Person: Gómez Castro, David
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First Name
David
Last Name
Gómez Castro
Affiliation
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Department
Análisis Matemático Matemática Aplicada
Area
Matemática Aplicada
Identifiers
10 results
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Now showing 1 - 10 of 10
Item Singular solutions for space-time fractional equations in a bounded domain(2023) Chan, Hardy; Gómez Castro, David; Vázquez, Juan LuisThis paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann-Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.- Project number: 214
Item Escenarios multimedia en formación de futuros profesores universitarios de matemáticas (ESCEMMAT-Univ)(2019) Gómez Chacón, Inés María; Baro González, Elías; Barbero, Marta; Capel Cuevas, Ángela; Caravantes Tortajada, Jorge; Contreras Tejada, Patricia; Díaz-Cano Ocaña, Antonio; Folgueira López, Marta; Gómez Castro, David; González Prieto, José Ángel; González Ortega, Jorge; Ivorra, Benjamín Pierre Paul; Juan Llamas, María Del Carmen; Melle Hernández, Alejandro; Pe Pereira, María; Prieto Yerro, María Ángeles; Sánchez Benito, María Mercedes; Ramos Del Olmo, Ángel ManuelPreparar al profesorado novel de matemáticas para una docencia universitaria de calidad, mediante el desarrollo de competencias y conocimiento estratégico para aprender a enseñar Matemáticas. Se desarrollan ejemplificaciones para ser implementadas. Item Homogenization and Shape Differentiation of Quasilinear Elliptic Equations(2018) Gómez Castro, David; Díaz Díaz, Jesús IldefonsoEsta tesis se ha divido en dos partes de tamaños desiguales. La primera parte es la componentecentral del trabajo del candidato. Se encarga de la optimización de reactores químicosde lecho fijo, y el estudio de su efectividad, como se expondrá en los siguientes párrafos. Lasegunda parte es el resultado de la visita del candidato al Prof. Häim Brezis en el InstitutoTecnológico de Israel (Technion) en Haifa, Israel. Se entra en una pregunta concreta sobrebases óptimas en L2, que es de importancia en Tratamiento de Imágenes, y que fue formuladopor el Prof. Brezis.La primera parte de la tesis, que estudio reactores químicos, se ha dividido en 4 capítulos.Estudia un modelo establecido que tiene aplicaciones directas en Ingeniería Química, y lanoción de efectividad. Una de las mayores dificultades con la que nos enfrentamos es elhecho que, por las aplicaciones en Ingeniería Química, estamos interesados en reacciones deorden menor que uni (de tipo raíz).El primer capítulo se centra en la modelización: obtener un modelo macroscópico (homogéneo)a partir de un comportamiento microscópico prescrito. A este método se le conocecomo homogeneización. La idea es considerar partículas periódicamente repetidas, de formafija G0, a una distancia ε, y que han sido reescaladas por un factor aε . La expresión habitualde este factor es aε = C0εα, donde α ≥ 1 y C0 es una constante positiva. El objetivo esestudiar los diferentes comportamientos cuando ε →0, y ya no se consideran las partículas.Primero, los casos de partículas grandes y partículas pequeños se tratan de formas distintas.Este segundo, que ha sido el central en esta tesis, se divide en subcrítico, crítico y supercrítico.En términos generales, existe un valor α∗ tal que los comportamientos de los casos α = 1(partículas grandes), 1 < α < α∗ (partículas subcríticos), α = α∗ (partículas críticos) yα > α∗ (partículas supercríticos) son significativamente distintos...Item Asymptotic Simplification of Aggregation-Diffusion Equations Towards the Heat kernel(Archive for Rational Mechanics and Analysis, 2023) Carrillo Menéndez, José; Gómez Castro, David; Yao, Yao; Zeng, Chongchun- Project number: 252
Item Escenarios Multimedia en Formación de Futuros Profesores Universitarios de Matemáticas (ESCEMMAT-Univ) (2ªFase)(2020) Gómez Chacón, Inés María; Díaz-Cano Ocaña, Antonio; Folgueira López, Marta; Gómez Castro, David; Ivorra, Benjamín Pierre Paul; Martínez Aguinaga, Francisco Javier; Ortuño, M. T.; Ramos del Olmo, Ángel Manuel; González Ortega, Jorge; González Prieto, José Ángel; Melle Hernández, Alejandro; Sanchez Benito, Mercedes; Barbero, Marta; Pe Pereira, María; Capel Cuevas, ÁngelaPreparar al profesorado novel de matemáticas para una docencia universitaria de calidad, en el desarrollo de competencias y conocimiento estratégico para aprender a enseñar Matemáticas. Ejemplificaciones e instrumentos de evaluación para el aula. Item Study of tumor growth indicates the existence of an “immunological threshold” separating states of pro- and antitumoral peritumoral inflammation(PLoS ONE, 2018) Bru Espino, Antonio Leonardo; Gómez Castro, David; Vila, Luis; Brú, Isabel; Souto, J. C.Background Peritumoral inflammation—a response mainly involving polimorphonuclear neutrophils—has traditionally been thought protumoral in its effects. In recent years, however, a number of studies have indicated that it may play an important antitumoral role. This discrepancy has been difficult to explain. Methods and findings This work describes a tool for simulating tumor growth that obeys the universal model of tumor growth dynamics, and shows through its use that low intensity peritumoral inflammation exerts a protumoral effect, while high intensity inflammation exerts a potent antitumoral effect. Indeed, the simulation results obtained indicate that a sufficiently strong antitumoral effect can reverse tumor growth, as has been suggested several times in the clinical literature. Conclusions The present result indicate that an ‘immunological threshold’ must exist, marking the boundary between states in which peritumoral inflammation is either harmful or beneficial. These findings lend support to the idea that stimulating intense peritumoral inflammation could be used as a treatment against solid tumors.Item Singular solutions for fractional parabolic boundary value problems(2020) Chan, Hardy; Gómez Castro, David; Vázquez, Juan LuisThe standard problem for the classical heat equation posed in a bounded domain Ω of Rn 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 nonzero boundary data must be singular, i.e., the solution u(t, x) blows up as x approaches ∂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 Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractionalLaplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kerne.Item A nonlocal memory strange term arising in the critical scale homogenization of diffusion equations with dynamic boundary conditions(Electronic Journal of Differential Equations, 2019) Díaz Díaz, Jesús Ildefonso; Gómez Castro, David; Shaposhnikova, Tatiana A.; Zubova, Maria N.Our main interest in this article is the study of homogenized 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.Item Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence(2022) Boccardo, L.; Gómez Castro, David; Díaz Díaz, Jesús IldefonsoIn this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem −div(M(x)∇u)=−div(E(x)u)+f in a bounded domain of RN with N≥3. We are particularly interested in singular E with divE≥0. We start by recalling known existence results when |E|∈LN that do not rely on the sign of divE. Then, under the assumption that divE≥0 distributionally, we extend the existence theory to |E|∈L2. 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 divE∼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 strong maximum principle fails in the presence of such singular drift terms E.Item Singular boundary behaviour and large solutions for fractional elliptic equations(Journal of the London Mathematical Society, 2022) Abatangelo, Nicola; Gómez Castro, David; Vázquez, Juan LuisWe perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace–Poisson equation with zero boundary data. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term 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, for equations driven by a wide class of nonlocal fractional operators, different blowup 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 operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function.