Person:
Díaz Díaz, Gregorio

First Name

Gregorio

Last Name

Díaz Díaz

Affiliation

Universidad Complutense de Madrid

Faculty / Institute

Ciencias Químicas

Department

Análisis Matemático Matemática Aplicada

Area

Matemática Aplicada

Identifiers

Full item page

Search Results

Now showing 1 - 10 of 12

On a stochastic parabolic PDE arising in Climatology
(Springer, 2002) Díaz Díaz, Gregorio; Díaz Díaz, Jesús Ildefonso
We study the existence and uniqueness of solutions of a nonlinear stochastic pde proposed by R. North and R. F. Cahalan in 1982 for the modeling of non-deterministic variability (as, for instance, the volcano actions) in the framework of energy balance climate models. The more delicate point concerns the uniqueness of solutions due to the presence of a multivalued graph β in the right hand side of the equation. In contrast with the deterministic case, it is possible to prove the uniqueness of a suitable weak solution associated to each given monotone (univalued and discontinuous) section b of the maximal monotone graph β. We get some stability results when the white noise converges to zero.
Remarks on the Monge-Ampère equation: some free boundary problems in geometry
(UCM, 2012) Díaz Díaz, Gregorio; Díaz Díaz, Jesús Ildefonso
This paper deals with several qualitative properties of solutions of some stationary and parabolic equations associated to the Monge-Ampère operator. Mainly, we focus our attention in the occurrence of a free boundary (separating the region where the solution u is locally a hyperplane, and so were the Hessian D2u is vanishing from the rest of the domain). Among other thinfs, we take advantage of these proceedings to give a detailed version of some results already announced long time ago when dealing wiht other fully nonlinear equations (see the 1979 àèr by the authors on other parabolic equations, remark 2.25 of the 1985 monograph by the second author and the 1985 paper by the first author. In particular, our results apply to suitable formulations of the Gauss curvature flow and of the worn stones problems intensively studied in the literature.
Uniqueness of the boundary behavior for large solutions to a degenerate elliptic equation involving the ∞–Laplacian
(Springer, 2003) Díaz Díaz, Gregorio; Díaz Díaz, Jesús Ildefonso
In this note we estimate the maximal growth rate at the boundary of viscosity solutions to −∆∞u + λ|u| m−1 u = f in Ω (λ > 0, m > 3).In fact, we prove that there is a unique explosive rate on the boundary for large solutions. A version of Liouville Theorem is also obtained when Ω = R N
Controlled boundary explosions: dynamics after blow-up for some semilinear problems with global controls
(2022) Casal, Alfonso C.; Díaz Díaz, Gregorio; Díaz Díaz, Jesús Ildefonso; Vegas, José Manuel
The main goal of this paper is to show that the blow up phenomenon (the explosion of the L ∞-norm) of the solutions of several classes of evolution problems can be controlled by means of suitable global controls α(t) (i.e. only dependent on time ) in such a way that the corresponding solution be well defined (as element of L1 loc(0, +∞ : X), for some functional space X) after the explosion time. We start by considering the case of an ordinary differential equation with a superlinear term and show that the controlled explosion property holds by using a delayed control (built through the solution of the problem and by generalizing the nonlinear variation of constants formula, due to V.M. Alekseev in 1961, to the case of neutral delayed equations (since the control is only in the space W−1,q′ loc (0, +∞ : R), for some q > 1). We apply those arguments to the case of an evolution semilinear problem in which the differential equation is a semilinear elliptic equation with a superlinear absorption and the boundary condition is dynamic and involves the forcing superlinear term giving rise to the blow up phenomenon. We prove that, under a suitable balance between the forcing and the absorption terms, the blow up takes place only on the boundary of the spatial domain which here is assumed to be a ball BR and for a constant as initial datum.
Solutions of reaction-diffusion equations blowing-up on the parabolic boundary
(Elsevier, 1994) Díaz Díaz, Jesús Ildefonso; Bandle, Chatherine; Díaz Díaz, Gregorio
We study the existence, the asymptotic behaviour near the parabolic boundary and the uniqueness of the solutions of nonlinear reaction-diffusion equations, which blow up on the parabolic boundary. We extend some results for elliptic problems given in ([1], [4]). A fondamental tool is the construction of suitable upper and lower solutions.
On an evolution problem associated to the modelling of incertitude into the environment
(Elsevier, 2007) Díaz Díaz, Gregorio; Díaz Díaz, Jesús Ildefonso; Faghloumi, C.
We consider a mathematical model, posed by J.E. Scheinkman, simulating that an industrial project takes place into the environment without destroying it. We introduce a change of variable leading the formulation to a nonlinear evolution problem which we study by means of L-infinity-accretive operator techniques. We prove that under suitable conditions there is extinction in finite time, which corresponds to some special behaviour of the solution of the original stochastic control problem.
problemas en ecuaciones en derivadas parciales con no linealidades sobre operadores diferenciales de segundo orden
(Universidad Complutense de Madrid, 2015) Díaz Díaz, Gregorio; Díaz Díaz, Jesús Ildefonso
On the free boundary associated to the stationary Monge–Ampère operator on the set of non strictly convex functions
(American Institute of Mathematical Sciences, 2015-04) Díaz Díaz, Gregorio; Díaz Díaz, Jesús Ildefonso
This paper deals with several qualitative properties of solutions of some stationary equations associated to the Monge-Ampere operator on the set of convex functions which are not necessarily understood in a strict sense. Mainly, we focus our attention on the occurrence of a free boundary (separating the region where the solution u is locally a hyperplane, thus, the Hessian D(2)u is vanishing from the rest of the domain). In particular, our results apply to suitable formulations of the Gauss curvature flow and of the worn stones problems intensively studied in the literature.
Expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation
(Elsevier, 2010-03) Díaz Díaz, Gregorio; Alarcón, S.; Letelier, René; Rey Cabezas, Jose María
The main goal of this paper is to study the asymptotic expansion near the boundary of the large solutions of the equation -Delta u + lambda u(m) = f in Omega, where lambda > 0, m > 1, f is an element of c(Omega), f >= 0, and Omega is an open bounded set of R-N, N > 1, with boundary smooth enough. Roughly speaking, we show that the number of explosive terms in the asymptotic boundary expansion of the solution is finite, but it goes to infinity as in goes to 1. We prove that the expansion consists in two eventual geometrical and non-geometrical parts separated by a term independent on the geometry of partial derivative Omega, but dependent on the diffusion. For low explosive sources the non-geometrical part does not exist; all coefficients depend on the diffusion and the geometry of the domain by means of well-known properties of the distance function dist(x, partial derivative Omega). For high explosive sources the preliminary coefficients, relative to the non-geometrical part, are independent on Omega and the diffusion. Finally, the geometrical part does not exist for very high explosive sources consists in two eventual geometrical and non-geometrical parts, separated by a term independent on the geometry of $\partial\Omega$∂Ω, but dependent on the diffusion. For low explosive sources the non-geometrical part does not exist; all coefficients depend on the diffusion and the geometry of the domain by means of well-known properties of the distance function ${\rm dist}(x,\partial\Omega)$dist(x,∂Ω). For high explosive sources the preliminary coefficients, relative to the non-geometrical part, are independent on $\Omega$Ω and the diffusion. Finally, the geometrical part does not exist for very high explosive sources.
Uniqueness and continuum of foliated solutions for a quasi-linear elliptic equation with a non-lipschitz nonlinearity
(Taylor & Francis, 1992-06-17) Díaz Díaz, Jesús Ildefonso; Barles, G.; Díaz Díaz, Gregorio