@book{diaz+gc+shaposhnikova2021book,address={Berlin, Boston},author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Shaposhnikova, T. A},doi={10.1515/9783110648997},isbn={978-3-11-064899-7},publisher={De Gruyter},title={{Nonlinear Reaction-Diffusion Processes for Nanocomposites}},url={https://www.degruyter.com/view/title/551325},year={2021},keywords={short-cv, selected},dimensions=true}
@online{CarrilloFernandez-JimenezGomez-Castro,title={Aggregation-Diffusion Equations with Saturation},author={Carrillo, Jos\'e Antonio and Fern\'andez-Jim\'enez, Alejandro and G\'omez-Castro, David},year={2024},arxivid={2410.10040},primaryclass={math.AP},keywords={Computer Science - Numerical Analysis,Mathematics - Analysis of PDEs,Mathematics - Numerical Analysis},dimensions=true}
P1
Aggregation-Diffusion Equations for Collective Behaviour in the Sciences
@online{Bailo+Carrillo+GC-ICIAM,title={Aggregation-Diffusion Equations for Collective Behaviour in the Sciences},author={Bailo, Rafael and Carrillo, José A. and Gómez-Castro, David},year={2024},arxivid={2405.16679},primaryclass={math.AP},dimensions=true}
Journal Articles (40)
J40
A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains
@article{BailoCarrilloFronzoniGC2025,title={A {{Finite-Volume Scheme}} for {{Fractional Diffusion}} on {{Bounded Domains}}},author={Bailo, Rafael and Carrillo, Jos\'e A. and Fronzoni, Stefano and G\'omez-Castro, David},journal={European Journal of Applied Mathematics},doi={10.1017/S0956792524000172},volume={36},number={2},year={2025},pages={398–418},arxivid={2309.08283},primaryclass={cs.NA},dimensions=true}
J39
Partial Mass Concentration for Fast-Diffusions with Non-Local Aggregation Terms
@article{Carrillo+Fernandez-Jimenez+GC2024JDE,title={Partial Mass Concentration for Fast-Diffusions with Non-Local Aggregation Terms},author={Carrillo, José A. and Gómez-Castro, David and Fernández-Jiménez, Alejandro},date={2024},journal={Journal of Differential Equations},volume={409},pages={700-773},year={2024},issn={0022-0396},doi={10.1016/j.jde.2024.08.013},arxivid={2304.04582},primaryclass={math.AP},keywords={short-cv,selected},dimensions=true}
J38
Interpreting systems of continuity equations in spaces of probability measures through PDE duality
@article{Carrillo+GC2024RACSAM,title={{Interpreting systems of continuity equations in spaces of probability measures through PDE duality}},author={Carrillo, José A. and Gómez-Castro, David},journal={Revista de la Real Academia de Ciencias Exactas, F{\'\i}sicas y Naturales. Serie A. Matem{\'a}ticas},year={2024},volume={118},number={127},doi={10.1007/s13398-024-01628-6},arxivid={2206.03968},primaryclass={math.AP},dimensions=true}
J37
Singular solutions for space-time fractional equations in a bounded domain
@article{Chan+GC+Vazquez2024,title={Singular solutions for space-time fractional equations in a bounded domain},author={Chan, Hardy and Gómez-Castro, David and Vázquez, Juan Luis},arxivid={2304.04431},primaryclass={math.AP},journal={Nonlinear Differential Equations and Applications},year={2024},volumen={31},doi={10.1007/s00030-024-00948-1},dimensions=true}
J36
Beginner’s guide to Aggregation-Diffusion Equations
In this article we study the existence, uniqueness, and integrability of solutions to the Dirichlet problem \(-\hbox{div}( M(x) ∇u ) = -\hbox{div} (E(x) u) + f\) in a bounded domain of \(\mathbb{R}N̂\) with \(N \ge 3\). We are particularly interested in singular \(E\) with \(\hbox{div} E \ge 0\). We start by recalling known existence results when \(|E| ∈LN̂\) that do not rely on the sign of \(\hbox{div} E \). Then, under the assumption that \(\hbox{div} E \ge 0\) distributionally, we extend the existence theory to \(|E| ∈L\^2\). For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of \(E\) singular at one point as \(Ax /|x|\^2\), or towards the boundary as \(\hbox{div} E ∼\hbox{dist}(x, ∂Ω)\^{-2-α}\). In these cases the singularity of \(E\) leads to \(u\) vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. \(∂u / ∂n < 0\), fails in the presence of such singular drift terms \(E\). For more information see https://ejde.math.txstate.edu/Volumes/2024/13/abstr.html
@article{BoccardoGCDiaz2024,title={Failure of the {{Hopf-Oleinik}} Lemma for a Linear Elliptic Problem with Singular Convection of Non-Negative Divergence},author={Boccardo, L. and Díaz, J.I. and Gómez-Castro, D.},year={2024},journal={Electronic Journal of Differential Equations},volume={2024},pages={13},issn={1072-6691},doi={10.58997/ejde.2024.13},url={https://ejde-ojs-txstate.tdl.org/ejde/article/view/605},arxivid={2211.10122},primaryclass={math.AP},dimensions=true}
J34
Asymptotic simplification of Aggregation-Diffusion equations towards the heat kernel
We give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at infinity, then the linear diffusion overcomes its effect, either attractive or repulsive, for large times independently of the initial data, and solutions behave like the fundamental solution of the heat equation with some rate. The potential W(x) ∼\log |x| for |x| ≫1 appears as the natural limiting case when the intermediate asymptotics change. In order to obtain such a result, we produce uniform-in-time estimates in a suitable rescaled change of variables for the entropy, the second moment, Sobolev norms and the C^αregularity with a novel approach for this family of equations using modulus of continuity techniques.
J33
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.
@article{abatangelo+gc+vazquez2023,archiveprefix={arXiv},arxivid={1910.00366},author={Abatangelo, Nicola and Gómez-Castro, D. and V{\'{a}}zquez, J. L.},pages={568-615},title={{Singular boundary behaviour and large solutions for fractional elliptic equations}},journal={Journal of the London Mathematical Society},year={2023},volume={107},issue={2},doi={10.1112/jlms.12692},dimensions=true}
J32
Singular solutions for fractional parabolic boundary value problems
The standard problem for the classical heat equation posed in a bounded domain }}\backslashOmega }}of }}{\backslashmathbb {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 }}\backslashpartial \backslashOmega }}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.
@article{Chan+GC+Vazquez2022RACSAM,title={Singular solutions for fractional parabolic boundary value problems},author={Chan, Hardy and G{\'o}mez-Castro, David and V{\'a}zquez, Juan Luis},doi={10.1007/s13398-022-01294-6},isbn={1579-1505},journal={Revista de la Real Academia de Ciencias Exactas, F{\'\i}sicas y Naturales. Serie A. Matem{\'a}ticas},number={4},pages={159},url={https://doi.org/10.1007/s13398-022-01294-6},volume={116},year={2022},arxivid={2007.13391},dimensions=true}
J31
A fast regularisation of a Newtonian vortex equation
We consider equations of the form u_t = ∇⋅( γ(u) ∇\mathrmN(u)), where \mathrmN is the Newtonian potential (inverse of the Laplacian) posed in the whole space \mathbb R^d, and γ(u) is the mobility. For linear mobility, γ(u)=u, the equation and some variations have been proposed as a model for superconductivity or superfluidity. In that case the theory leads to uniqueness of bounded weak solutions having the property of compact space support, and in particular there is a special solution in the form of a disk vortex of constant intensity in space u=c_1t^-1 supported in a ball that spreads in time like c_2t^1/d, thus showing a discontinuous leading front. In this paper we propose the model with sublinear mobility γ(u)=u^α, with 0<α<1, and prove that nonnegative solutions recover positivity everywhere, and moreover display a fat tail at infinity. The model acts in many ways as a regularization of the previous one. In particular, we find that the equivalent of the previous vortex is an explicit self-similar solution decaying in time like u=O(t^-1/α) with a space tail with size u=O(|x|^- d/(1-α)). We restrict the analysis to radial solutions and construct solutions by the method of characteristics. We introduce the mass function, which solves an unusual variation of Burger’s equation, and plays an important role in the analysis. We show well-posedness in the sense of viscosity solutions. We also construct numerical finite-difference convergent schemes.
@article{Carrillo+GC+Vazquez2022AIHP,archiveprefix={arXiv},arxivid={1912.00912},author={Carrillo, Jos{\'{e}} A. and Gómez-Castro, D. and V{\'{a}}zquez, J. L.},title={{A fast regularisation of a Newtonian vortex equation}},url={http://arxiv.org/abs/1912.00912},year={2022},volume={39},pages={705-747},journal={Annales de l’Institut Henri Poincaré C, Analyse non linéaire},doi={10.4171/AIHPC/17},dimensions=true}
J30
Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility
We consider density solutions for gradient flow equations of the form u_t = ∇⋅( γ(u) ∇\mathrm N(u)), where \mathrm N is the Newtonian repulsive potential in the whole space \mathbb R^d with the nonlinear convex mobility γ(u)=u^α, and α>1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u)=u. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u=c_1t^-1 supported in a ball that spreads in time like c_2t^1/d, thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ(u)=u^α, with 0<α<1 studied in [9]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detail analysis allowing us to show a waiting time phenomena. This is a typical behavior for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations ilustrating the proven results and showcasing some open problems.
@article{Carrillo+GC+Vazquez2022ANONA,author={Carrillo, Jose A. and G{\'{o}}mez-Castro, David and V{\'{a}}zquez, Juan Luis},journal={Advances in Nonlinear Analysis},title={{Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility}},year={2022},number={1},pages={937--967},volume={11},archiveprefix={arXiv},arxivid={2007.01185},doi={10.1515/anona-2021-0231},publisher={Walter de Gruyter {GmbH}},dimensions=true}
J29
Infinite-time concentration in aggregation–diffusion equations with a given potential
Typically, aggregation-diffusion is modeled by parabolic equations that combine linear or nonlinear diffusion with a Fokker-Planck convection term. Under very general suitable assumptions, we prove that radial solutions of the evolution process converge asymptotically in time towards a stationary state representing the balance between the two effects. Our parabolic system is the gradient flow of an energy functional, and in fact we show that the stationary states are minimizers of a relaxed energy. Here, we study radial solutions of an aggregation-diffusion model that combines nonlinear fast diffusion with a convection term driven by the gradient of a potential, both in balls and the whole space. We show that, depending on the exponent of fast diffusion and the potential, the steady state is given by the sum of an explicit integrable function, plus a Dirac delta at the origin containing the rest of the mass of the initial datum. Furthermore, it is a global minimizer of the relaxed energy. This splitting phenomenon is an uncommon example of blow-up in infinite time.
@article{Carrillo+GC+Vazquez2022JMPA,archiveprefix={arXiv},arxivid={2103.12631},author={Carrillo, J. A. and Gómez-Castro, D. and V{\'{a}}zquez, J. L.},doi={10.1016/j.matpur.2021.11.002},issn={00217824},journal={Journal de Math{\'{e}}matiques Pures et Appliqu{\'{e}}es},pages={346--398},title={{Infinite-time concentration in aggregation--diffusion equations with a given potential}},url={https://linkinghub.elsevier.com/retrieve/pii/S002178242100163X},volume={157},year={2022},keywords={short-cv},dimensions=true}
J28
Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas
We introduce three representation formulas for the fractional p -Laplace operator in the whole range of parameters 0 < s < 1 and 1 < p < ∞. Note that for p ≠ 2 this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the nonlinearity to the Caffarelli-Silvestre linear extension technique. The third one is the corresponding nonlinear version of the Balakrishnan formula. We also discuss the correct choice of the constant of the fractional p -Laplace operator in order to have continuous dependence as p → 2 and s → 0 + , 1 − .
@article{delTeso+GC+Vazquez2020pLap,author={{del Teso}, F{\'{e}}lix and Gómez-Castro, D. and V{\'{a}}zquez, J. L.},arxivid={2010.06933},doi={10.1515/fca-2021-0042},issn={1311-0454},journal={Fract. Calc. Appl. Anal.},number={4},pages={966--1002},title={{Three representations of the fractional $p$-Laplacian: Semigroup, extension and Balakrishnan formulas}},url={https://www.degruyter.com/document/doi/10.1515/fca-2021-0042/html},volume={24},year={2021},dimensions=true}
J27
Characterisation of homogeneous fractional Sobolev spaces
Lorenzo
Brasco
, D.
Gómez-Castro
, and J. L.
Vázquez
Calculus of Variations and Partial Differential Equations, 2021
Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces $\mathcal D^s,p (\mathbb R^n) D s , p ( R n ) and their embeddings, for s ∈(0,1] s ∈ ( 0 , 1 ] and p\ge 1 p ≥ 1 . They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodeckiĭ seminorms. For s p < n s p < n or s = p = n = 1 s = p = n = 1 we show that \mathcal D^s,p(\mathbb R^n) D s , p ( R n ) is isomorphic to a suitable function space, whereas for s p \ge n$ s p ≥ n it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we present a Morrey–Campanato inequality where the Gagliardo–Slobodeckiĭ seminorm controls from above a suitable Campanato seminorm.
@article{Brasco+GC+Vazquez2021,archiveprefix={arXiv},arxivid={2007.08000},author={Brasco, Lorenzo and Gómez-Castro, D. and V{\'{a}}zquez, J. L.},doi={10.1007/s00526-021-01934-6},issn={0944-2669},journal={Calculus of Variations and Partial Differential Equations},number={2},title={{Characterisation of homogeneous fractional Sobolev spaces}},url={http://link.springer.com/10.1007/s00526-021-01934-6},volume={60},year={2021},keywords={short-cv,selected},dimensions=true}
J26
Steiner symmetrization for anisotropic quasilinear equations via partial discretization
In this paper we obtain comparison results for the quasilinear equation -\Delta_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/ns^1/n’ U_ss)^p-1. We show that this operator is T-accretive in L^∞. We extend our results for -\Delta_p,x to general operators of the form -\textdiv (a(|\nabla_x u|) \nabla_x u) where a is non-decreasing and behaves like | ⋅|^p-2 at infinity.
@article{Brock+Diaz+GC+Mercaldo+Ferone2021,archiveprefix={arXiv},arxivid={1912.02080},author={Brock, Friedemann and D{\'{i}}az, J. I. and Ferone, Adele and Gómez-Castro, D. and Mercaldo, Anna},doi={10.1016/j.anihpc.2020.07.005},issn={02941449},journal={Annales de l'Institut Henri Poincar{\'{e}} C, Analyse non lin{\'{e}}aire},keywords={And phrases. Integro-differential operators,Eigenvalue problems,Fractional Laplacian,Large solutions},number={2},pages={347--368},title={{Steiner symmetrization for anisotropic quasilinear equations via partial discretization}},url={https://linkinghub.elsevier.com/retrieve/pii/S0294144920300706},volume={38},year={2021},dimensions=true}
J25
Blow-up phenomena in nonlocal eigenvalue problems: When theories of L^1 and L^2 meet
We develop a linear theory of very weak solutions for nonlocal eigenvalue problems \mathcal L u = λu + f involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and without singular boundary data. We consider mild hypotheses on the Green’s function and the standard eigenbasis of the operator. The main examples in mind are the fractional Laplacian operators. Without singular boundary datum and when λis not an eigenvalue of the operator, we construct an L^2-projected theory of solutions, which we extend to the optimal space of data for the operator \mathcal L. We present a Fredholm alternative as λtends to the eigenspace and characterise the possible blow-up limit. The main new ingredient is the transfer of orthogonality to the test function. We then extend the results to singular boundary data and study the so-called large solutions, which blow up at the boundary. For that problem we show that, for any regular value λ, there exist "large eigenfunctions" that are singular on the boundary and regular inside. We are also able to present a Fredholm alternative in this setting, as λapproaches the values of the spectrum. We also obtain a maximum principle for weighted L^1 solutions when the operator is L^2-positive. It yields a global blow-up phenomenon as the first eigenvalue is approached from below. Finally, we recover the classical Dirichlet problem as the fractional exponent approaches one under mild assumptions on the Green’s functions. Thus "large eigenfunctions" represent a purely nonlocal phenomenon.
@article{chan+gc+vazquez2021,archiveprefix={arXiv},arxivid={2004.04579},author={Chan, Hardy and Gómez-Castro, D. and V{\'{a}}zquez, J. L.},doi={10.1016/j.jfa.2020.108845},issn={00221236},journal={Journal of Functional Analysis},keywords={short-cv},number={7},pages={108845},title={{Blow-up phenomena in nonlocal eigenvalue problems: When theories of $L^1$ and $L^2$ meet}},url={http://arxiv.org/abs/2004.04579 https://linkinghub.elsevier.com/retrieve/pii/S0022123620303888},volume={280},year={2021},dimensions=true}
J24
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 (\mathbbR^n) control translations. In particular, we prove that \|u(⋅+ y) - u \|_L^p (\mathbbR^n) \le C [ u ] _W^s,p (\mathbbR^n) |y|^s for n\geq1, s ∈[0,1] and p ∈[1,+∞], 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(\mathbbR^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 (\mathbbR^n)).
@article{dT+GC+V2020Taylor,archiveprefix={arXiv},arxivid={2004.12196},author={{del Teso}, F{\'{e}}lix and Gómez-Castro, D. and V{\'{a}}zquez, J. L.},doi={10.1016/j.na.2020.111995},issn={0362546X},journal={Nonlinear Anal.},keywords={convolution,discretization error,fractional sobolev spaces,gagliardo norms,interpolation,taylor expansions,translation estimates},title={{Estimates on translations and Taylor expansions in fractional Sobolev spaces}},url={https://linkinghub.elsevier.com/retrieve/pii/S0362546X20302169},volume={200},year={2020},dimensions=true}
J23
A Time-Dependent Strange Term Arising in Homogenization of an Elliptic Problem with Rapidly Alternating Neumann and Dynamic Boundary Conditions Specified at the Domain Boundary: The Critical Case
J. I.
Díaz
, D.
Gómez-Castro
, T. A.
Shaposhnikova
, and M. N.
Zubova
Abstract: A strange term arising in the homogenization of elliptic (and parabolic) equations with dynamic boundary conditions given on some boundary parts of critical size is considered. A problem with dynamic boundary conditions given on the union of some boundary subsets of critical size arranged ε-periodically along the boundary and with homogeneous Neumann conditions given on the rest of the boundary is studied. It is proved that the homogenized boundary condition is a Robin-type containing a nonlocal term depending on the trace of the solution u(x, t) on the boundary ∂Ω.
@article{diaz+gc+shaposh+zubova2020boundary+critical+time,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Shaposhnikova, T. A. and Zubova, M. N.},doi={10.1134/S106456242002009X},issn={1064-5624},journal={Dokl. Math.},keywords={dynamic boundary conditions,homogenization,rapidly oscillating boundary conditions},number={2},pages={96--101},title={{A Time-Dependent Strange Term Arising in Homogenization of an Elliptic Problem with Rapidly Alternating Neumann and Dynamic Boundary Conditions Specified at the Domain Boundary: The Critical Case}},url={http://link.springer.com/10.1134/S106456242002009X},volume={101},year={2020},dimensions=true}
J22
Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes
J. I.
Díaz
, D.
Gómez-Castro
, Alexander V
Podolskiy
, and T. A
Shaposhnikova
@article{diaz+gomez-castro+podolskii+shaposhnikova2020nano-composites,archiveprefix={arXiv},arxivid={1807.07361},author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Podolskiy, Alexander V and Shaposhnikova, T. A},doi={10.1515/anona-2018-0158},issn={2191-9496},journal={Adv. Nonlinear Anal.},mendeley-groups={homogenisation/boundary particles/critical},number={1},pages={193--227},title={{Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes}},url={http://arxiv.org/abs/1807.07361 http://www.degruyter.com/view/j/anona.ahead-of-print/anona-2017-0140/anona-2017-0140.xml http://www.degruyter.com/view/j/anona.2020.9.issue-1/anona-2018-0158/anona-2018-0158.xml},volume={9},year={2020},dimensions=true}
J21
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 (-∆)^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 (-∆)^s and prove well-posedness for functions as data.If V is bounded or mildly singular a unique solution of (-∆)^s u + V u = μexists 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 (-∆)^s u + V u = \delta_x, where \delta_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 ∈Ω: \textrmno solution of (-∆)^s u + Vu = \delta_x \textrm exists} is relevant in the following sense: a solution of (-∆)^s u + V u = μexists 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 (-∆)^s u + Vu = f vanishes on Z.
@article{GC+Vazquez2018,archiveprefix={arXiv},arxivid={1812.02120},author={Gómez-Castro, D. and V{\'{a}}zquez, J. L.},doi={10.3934/dcds.2019298},issn={1553-5231},journal={Discret. Contin. Dyn. Syst. - A},keywords={35D30,35J10,35J67,35J75,Nonlocal elliptic equations,Schr{\"{o}}dinger operators,bounded domains,measure data Mathematics Subject Classification 35,singular potentials},number={12},pages={7113--7139},title={{The fractional Schr{\"{o}}dinger equation with singular potential and measure data}},url={http://aimsciences.org//article/doi/10.3934/dcds.2019298},volume={39},year={2019},dimensions=true}
J20
A nonlocal memory strange term arising in the critical scale homogenisation of a diffusion equation with a dynamic boundary condition
J. I.
Díaz
, D.
Gómez-Castro
, T. A.
Shaposhnikova
, and Maria N.
Zubova
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.
@article{diaz+gc+shaposhnikova+zubova2019nonlocal+memory,archiveprefix={arXiv},arxivid={1905.11709},author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Shaposhnikova, T. A. and Zubova, Maria N.},journal={Electron. J. Differ. Equ.},number={77},pages={1--13},title={{A nonlocal memory strange term arising in the critical scale homogenisation of a diffusion equation with a dynamic boundary condition}},url={http://ejde.math.txstate.edu},volume={2019},year={2019},dimensions=true}
J19
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.
@article{diaz+gc+ramos2018,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Ramos, Angel M.},doi={10.1515/anona-2018-0041},issn={2191-9496},journal={Adv. Nonlinear Anal.},keywords={Browder-Minty existence results,Green operators,Lithium-ion battery cell,fixed point theory,multiscale mathematical model,super and sub solutions},pages={1132--1157},title={{On the well-posedness of a multiscale mathematical model for Lithium-ion batteries}},url={http://www.degruyter.com/view/j/anona.ahead-of-print/anona-2018-0041/anona-2018-0041.xml http://www.degruyter.com/view/j/anona.2019.8.issue-1/anona-2018-0041/anona-2018-0041.xml},volume={8},number={1},year={2019},dimensions=true}
J18
Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition
J. I.
Díaz
, D.
Gómez-Castro
, Alexander V
Podol’skii
, and T. A
Shaposhnikova
The aim of this paper is to consider the asymptotic behavior of boundary value problems in
@article{diaz+gc+podolskii+shaposhnikova2019anona,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Podol'skii, Alexander V and Shaposhnikova, T. A},doi={10.1515/anona-2017-0140},issn={2191-9496},journal={Adv. Nonlinear Anal.},number={1},pages={679--693},title={{Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition}},url={http://www.degruyter.com/view/j/anona.ahead-of-print/anona-2017-0140/anona-2017-0140.xml},volume={8},year={2019},dimensions=true}
J17
Classification of homogenized limits of diffusion problems with spatially dependent reaction over critical-size particles
J. I.
Díaz
, D.
Gómez-Castro
, T. A.
Shaposhnikova
, and M. N.
Zubova
\textcopyright 2018 Informa UK Limited, trading as Taylor & Francis Group The main goal of this paper is to characterize the change of structural behavior (i.e. the appearance of the so-called ‘strange terms’) arising in the homogenization process when applied to distributed microscopic chemical reactions taking place on fixed-bed nanoreactors, at the microscopic level, on the boundary of the particles of critical size. The presence of non-homogeneous distributed functions (Formula presented.) of the reaction kinetics may be originated by many different reasons. The case of quick oscillation is often due to own structure of the fixed bed reactor since the flux of the fluid acts on each particle in a non-homogeneous way. In some other cases, the non-homogeneous distributed functions (Formula presented.) of the reaction kinetics is artificially provoked in order to control a certain desired global effect. Our main result gives a complete classification of the strange terms according the assumed periodicity on the distributed functions (Formula presented.) of the reaction kinetic.
@article{diaz-gc-shaposhnikova-zubova2018,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Shaposhnikova, T. A. and Zubova, M. N.},doi={10.1080/00036811.2018.1441997},issn={0003-6811},journal={Appl. Anal.},keywords={Homogenization,critical sizes,diffusion processes,microscopic distributed boundary reaction,periodic asymmetric particles},number={1-2},pages={232--255},title={{Classification of homogenized limits of diffusion problems with spatially dependent reaction over critical-size particles}},url={https://www.tandfonline.com/doi/full/10.1080/00036811.2018.1441997},volume={98},year={2019},dimensions=true}
J16
The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach
@article{diaz-gc-vazquez2018,archiveprefix={arXiv},arxivid={1804.08398},author={D{\'{i}}az, J. I. and Gómez-Castro, D. and V{\'{a}}zquez, J. L.},doi={10.1016/j.na.2018.05.001},issn={0362546X},journal={Nonlinear Anal.},keywords={Bounded domains,Nonlocal elliptic equations,Schr{\"{o}}dinger operators,Super-singular potentials,Very weak solutions,Weighted spaces},pages={325--360},title={{The fractional Schr{\"{o}}dinger equation with general nonnegative potentials. The weighted space approach}},url={https://linkinghub.elsevier.com/retrieve/pii/S0362546X18301160},volume={177},year={2018},dimensions=true}
J15
Homogenization of Boundary Value Problems in Plane Domains with Frequently Alternating Type of Nonlinear Boundary Conditions: Critical Case
J. I.
Díaz
, D.
Gómez-Castro
, A. V.
Podolskiy
, and T. A.
Shaposhnikova
\textcopyright 2018, Pleiades Publishing, Ltd. In the present paper we consider a boundary homogenization problem for the Poisson’s equation in a bounded domain and with a part of the boundary conditions of highly oscillating type (alternating between homogeneous Neumman condition and a nonlinear Robin type condition involving a small parameter). Our main goal in this paper is to investigate the asymptotic behavior as ε→ 0 of the solution to such a problem in the case when the length of the boundary part, on which the Robin condition is specified, and the coefficient, contained in this condition, take so-called critical values. We show that in this case the character of the nonlinearity changes in the limit problem. The boundary homogenization problems were investigate for example in [1, 2, 4]. For the first time the effect of the nonlinearity character change via homogenization was noted for the first time in [5]. In that paper an effective model was constructed for the boundary value problem for the Poisson’s equation in the bounded domain that is perforated by the balls of critical radius, when the space dimension equals to 3. In the last decade a lot of works appeared, e.g., [6–10], in which this effect was studied for different geometries of perforated domains and for different differential operators. We note that in [6–10] only perforations by balls were considered. In papers [11, 12] the case of domains perforated by an arbitrary shape sets in the critical case was studied.
@article{diaz-gc-podolskii-shaposhikova2018doklady,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Podolskiy, A. V. and Shaposhnikova, T. A.},doi={10.1134/S1064562418030225},issn={1064-5624},journal={Dokl. Math.},mendeley-groups={homogenisation/boundary particles/critical},number={3},pages={271--276},title={{Homogenization of Boundary Value Problems in Plane Domains with Frequently Alternating Type of Nonlinear Boundary Conditions: Critical Case}},url={http://link.springer.com/10.1134/S1064562418030225},volume={97},year={2018},dimensions=true}
J14
Study of tumor growth indicates the existence of an “immunological threshold” separating states of pro- and antitumoral peritumoral inflammation
Antonio
Brú
, D.
Gómez-Castro
, Luis
Vila
, Isabel
Brú
, and Juan Carlos
Souto
@article{bru-gc2018,author={Br{\'{u}}, Antonio and Gómez-Castro, D. and Vila, Luis and Br{\'{u}}, Isabel and Souto, Juan Carlos},doi={10.1371/journal.pone.0202823},issn={1932-6203},journal={PLoS One},number={11},pages={e0202823},title={{Study of tumor growth indicates the existence of an “immunological threshold” separating states of pro- and antitumoral peritumoral inflammation}},url={http://dx.plos.org/10.1371/journal.pone.0202823},volume={13},year={2018},dimensions=true}
J13
Non existence of critical scales in the homogenization of the problem with p-Laplace diffusion and nonlinear reaction in the boundary of periodically distributed particles in n-dimensional domains when p>n
J. I.
Díaz
, D.
Gómez-Castro
, A. V.
Podolskii
, and T. A.
Shaposhnikova
Rev. la Real Acad. Ciencias Exactas, Físicas y Nat. Ser. A. Matemáticas, 2018
\textcopyright 2017, Springer-Verlag Italia. In previous works, the homogenization of the problem with p-Laplace diffusion and nonlinear reaction in the boundary of periodically distributed particles in n-dimensional domains has been studied in the cases where p≤ n. The main trait of the cases p≤ n is the existence of a critical size of the particles, for which the nonlinearity arising of the limit problem does not coincide with the non linear term of the microscopic reaction. The main result of this paper proves that in the case p> n there exists no critical size.
@article{diaz-gc-podolskii-shaposhnikova2018,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Podolskii, A. V. and Shaposhnikova, T. A.},doi={10.1007/s13398-017-0381-z},issn={1578-7303},journal={Rev. la Real Acad. Ciencias Exactas, F{\'{i}}sicas y Nat. Ser. A. Matem{\'{a}}ticas},keywords={Homogenization,Non-critical sizes,Non-linear boundary reaction,p-Laplace diffusion},number={2},pages={331--340},title={{Non existence of critical scales in the homogenization of the problem with p-Laplace diffusion and nonlinear reaction in the boundary of periodically distributed particles in n-dimensional domains when $p>n$}},url={http://link.springer.com/10.1007/s13398-017-0381-z},volume={112},year={2018},dimensions=true}
J12
Existence and uniqueness of solutions of Schrödinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications
J. I.
Díaz
, D.
Gómez-Castro
, and J.-M.
Rakotoson
Motivated mainly by the localization over an open bounded set Ωof \mathbb R^n of solutions of the Schrödinger equations, we consider the Schrödinger equation over Ωwith a very singular potential V(x) \ge C d (x, ∂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(⋅, ∂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 ∂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(⋅, ∂Ω)^ α), where α∈[0,1], of the associated operator, the corresponding parabolic problem and the study of the complex evolution Schrödinger equation in \mathbb R^n.
@article{diaz+gc+rakotoson2017schrodinger,archiveprefix={arXiv},arxivid={1710.06679},author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Rakotoson, J.-M.},doi={10.7153/dea-2018-10-04},issn={1847-120X},journal={Differ. Equations Appl.},keywords={accretive operator,complex evolution,local kato inequality,no boundary conditions,schr{\"{o}}dinger equation,very,very singular potential,weak distributional solution},number={1},pages={47--74},title={{Existence and uniqueness of solutions of Schr{\"{o}}dinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications}},url={http://arxiv.org/abs/1710.06679 http://dea.ele-math.com/10-04},volume={10},year={2018},dimensions=true}
J11
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 }\backslashmathbb 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.
@article{diaz+gc+rakotoson+temam:2018veryweak,archiveprefix={arXiv},arxivid={1710.07048},author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Rakotoson, J. M. and Temam, R.},doi={10.3934/dcds.2018023},issn={1078-0947},journal={Discrete and Continuous Dynamical Systems},number={2},pages={509--546},title={{Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach}},url={http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=14731},volume={38},year={2018},keywords={short-cv},dimensions=true}
J10
Change of homogenized absorption term in diffusion processes with reaction on the boundary of periodically distributed asymmetric particles of critical size
J. I.
Díaz
, D.
Gómez-Castro
, T. A.
Shaposhnikova
, and M. N.
Zubova
Electronic Journal of Differential Equations, 2017
@article{diaz+gc+shaposhnikova+zubova2017ejde,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Shaposhnikova, T. A. and Zubova, M. N.},journal={Electronic Journal of Differential Equations},keywords={2017,and phrases,c 2017 texas state,critical sizes,diffusion processes,homogenization,microscopic non-linear boundary reaction,periodic asymmetric particles,published july 13,submitted june 12,university},number={178},pages={1--25},title={{Change of homogenized absorption term in diffusion processes with reaction on the boundary of periodically distributed asymmetric particles of critical size}},volume={2017},year={2017},sortname={010},url={https://ejde.math.txstate.edu/},dimensions=true}
J09
On the asymptotic limit of the effectiveness of reaction–diffusion equations in periodically structured media
J. I.
Díaz
, D.
Gómez-Castro
, A.V.
Podolskii
, and T.A.
Shaposhnikova
Journal of Mathematical Analysis and Applications, 2017
@article{diaz+gomez-castro+podolskii+shaposhnikova2017jmaa,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Podolskii, A.V. and Shaposhnikova, T.A.},doi={10.1016/j.jmaa.2017.06.036},issn={0022247X},journal={Journal of Mathematical Analysis and Applications},keywords={Effectiveness factor,Homogenization,Non-critical sizes,Non-linear boundary reaction,p-Laplace diffusion},number={2},pages={1597--1613},title={{On the asymptotic limit of the effectiveness of reaction–diffusion equations in periodically structured media}},url={http://www.sciencedirect.com/science/article/pii/S0022247X17305863 http://linkinghub.elsevier.com/retrieve/pii/S0022247X17305863},volume={455},year={2017},dimensions=true}
J08
Homogenization of variational inequalities of Signorini type for the p-Laplacian in perforated domains when p ∈(1, 2)
J. I.
Díaz
, D.
Gómez-Castro
, A. V.
Podol’skii
, and T. A.
Shaposhnikova
@article{diaz+gomez-castro+podolskii+shaposhnikova2017signorini+p-Laplace+1-2,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Podol'skii, A. V. and Shaposhnikova, T. A.},doi={10.1134/S1064562417020132},journal={Doklady Mathematics},number={2},pages={151--156},title={{Homogenization of variational inequalities of Signorini type for the $p$-Laplacian in perforated domains when $p \in (1, 2)$}},volume={95},year={2017},dimensions=true}
J07
Visibility to discern local from nonlocal dynamic processes
Antonio
Brú
, D.
Gómez-Castro
, and J.C.
Nuño
Physica A: Statistical Mechanics and its Applications, 2017
We compare using visibility the usual Kardar-Parisi-Zhang (KPZ) universality class and a fractional Edward-Wilkinson (EWf) equation with correlated noise, which share the same kinetic roughening exponents. The KPZ universality class is described by an equation in terms of the usual derivatives, uncorrelated noise and therefore is intrinsically local. The second model includes fractional powers of the Laplace operator and correlated noise, both of which are nonlocal. From their scaling properties, one could be tempted to conclude that both dynamics belong to the same universality class, specifically, to the KPZ universality class. However, this is a wrong conclusion that calls the attention against the indiscriminate application of this approach in real systems without taking into consideration basic physical assumptions (e.g. locality). These examples reveal the necessity of finding new algorithms for detecting characteristics that remain unnoticed to classical scaling analysis, where only the two first moments of the interface distribution (mean and variance) are used to classify the dynamics. We show that visibility and, in particular, the kinetic roughening exponents of the visibility interface, are able to distinguish between these two dynamics which are confused by standard techniques.
@article{bru+gomez-castro+nuno2016,author={Br{\'{u}}, Antonio and Gómez-Castro, D. and Nu{\~{n}}o, J.C.},doi={10.1016/j.physa.2016.12.078},issn={03784371},journal={Physica A: Statistical Mechanics and its Applications},pages={718--723},title={{Visibility to discern local from nonlocal dynamic processes}},url={http://linkinghub.elsevier.com/retrieve/pii/S0378437116310585},volume={471},year={2017},dimensions=true}
J06
Shape differentiation of a steady-state reaction-diffusion problem arising in Chemical Engineering: the case of non-smooth kinetic with dead core
D.
Gómez-Castro
Electronic Journal of Differential Equations, 2017
@article{Gomez-Castro2017,archiveprefix={arXiv},arxivid={1708.01041},author={Gómez-Castro, D.},journal={Electronic Journal of Differential Equations},keywords={chemical engineering,dead core,reaction-diffusion,shape differentiation},number={221},pages={1--11},title={{Shape differentiation of a steady-state reaction-diffusion problem arising in Chemical Engineering: the case of non-smooth kinetic with dead core}},url={https://ejde.math.txstate.edu/},volume={2017},year={2017},dimensions=true}
@article{brezis+gc2017cras,author={Brezis, Ha{\"{i}}m and Gómez-Castro, D.},doi={10.1016/j.crma.2017.06.004},issn={1631073X},journal={Comptes Rendus Mathematique},number={7},pages={780--785},publisher={Elsevier Masson SAS},title={{Rigidity of optimal bases for signal spaces}},url={http://linkinghub.elsevier.com/retrieve/pii/S1631073X17301723},volume={355},year={2017},dimensions=true}
J04
Homogenization of the p-Laplace operator with nonlinear boundary condition on critical size particles: identifying the strange terms for some non smooth and multivalued operators
J. I.
Díaz
, D.
Gómez-Castro
, A. V.
Podol’skii
, and T. A.
Shaposhnikova
@article{Diaz+Gomez-Castro+Podolski+Shaposhnikova:2016homogenization+Heaviside,annote={Mathematics: Q3},author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Podol'skii, A. V. and Shaposhnikova, T. A.},doi={10.1134/S1064562416040098},issn={10645624},journal={Doklady Mathematics},mendeley-groups={homogenization/critical holes},number={1},pages={387--392},title={{Homogenization of the p-Laplace operator with nonlinear boundary condition on critical size particles: identifying the strange terms for some non smooth and multivalued operators}},volume={94},year={2016},dimensions=true}
J03
On the Effectiveness of Wastewater Cylindrical Reactors: an Analysis Through Steiner Symmetrization
@article{Diaz+Gomez-Castro:2014pageoph,annote={1.59
Geochemistry {\&} Geophysics: Q3},author={D{\'{i}}az, J. I. and Gómez-Castro, D.},doi={10.1007/s00024-015-1124-8},issn={0033-4553},journal={Pure and Applied Geophysics},keywords={Steiner symmetrization,Wastewater treatment,chemical reactor tanks,effectiveness},number={3},pages={923--935},publisher={Springer Basel},title={{On the Effectiveness of Wastewater Cylindrical Reactors: an Analysis Through Steiner Symmetrization}},url={http://dx.doi.org/10.1007/s00024-015-1124-8 http://link.springer.com/10.1007/s00024-015-1124-8},volume={173},year={2016},dimensions=true}
J02
The Effectiveness Factor of Reaction-Diffusion Equations: Homogenization and Existence of Optimal Pellet Shapes
J. I.
Díaz
, D.
Gómez-Castro
, and Claudia
Timofte
@article{diaz+gomez-castro+timofte2017optimal+shapes,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Timofte, Claudia},doi={10.1007/BF03377396},issn={2296-9020},journal={Journal of Elliptic and Parabolic Equations},number={1-2},pages={119--129},title={{The Effectiveness Factor of Reaction-Diffusion Equations: Homogenization and Existence of Optimal Pellet Shapes}},url={http://link.springer.com/10.1007/BF03377396},volume={2},year={2016},dimensions=true}
J01
An Application of Shape Differentiation to the Effectiveness of a Steady State Reaction-Diffusion Problem Arising in Chemical Engineering
@article{Diaz+Gomez-Castro:2015shapediff,annote={0.79
Q2: Mathematics
Q3: Mathematics, Applied},author={D{\'{i}}az, J. I. and Gómez-Castro, D.},issn={1072-669},journal={Electronic Journal of Differential Equations},pages={31--45},title={{An Application of Shape Differentiation to the Effectiveness of a Steady State Reaction-Diffusion Problem Arising in Chemical Engineering}},url={http://ejde.math.txstate.edu},volume={22},year={2015},dimensions=true}
Conference Proceedings
C03
On the influence of pellet shape on the effectiveness factor of homogenized chemical reactions
J. I.
Díaz
, D.
Gómez-Castro
, and Claudia
Timofte
In Proceedings Of The XXIV Congress On Differential Equations And Applications XIV Congress On Applied Mathematics, 2015
@inproceedings{Diaz+Gomez-Castro+Timofte:2015cedya,author={D{\'{i}}az, J. I. and Gómez-Castro, D. and Timofte, Claudia},booktitle={Proceedings Of The XXIV Congress On Differential Equations And Applications XIV Congress On Applied Mathematics},mendeley-groups={homogenization},pages={571--576},title={{On the influence of pellet shape on the effectiveness factor of homogenized chemical reactions}},year={2015},dimensions=true}
C02
Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem
@inproceedings{Diaz+Gomez-Castro:2014aims,author={Gómez-Castro, D. and D{\'{i}}az, J. I.},booktitle={Dynamical Systems and Differential Equations, Proceedings of the 10th AIMS International Conference (Madrid, Spain)},doi={10.3934/proc.2015.0379},isbn={1-60133-018-9},keywords={and phrases,semilinear concave parabolic equations,steiner rearrangement,trotter-kato},pages={379--386},publisher={American Institute of Mathematical Sciences},title={{Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem}},url={http://www.aimsciences.org/journals/displayPaperPro.jsp?paperID=11893},year={2015},dimensions=true}
C01
A mathematical proof in nanocatalysis: better homogenized results in the diffusion of a chemical reactant through critically small reactive particles
@inproceedings{diaz+gomez-castro2016ecmi,author={D{\'{i}}az, J. I. and Gómez-Castro, D.},booktitle={Progress in Industrial Mathematics at ECMI 2016},editor={Quintela, Peregrina and Barral, Patricia and G{\'{o}}mez, Dolores and Pena, Francisco J. and Rodr{\'{i}}guez, Jer{\'{o}}nimo and Salgado, Pilar and V{\'{a}}zquez-Mendez, Miguel E.},isbn={978-3-319-63082-3},publisher={Springer},title={{A mathematical proof in nanocatalysis: better homogenized results in the diffusion of a chemical reactant through critically small reactive particles}},year={2017},doi={10.1007/978-3-319-63082-3},dimensions=true}
Other
Foreword to Special Issue in Honour Juan Luis Vázquez. Discrete and Continuous Dynamical Systems 43 (3&4)
@misc{CarrilloGCTeso2023DCDSForeword,title={Foreword to Special Issue in Honour Juan Luis Vázquez. Discrete and Continuous Dynamical Systems 43 (3\&4)},year={2023},doi={10.3934/dcds.2023005},author={Carrillo, José A. and {del Teso}, Félix and Gómez-Castro, David},dimensions=true}
Aggregation-Diffusion Equations for Collective Behaviour in the Sciences
@misc{Bailo+Carrillo+GC-ICIAN,title={Aggregation-Diffusion Equations for Collective Behaviour in the Sciences},author={Bailo, Rafael and Carrillo, José A. and Gómez-Castro, David},year={2024},url={https://www.siam.org/publications/siam-news/articles/aggregation-diffusion-equations-for-collective-behavior-in-the-sciences/},note={SIAM News Blog},dimensions=true}
Homogenization and Shape Differentiation of Quasilinear Elliptic Equations
D.
Gómez-Castro
Facultad de Matemáticas, Universidad Complutense de Madrid, 2017
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, }-\backslashDelta u + \backslashnabla u \backslashcdot 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 }-\backslashDelta} with Dirichlet boundary conditions is the unique basis to approximate functions in }H_0\^1} in }L\^2} in an optimal way.
