss some interesting consequences and examples such as extensions of the Olivier's theorem and Raabe, Bertrand and Gauss's test. In this paper, we propose extensions for the classical Kummer test, which is a very far-reaching criterion that provides sufficient and necessary conditions for convergence and divergence of series of positive terms. Furthermore, we present and discuss some interesting consequences and examples such as extensions of the Olivier's theorem and Raabe, Bertrand and Gauss's test. △ Less

Submitted 6 April, 2022; v1 submitted 9 January, 2021; originally announced January 2021.

Comments: 12 pages (Accepted for publication in Communications in Mathematics in June 17, 2020 )

MSC Class: 40A05; 40C99

Journal ref: Communications in Mathematics, Volume 30 (2022), Issue 1 (May 12, 2022) cm:9290

Stationary fully nonlinear mean-field games

Authors: Pêdra D. S. Andrade, Edgard A. Pimentel

Abstract: In this paper we examine fully nonlinear mean-field games associated with a minimization problem. The variational setting is driven by a functional depending on its argument through its Hessian matrix. We work under fairly natural conditions and establish improved (sharp) regularity for the solutions in Sobolev spaces. Then, we prove the existence of minimizers for the variational problem and the… ▽ More In this paper we examine fully nonlinear mean-field games associated with a minimization problem. The variational setting is driven by a functional depending on its argument through its Hessian matrix. We work under fairly natural conditions and establish improved (sharp) regularity for the solutions in Sobolev spaces. Then, we prove the existence of minimizers for the variational problem and the existence of solutions to the mean-field games system. We also investigate a unidimensional example and unveil new information on the explicit solutions. Our findings can be generalized to a larger class of operators, yielding information on a broader range of examples. △ Less

Submitted 29 October, 2020; originally announced October 2020.

Comments: Accepted for publication in Journal d'Analyse Mathématique

MSC Class: 35B65; 35J35; 35A01

Geometric regularity theory for a time-dependent Isaacs equation

Authors: Pêdra D. S. Andrade, Giane C. Rampasso, Makson S. Santos

Abstract: The purpose of this work is to produce a regularity theory for a class of parabolic Isaacs equations. Our techniques are based on approximation methods which allow us to connect our problem with a Bellman parabolic model. An approximation regime for the coefficients, combined with a smallness condition on the source term unlocks new regularity results in Sobolev and Hölder spaces. The purpose of this work is to produce a regularity theory for a class of parabolic Isaacs equations. Our techniques are based on approximation methods which allow us to connect our problem with a Bellman parabolic model. An approximation regime for the coefficients, combined with a smallness condition on the source term unlocks new regularity results in Sobolev and Hölder spaces. △ Less

Submitted 12 January, 2022; v1 submitted 28 April, 2020; originally announced April 2020.

MSC Class: 35B65; 35K55; 35Q91

Orbital stability of one-parameter periodic traveling waves for dispersive equations and applications

Authors: Thiago Pinguello de Andrade, Ademir Pastor

Abstract: This paper establishes suficient conditions for the orbital stability of one-parameter spatially periodic traveling-wave solutions for one-dimensional dispersive equations. Our method of proof combines known techniques with some new ideas. As a consequence of our result, we give several applications for well known dispersive equations. Extension of the theory to regularized equations is also estab… ▽ More This paper establishes suficient conditions for the orbital stability of one-parameter spatially periodic traveling-wave solutions for one-dimensional dispersive equations. Our method of proof combines known techniques with some new ideas. As a consequence of our result, we give several applications for well known dispersive equations. Extension of the theory to regularized equations is also established. △ Less

Submitted 27 April, 2020; v1 submitted 21 April, 2020; originally announced April 2020.

Comments: 31 pages, 12 figures

Journal ref: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2019

Orbital stability of periodic traveling-wave solutions for the regularized Schamel equation

Authors: Thiago Pinguello de Andrade, Ademir Pastor

Abstract: In this work we study the orbital stability of periodic traveling-wave solutions for dispersive models. The study of traveling waves started in the mid-18th century when John S. Russel established that the flow of water waves in a shallow channel has constant evolution. In recent years, the general strategy to obtain orbital stability consists in proving that the traveling wave in question minimiz… ▽ More In this work we study the orbital stability of periodic traveling-wave solutions for dispersive models. The study of traveling waves started in the mid-18th century when John S. Russel established that the flow of water waves in a shallow channel has constant evolution. In recent years, the general strategy to obtain orbital stability consists in proving that the traveling wave in question minimizes a conserved functional restricted to a certain manifold. Although our method can be applied to other models, we deal with the regularized Schamel equation, which contains a fractional nonlinear term. We obtain a smooth curve of periodic traveling-wave solutions depending on the Jacobian elliptic functions and prove that such solutions are orbitally stable in the energy space. In our context, instead of minimizing the augmented Hamiltonian in the natural codimension two manifold, we minimize it in a "new" manifold, which is suitable to our purposes. △ Less

Submitted 22 December, 2015; originally announced December 2015.

Comments: to appear in Phys. D