Papers by Julian Fernandez Bonder
Advances in Nonlinear Analysis, Mar 31, 2020
We consider an optimal rearrangement minimization problem involving the fractional Laplace operat... more We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (−∆) s , < s < , and the Gagliardo seminorm |u|s. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satis es −(−∆) s U − χ {U≤ } min{−(−∆) s U + ; } = χ {U> } , which happens to be the fractional analogue of the normalized obstacle problem ∆u = χ {u> } .
arXiv (Cornell University), Dec 13, 2018
In this paper we define the notion of nonlocal magnetic Sobolev spaces with nonstandard growth fo... more In this paper we define the notion of nonlocal magnetic Sobolev spaces with nonstandard growth for Lipschitz magnetic fields. In this context we prove a Bourgain-Brezis-Mironescu type formula for functions in this space as well as for sequences of functions. Finally, we deduce some consequences such as the Γ−convergence of modulars and convergence of solutions for some non-local magnetic Laplacian allowing non-standard growth laws to its local counterpart.
arXiv: Analysis of PDEs, 2016
In this paper we analyze the shape derivative of a cost functional appearing in image restoration.
En esta Tesis consideramos el siguiente problema de perturbacion singular que se presenta en teor... more En esta Tesis consideramos el siguiente problema de perturbacion singular que se presenta en teoria de combustion Δuᵋ - uᵋt = Yᵋƒe(uᵋ) en D, ΔYᵋ - Yᵋt = Yᵋƒe(uᵋ) en D, donde D C Rᴺ+¹, ƒe(s) = 1/e² ƒ(s/e) con ƒ una funcion Lipschitz soportada en (-∞, 1]. En este sistema Yᵋ es la fraccion de masa de algun reactante, uᵋ la temperatura rescalada de la mezcla y e es esencialmente el inverso de la energia de activacion. Este modelo es derivado en el contexto de la teoria de llamas premezcladas equidifusionales para numero de Lewis 1. Probamos que, bajo hipotesis adecuadas sobre las funciones uᵋ e Yᵋ, podemos pasar al limite (e → 0) — llamado limite de alta energia de activacion — y que la funcion limite u = lim uᵋ = lim Yᵋ es una solucion del siguiente problema de frontera libre (P) Δu - ut = 0 en {u>0} │Vu│ = √2M(x,t) en ∂{u>0} en un sentido puntual en los puntos regulares de la frontera libre y en el sentido de la viscosidad. En (P), M(x,t) = ƒ¹w˳(x,t) (s+w˳(x,t))ƒ(s)ds y -1 0} es...
arXiv (Cornell University), Jul 11, 2017
In this paper we define the fractional order Orlicz-Sobolev spaces, and prove its convergence to ... more In this paper we define the fractional order Orlicz-Sobolev spaces, and prove its convergence to the classical Orlicz-Sobolev spaces when the fractional parameter s ↑ 1 in the spirit of the celebrated result of Bourgain-Brezis-Mironescu. We then deduce some consequences such as Γ−convergence of the modulars and convergence of solutions for some fractional versions of the ∆g operator as the fractional parameter s ↑ 1.
arXiv (Cornell University), Aug 12, 2020
We establish interior and up to the boundary Hölder regularity estimates for weak solutions of th... more We establish interior and up to the boundary Hölder regularity estimates for weak solutions of the Dirichlet problem for the fractional g−Laplacian with bounded right hand side and g convex. These are the first regularity results available in the literature for integro-differential equations in the context of fractional Orlicz-Sobolev spaces.
arXiv (Cornell University), Nov 25, 2021
In this paper we extend the well-known concentration-compactness principle of P.L. Lions to Orlic... more In this paper we extend the well-known concentration-compactness principle of P.L. Lions to Orlicz spaces. As an application we show an existence result to some critical elliptic problem with nonstandard growth.
Studia Mathematica, 2021
In this paper we define the notion of nonlocal magnetic Sobolev spaces with nonstandard growth fo... more In this paper we define the notion of nonlocal magnetic Sobolev spaces with nonstandard growth for Lipschitz magnetic fields. In this context we prove a Bourgain-Brezis-Mironescu type formula for functions in this space as well as for sequences of functions. Finally, we deduce some consequences such as the Γ−convergence of modulars and convergence of solutions for some non-local magnetic Laplacian allowing non-standard growth laws to its local counterpart.
Siam Journal on Mathematical Analysis, 2017
In this work we obtain a compactness result for the H−convergence of a family of nonlocal and non... more In this work we obtain a compactness result for the H−convergence of a family of nonlocal and nonlinear monotone elliptic-type problems by means of Tartar's method of oscillating test functions.
arXiv (Cornell University), Jan 26, 2016
In this work we study the convergence of an homogenization problem for half-eigenvalues and Fučík... more In this work we study the convergence of an homogenization problem for half-eigenvalues and Fučík eigencurves. We provide quantitative bounds on the rate of convergence of the curves for periodic homogenization problems.
arXiv (Cornell University), Dec 4, 2019
In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by mea... more In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by means of Gamma-convergence methods. This method allows us to treat different eigenvalue problems under a unified framework. We are able to recover some known results for the behavior of the eigenvalues of the p−fractional laplacian when the fractional parameter s goes to 1, and to extend some known results for the behavior of the same eigenvalue problem when p goes to ∞. Finally we analyze other eigenvalue problems not previously covered in the literature.
arXiv (Cornell University), Feb 26, 2018
In this paper we extend the well-known concentration-compactness principle for the Fractional Lap... more In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional p−laplacian in the whole R n .
Proceedings of the American Mathematical Society, Sep 15, 2015
In this paper we prove a Lyapunov type inequality for quasilinear problems with indefinite weight... more In this paper we prove a Lyapunov type inequality for quasilinear problems with indefinite weights. We show that the first eigenvalue is bounded below in terms of the integral of the weight, instead of the integral of its positive part. We apply this inequality to some eigenvalue homogenization problems with indefinite weights.
arXiv (Cornell University), Oct 10, 2019
In this paper we deal with the stability of solutions to fractional p−Laplace problems with nonli... more In this paper we deal with the stability of solutions to fractional p−Laplace problems with nonlinear sources when the fractional parameter s goes to 1. We prove a general convergence result for general weak solutions which is applied to study the convergence of ground state solutions of p−fractional problems in bounded and unbounded domains as s goes to 1. Moreover, our result applies to treat the stability of p−fractional eigenvalues as s goes to 1.
Mediterranean Journal of Mathematics, Mar 11, 2017
In this work we study the convergence of an homogenization problem for half-eigenvalues and Fučík... more In this work we study the convergence of an homogenization problem for half-eigenvalues and Fučík eigencurves. We provide quantitative bounds on the rate of convergence of the curves for periodic homogenization problems.
arXiv (Cornell University), May 14, 2019
We consider an optimal rearrangement minimization problem involving the fractional Laplace operat... more We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (−∆) s , 0 < s < 1, and Gagliardo seminorm |u|s. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satisfies −(−∆) s U − χ {U ≤0} min{−(−∆) s U + ; 1} = χ {U >0} , which happens to be the fractional analogue of the normalized obstacle problem ∆u = χ {u>0} .
arXiv (Cornell University), Feb 18, 2009
We consider the optimization problem of minimizing R Ω |∇u| p(x) + λχ {u>0} dx in the class of fu... more We consider the optimization problem of minimizing R Ω |∇u| p(x) + λχ {u>0} dx in the class of functions W 1,p(•) (Ω) with u − ϕ0 ∈ W 1,p(•) 0 (Ω), for a given ϕ0 ≥ 0 and bounded. W 1,p(•) (Ω) is the class of weakly differentiable functions with R Ω |∇u| p(x) dx < ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω ∩ ∂{u > 0}, is a regular surface.
arXiv (Cornell University), Feb 17, 2006
We consider the optimization problem of minimizing Ω |∇u| p dx with a constrain on the volume of ... more We consider the optimization problem of minimizing Ω |∇u| p dx with a constrain on the volume of {u > 0}. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution u is locally Lipschitz continuous and that the free boundary, ∂{u > 0} ∩ Ω, is smooth.
arXiv (Cornell University), Aug 28, 2012
We study the rate of convergence for (variational) eigenvalues of several non-linear problems inv... more We study the rate of convergence for (variational) eigenvalues of several non-linear problems involving oscillating weights and subject to different kinds of boundary conditions in bounded domains.
arXiv (Cornell University), Mar 8, 2012
In this paper we study the rate of convergence of the eigenvalues of 1−dimensional rapidly oscill... more In this paper we study the rate of convergence of the eigenvalues of 1−dimensional rapidly oscillating p−laplacian type problems and find explicit order of convergence both in k and in ε. Moreover, explicit bounds on the constant entering in the estimate are also obtained.
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Papers by Julian Fernandez Bonder