arXiv subjects

David Gómez-Castro

Publications and source records attributed to David Gómez-Castro.

31 records · Page 2Linked to original sources

arXiv2020

Singular solutions for fractional parabolic boundary value problems

The standard problem for the classical heat equation posed in a bounded domain $Ω$ of $\mathbb R^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the non-zero boundary data must be singular, i.e., the solution $u(t,x)$ blows up as $x$ approaches $\partial Ω$ in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admitting initial data and a forcing term. When the boundary data are zero we recover the standard fractional heat semigroup. A general class of integro-differential operators may replace the classical fractional Laplacian operators, thus enlarging the scope of the work. As further results on the spectral theory of the fractional heat semigroup, we show that a one-sided Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractional Laplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kernel.

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arXiv2020

Steiner symmetrization for anisotropic quasilinear equations via partial discretization

In this paper we obtain comparison results for the quasilinear equation $-Δ_{p,x} u - u_{yy} = f$ with homogeneous Dirichlet boundary conditions by Steiner rearrangement in variable $x$, thus solving a long open problem. In fact, we study a broader class of anisotropic problems. Our approach is based on a finite-differences discretization in $y$, and the proof of a comparison principle for the discrete version of the auxiliary problem $A U - U_{yy} \le \int_0^s f^*$, where $AU = (nω^{1/n}s^{1/n'} )^p (- U_{ss})^{p-1}$. We show that this operator is T-accretive in $L^\infty$. We extend our results for $-Δ_{p,x}$ to general operators of the form $-\mathrm{div} (a(|\nabla_x u|) \nabla_x u)$ where $a$ is non-decreasing and behaves like $| \cdot |^{p-2}$ at infinity.

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arXiv2020

Estimates on translations and Taylor expansions in fractional Sobolev spaces

In this paper we study how the (normalised) Gagliardo semi-norms $[u]_{W^{s,p} (\mathbb{R}^n)}$ control translations. In particular, we prove that $\| u(\cdot + y) - u \|_{L^p (\mathbb{R}^n)} \le C [ u ] _{W^{s,p} (\mathbb{R}^n)} |y|^s$ for $n\geq1$, $s \in [0,1]$ and $p \in [1,+\infty]$, where $C$ depends only on $n$. We then obtain a corresponding higher-order version of this result: we get fractional rates of the error term in the Taylor expansion. We also present relevant implications of our two results. First, we obtain a direct proof of several compact embedding of $W^{s,p}(\mathbb{R}^n)$ where the Fréchet-Kolmogorov Theorem is applied with known rates. We also derive fractional rates of convergence of the convolution of a function with suitable mollifiers. Thirdly, we obtain fractional rates of convergence of finite-difference discretizations for $W^{s,p} (\mathbb{R}^n))$.

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arXiv2019

Singular boundary behaviour and large solutions for fractional elliptic equations

We show that the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains strongly differs from the one of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data. In this classical case it is known that, at least in a suitable weak sense, solutions of non-homogeneous Dirichlet problem are unique and tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that such results are false for equations driven by a wide class of nonlocal fractional operators, extending previous findings for some models of the fractional Laplacian operator. Actually, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal pperators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function.

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arXiv2019

A nonlocal memory strange term arising in the critical scale homogenisation of a diffusion equation with a dynamic boundary condition

Our main interest in this paper is the study of homogenised limit of a parabolic equation with a nonlinear dynamic boundary condition of the micro-scale model set on a domain with periodically place particles. We focus on the case of particles (or holes) of critical diameter with respect to the period of the structure. Our main result proves the weak convergence of the sequence of solutions of the original problem to the solution of a reaction-diffusion parabolic problem containing a `strange term'. The novelty of our result is that this term is a nonlocal memory solving an ODE. We prove that the resulting system satisfies a comparison principle.

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arXiv2019

The fractional Schrödinger equation with singular potential and measure data

We consider the steady fractional Schrödinger equation $L u + V u = f$ posed on a bounded domain $Ω$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-Δ)^s$; $V\ge 0$ is a potential with possible singularities, and the right-hand side are integrable functions or Radon measures. We reformulate the problem via the Green function of $(-Δ)^s$ and prove well-posedness for functions as data.If $V$ is bounded or mildly singular a unique solution of $(-Δ)^s u + V u = μ$ exists for every Borel measure $μ$. On the other hand, when $V$ is allowed to be more singular, but only on a finite set of points, a solution of $(-Δ)^s u + V u = δ_x$, where $δ_x$ is the Dirac measure at $x$, exists if and only if $h(y) = V(y) |x - y|^{-(n+2s)}$ is integrable on some small ball around $x$. We prove that the set $Z = \{x \in Ω: \textrm{no solution of } (-Δ)^s u + Vu = δ_x \textrm{ exists}\}$ is relevant in the following sense: a solution of $(-Δ)^s u + V u = μ$ exists if and only if $|μ| (Z) = 0$. Furthermore, $Z$ is the set points where the strong maximum principle fails, in the sense that for any bounded $f$ the solution of $(-Δ)^s u + Vu = f$ vanishes on $Z$.

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arXiv2018

Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis "nano-composite" membranes

One of the main goals of this paper is to extend some of the mathematical techniques of some previous papers by the authors showing that some very useful phenomenological properties which can be observed to the nano-scale can be simulated and justified mathematically by means of some homogenization processes when a certain critical scale is used in the corresponding framework. Here the motivating problem in consideration is formulated in the context of the reverse osmosis. We consider, on a part of the boundary of a domain $Ω\subset \mathbb R^n$, a set of very small periodically distributed semipermeable membranes having an ideal infinite permeability coefficient (which leads to Signorini type boundary conditions) on a part $Γ_1$ of the boundary. We also assume that a possible chemical reaction may take place on the membranes. We obtain the rigorous convergence of the problems to a homogenized problem in which there is a change in the constitutive nonlinearities. Changes of this type are the reason for the big success of the nanocomposite materials. Our proof is carried out for membranes not necessarily of radially symmetric shape. The definition of the associated critical scale depends on the dimension of the space (and it is quite peculiar for the special case of $n=2$). Roughly speaking, our result proves that the consideration of the critical case of the scale leads to an homogenized formulation which is equivalent to have a global semipermeable membrane, at the whole part of the boundary $Γ_1$, with a "finite permeability coefficient of this virtual membrane" which is the best we can get, even if the original problem involves a set of membranes of any arbitrary finite permeability coefficients.

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arXiv2018

The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach

We study the Dirichlet problem for the stationary Schrödinger fractional Laplacian equation $(-Δ)^s u + V u = f$ posed in bounded domain $ Ω\subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative potentials $V\in L^1_{loc}(Ω)$ and prove well-posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions $f$. Important properties of the solutions, such as its boundary behaviour, are derived. The case of super singular potentials that blow up near the boundary is given special consideration. Related literature is commented.

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arXiv2018

On the well-posedness of a multiscale mathematical model for Lithium-ion batteries

We consider the mathematical treatment of a system of nonlinear partial differential equations based on a model, proposed in 1972 by J. Newman, in which the coupling between the Lithium concentration, the phase potentials and temperature in the electrodes and the electrolyte of a Lithium battery cell is considered. After introducing some functional spaces well-adapted to our framework we obtain some rigorous results showing the well-posedness of the system, first for some short time and then, by considering some hypothesis on the nonlinearities, globally in time. As far as we know, this is the first result in the literature proving existence in time of the full Newman model, which follows previous results by the third author in 2016 regarding a simplified case.

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arXiv2018

Existence and uniqueness of solutions of Schrödinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications

Motivated mainly by the localization over an open bounded set $Ω$ of $\mathbb R^n$ of solutions of the Schrödinger equations, we consider the Schrödinger equation over $Ω$ with a very singular potential $V(x) \ge C d (x, \partial Ω)^{-r}$ with $r\ge 2$ and a convective flow $\vec U$. We prove the existence and uniqueness of a very weak solution of the equation, when the right hand side datum $f(x)$ is in $L^1 (Ω, d(\cdot, \partial Ω))$, even if no boundary condition is a priori prescribed. We prove that, in fact, the solution necessarily satisfies (in a suitable way) the Dirichlet condition $u = 0$ on $\partial Ω$. These results improve some of the results of the previous paper by the authors in collaboration with Roger Temam. In addition, we prove some new results dealing with the $m$-accretivity in $L^1 (Ω, d(\cdot, \partial Ω)^ α)$, where $α\in [0,1]$, of the associated operator, the corresponding parabolic problem and the study of the complex evolution Schrödinger equation in $\mathbb R^n$.

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arXiv2017

Homogenization and Shape Differentiation of Quasilinear Elliptic Equations

This thesis is divided into five chapters. The aim is the study of the effectiveness of a chemical as defined by R. Aris for semilinear elliptic equations. The first chapter focuses on homogenization on quasilinear diffusion-reaction problems in domains with small particles. The classical results are extend to less smooth nonlinearities, and more general shapes, specially new for the critical case. The second chapter deals with Steiner symmetrisation of semilinear elliptic and parabolic equations. The third chapter deals with shape differentiation, with smooth and non smooth nonlinearities. The fourth chapter deals with linear elliptic equations with a potential, $-Δu + \nabla u \cdot b + V u$, where the potential, $V$, "blows up" near the boundary. This kind of equations appear as a result of the shape differentiation process, in the non-smooth case. The fifth chapter develops the second part of the thesis, and includes results obtained during 2017 visit to Prof. Brezis. We showed that the basis of eigenvalues of $-Δ$ with Dirichlet boundary conditions is the unique basis to approximate functions in $H_0^1$ in $L^2$ in an optimal way.

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arXiv2017

Linear diffusion with singular absorption potential and/or unbounded convective flow: the weighted space approach

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\mathbb R^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain.

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arXiv2017

Shape differentiation of a steady-state reaction-diffusion problem arising in Chemical Engineering: the case of non-smooth kinetic with dead core

In this paper we consider an extension of the results in shape differentiation of semilinear equations with smooth nonlinearity presented in J.I. Díaz and D. Gómez-Castro: An Application of Shape Differentiation to the Effectiveness of a Steady State Reaction-Diffusion Problem Arising in Chemical Engineering. Electron. J. Differ. Equations in 2015 to the case in which the nonlinearities might be less smooth. Namely we will show that Gateaux shape derivatives exists when the nonlinearity is only Lipschitz continuous, and we will give a definition of the derivative when the nonlinearity has a blow up. In this direction, we will study the case of root-type nonlinearities.

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