arXiv subjects
Publications and source records attributed to Juan Casado-Díaz.
In this paper we consider some optimal control problems governed by elliptic partial differential equations. The solution is the state variable, while the control variable is, depending on the case, the coefficient of the PDE, the potential, the right-hand side. The cost functional is of integral type and involves both the state and control variables.
We consider an optimization problem related to elliptic PDEs of the form $-{\rm div}(a(x)\nabla u)=f$ with Dirichlet boundary condition on a given domain $Ω$. The coefficient $a(x)$ has to be determined, in a suitable given class of admissible choices, in order to optimize a given criterion. We first deal with the case when the cost is the so-called elastic compliance, and then we discuss the more general case when the problem is written as an optimal control problem.
We investigate optimal control problems governed by the elliptic partial differential equation $-Δu=f$ subject to Dirichlet boundary conditions on a given domain $Ω$. The control variable in this setting is the right-hand side $f$, and the objective is to minimize a cost functional that depends simultaneously on the control $f$ and on the associated state function $u$. We establish the existence of optimal controls and analyze their qualitative properties by deriving necessary conditions for optimality. In particular, when pointwise constraints of the form $α\le f\leβ$ are imposed a priori on the control, we examine situations where a {\it bang-bang} phenomenon arises, that is where the optimal control $f$ assumes only the extremal values $α$ and $β$. More precisely, the control takes the form $f=\alpha1_E+\beta1_{Ω\setminus E}$, thereby placing the problem within the framework of shape optimization. Under suitable assumptions, we further establish certain regularity properties for the optimal sets $E$. Finally, in the last part of the paper, we present numerical simulations that illustrate our theoretical findings through a selection of representative examples.
We consider a Maxwell system on $\mathbb{R}^3$ with periodic and highly oscillating coefficients. It is known that the solutions converge in the weak-$\ast$ topology of $L^\infty(0,T;\,L^2(\mathbb{R}^3))$ to the solution of a similar problem with constant coefficients given as the $H$-limits of the electric permittivity and the magnetic permeability respectively, i.e. the limit in the sense of the homogenization of linear elliptic equations with varying coefficients. However, it is not true that the elliptic corrector also provides a corrector for the solution of the Maxwell system, i.e. an approximation of the solutions in the strong topology of $L^2$. We shall prove that the oscillations in the space variable also produce oscillations in the time variable. We get a corrector consisting of adding to the elliptic corrector the sum of infinitely plane waves in the fast variable. Note that related results have been previously proved for the wave equation. Our proof is based on the two-scale convergence theory for almost periodic functions. One of the novelties is to show how this contains the classical Block decomposition.
We carry out the homogenization of a fluid-structure interaction problem consisting in the periodic inclusions of a viscous fluid in an elastic body. We get a macrostructure model where the body behaves as a viscoelastic material with a long-range memory term. Our aim is not only to get this limit problem but also to study its main properties. Using the micro-structure variables it is simple to check that it satisfies an energy conservation law assuring in particular the existence and uniqueness of solution. The difficulty is to characterize these properties using only the macroscopic system. We prove that the nonlocal term is given through a convolution kernel which exponentially decreases to zero and satisfies some positive conditions which we write in terms of the Laplace transform. These conditions can be used to directly prove the existence and uniqueness of solution. The results apply to modeling the mechanical behavior of skin as an indicator of what kind of models we should use. In a naive interpretation the fluid inclusions represent the cells and the elastic medium the extracellular matrix.
The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in $L^N(Ω)$, with $N$ the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space $H^1_0(Ω)$. However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in \cite{boc1}, and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.
We consider the optimal arrangement of two diffusion materials in a bounded open set $Ω\subset \mathbb{R}^N$ in order to maximize the energy. The diffusion problem is modeled by the $p$-Laplacian operator. It is well known that this type of problems has no solution in general and then that it is necessary to work with a relaxed formulation. In the present paper we obtain such relaxed formulation using the homogenization theory, i.e. we replace both materials by microscopic mixtures of them. Then we get some uniqueness results and a system of optimality conditions. As a consequence we prove some regularity properties for the optimal solutions of the relaxed problem. Namely, we show that the flux is in the Sobolev space $H^1(Ω)^N$ and that the optimal proportion of the materials is derivable in the orthogonal direction to the flux. This will imply that the unrelaxed problem has no solution in general. Our results extend those obtained by the first author for the Laplace operator.