\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small 2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal,\\ \emph{Electronic Journal of Differential Equations}, Conference 22 (2015), pp. 31--45.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{31} \title[\hfilneg EJDE-2015/Conf/22 \hfil An application of shape differentiation] {An application of shape differentiation to the effectiveness of a steady state reaction-diffusion problem arising in chemical engineering} \author[J. I. D\'iaz, D. G\'omez-Castro \hfil EJDE-2015/conf/22 \hfilneg] {Jes\'us Ildefonso D\'iaz, David G\'omez-Castro} \dedicatory{Dedicated to our colleague and good friend Alfonso Casal on his 70th birthday} \address{Jes\'us Ildefonso D\'iaz \newline Instituto de Matem\'atica Interdisciplinar and Dpto. de Matem\'a tica Aplicada\\ Facultad de Ciencias Matem\'aticas.Universidad Complutense de Madrid\\ Plaza de las Ciencias 3, 28040 Spain} \email{jidiaz@ucm.es} \address{David G\'omez-Castro \newline Instituto de Matem\'atica Interdisciplinar and Dpto. de Matem\'a tica Aplicada\\ Facultad de Ciencias Matem\'aticas, Universidad Complutense de Madrid\\ Plaza de las Ciencias 3, 28040 Spain} \email{dgcastro@ucm.es} \thanks{Published November 20, 2015} \subjclass[2010]{35J61, 46G05, 35B30} \keywords{Shape differentiation; effectiveness factor; reaction-diffusion; \hfill\break\indent chemical engineering; numerical experiments} \begin{abstract} In applications it is common to arrive at a problem where the choice of an optimal domain is considered. One such problem is the one associated with the steady state reaction diffusion equation given by a semilinear elliptic equation with a monotone nonlinearity $g$. In some contexts, in particular in chemical engineering, it is common to consider the functional given by the integral of this nonlinear term of the solution dived by the measure of the domain $\Omega $ in which the pde takes place. This is often related with the effectiveness of the reaction. In this paper our aim is to study the differentiability of such functional as study connected to the optimality of the best chemical reactor. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{hyp}[theorem]{Hypothesis} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction and statement of results} The main goal of this article is to analyze the differentiability, with respect to the domain $\Omega $, of the \emph{effectiveness factor} \begin{equation*} \mathcal E ( \Omega) = \frac{1}{|\Omega |}\int_{\Omega }\beta (w_{\Omega })dx \end{equation*} where $w_{\Omega }$ is the solution of the problem arising in chemical catalysis \cite{Aris:1975,Aris+Strieder:1973} \begin{equation} \begin{gathered} -\Delta w+\beta (w)=\hat{f}, \quad \text{in }\Omega , \\ w=1, \quad \text{on }\partial \Omega . \end{gathered} \label{eq:model problem w} \end{equation} The model can be obtained in different ways, including homogenization techniques: see, e.g. \cite{Conca+Diaz+Timofte:2003} and \cite{Conca+Diaz+Linan+Timofte:2004}. By introducing the change in variable $u=1-w $ the problem can be reformulated as \begin{equation} \begin{gathered} -\Delta u+g(u)=f, \quad \text{in }\Omega , \\ u=0, \quad \text{on }\partial \Omega . \end{gathered} \label{eq:model problem u} \end{equation} where $g(u)=\beta (1)-\beta (1-u)$ and $f=\beta (1)-\hat{f}$. In this case instead of the effectiveness factor we can study $\eta (\Omega) = 1 - \mathcal E (\Omega)$ \begin{equation} \eta (\Omega )=\frac{1}{|\Omega |}\int_{\Omega }g(u_{\Omega })dx\,, \label{eq:ineffectiveness} \end{equation} where $u_{\Omega }$ is the solution of \eqref{eq:model problem u}. In the chemical context this factor represents the amount of reaction taking place. This kind of problems fall with the family of problems studied by several authors in the literature (see, e.g. \cite{Murat+Simon:1976,Pironneau:2012optimal,Simon:1980differentiation} and the references therein). In the most general case this family of problems may be described by: \begin{equation} \begin{gathered} A(u(D))=f, \quad \text{in }D, \\ B(u(D))=g, \quad \text{on }\partial D \end{gathered} \label{eq:formulation Simon} \end{equation} and the functional can by given generally as \begin{equation*} J(D)=\int_DC(u_D)dx, \end{equation*} where $A,B,C$ may contain also some derivatives of $u_D$. In this paper we shall concentrate our attention in problem \eqref{eq:model problem u} and we shall provide elementary and direct proofs of results which could be obtained from the general theory but under stronger assumptions (see, for instance, the statement taken from \cite{Simon:1980differentiation} which is reproduced here in Section 2). As mentioned before, our aim is to study the differentiability of functional \eqref{eq:ineffectiveness}. We consider a fixed domain open bounded regular set of $\mathbb{R}^n$, $\Omega _0$, and study its deformations given by a function $\theta :\mathbb{R}^n\to \mathbb{R}^n$, so that the new domain is $\Omega =(Id+\theta )\Omega _0$. We consider, as it is the case in chemistry catalysis, $g$ and $f$ such that $0\leq u\leq 1$. We also mention that this kind of differentiation result also appears in many other contexts. Besides the above mentioned references we recall here the articles \cite{Dervieux:1980perturbationplasma} for a linear problem with a Dirichlet constant boundary condition and \cite{Mignot+Murat+Puel:1979variation} were a semilinear equation arising in combustion was considered (corresponding, in our formulation to take $g(u)=-e^{u}$). To obtain this properties in the sense of derivatives, we consider two approaches, mimicking the approach in differential geometry. We first consider the global differentiability of solutions (as it was done in the linear cases in \cite{Henrot+Pierre:2005,Allaire:2007} and the most general case in \cite{Simon:1980differentiation}), which unfortunately requires derivatives in spaces of too regular functions, and then we take advantage of the differentiation along curves (the approach followed in \cite {Sokolowski+Zolesio:1991}). Let us call, for simplicity, $u_{\Omega }$ the solution of \eqref{eq:model problem u}. This corresponds to the Lagrangian understanding of the problem in the sense that the functional under study is study in terms of the direct domain $\Omega $. However, we can consider the Eulerian understanding of the problem by recalling that in this family of domains, $\Omega =(Id+\theta )\Omega _0,$ we can introduce a new function $v_{\theta }:\Omega_0\to \mathbb{R}$ defined by \begin{equation} v_{\theta }=(I+\theta )^{\ast }u_{(I+\theta )\Omega _0}=u_{(I+\theta )\Omega _0}\circ (I+\theta ), \label{eq:defn v theta} \end{equation} simplifying the study of the differentiability of $u_{\Omega }$ and the functional $\eta (\Omega )$ with respect to $\Omega$. Our proof relies heavily on the Implicit Function Theorem. The application of this theorem requires an uniform choice of functional space, which would require some additional information on $u$. This kind of problems in the functional setting is well portrayed in \cite{Brezis:1999}. For the nonlinearity $g$ we shall consider the following assumptions: \begin{hyp} \label{hyp:g} \rm $g$ is nondecreasing \end{hyp} \begin{hyp}\label{hyp:g Nemitskij} \rm The Nemitskij operator for $g$ (which we will denote again by $g$ in some circumstances, as a widely accepted abuse of notation) $G:H^{1}(\Omega ) \to L^{2}(\Omega )$ defined by \begin{equation} G(u) = g\circ u \label{eq:Netmiskij operator for g} \end{equation} is of class $C^{m}$ for some $m\geq 1$. \end{hyp} We recall that Hypothesis \ref{hyp:g Nemitskij} immediately implies that $[DG](v)\varphi =g'(v)\varphi $ for $\varphi ,v\in H^{1}(\Omega )$ and that if $G$ is of class $\mathcal{C}^{k}$ with $k>1$ then necessarily $ g(s)=as+b$ for some $a,b\in \mathbb{R}$. Our first result collects some general results on the differentiability of the solution $u_{\Omega }$ with respect to $\Omega $: \begin{theorem}\label{thm:differentiability of solution} Let $g$ satisfy Hypothesis \ref{hyp:g} and \ref{hyp:g Nemitskij}. Then, the map $W^{1,\infty }(\mathbb{R}^n,\mathbb{R}^n) \to H_0^{1}(\Omega _0)$, \[ \theta \mapsto v_{\theta } \] (where $v_{\theta }$ is defined by \eqref{eq:defn v theta}) is of class $\mathcal{C}^{l}$ in a neighbourhood of 0 if $f\in H^{k}(\mathbb{R}^n)$ where $l=\min \{k,l\}$. Furthermore, the application $u:W^{1,\infty }(\mathbb{R}^n,\mathbb{R}^n) \to L^{2}(\mathbb{R} ^n)$, \[ \theta \mapsto u_{(I+\theta (\Omega _0))} \] (where $u_{\theta }$ is extended by zero outside $(I+\theta )(\Omega _0)$) is differentiable at $0$. In fact $u':W^{1,\infty }(\mathbb{R}^n, \mathbb{R}^n)\to H^{1}(\Omega )$ and \begin{equation*} u'(0)\theta +\nabla u_{\Omega _0}\cdot \theta \in H_0^{1}(\Omega ). \end{equation*} \end{theorem} As in differential geometry, to compute a derivative we can take two routes. The first one is to show the existence of a global derivative, and this allows to compute some properties of our functions. The other one, is to compute the derivative along curves. \begin{definition} \label{def1} \rm We say that $\Phi$ is a curve of deformations if $\Phi: [0,T) \to W^{1,\infty} (\Omega_0)$ with $\det \Phi (\tau) > 0$. \end{definition} \begin{hyp} \label{hyp:Phi} \rm We will say that $\theta$ is a curve of small perturbations of the identity (with direction $V$) if $\Phi (\tau) = I + \theta(\tau)$ is a curve of deformations and \begin{enumerate} \item $\theta:[0,T) \to W^{1,\infty} (\mathbb{R}^n)$ is differentiable at $0$, \item $\theta(0) = 0$, \item $\theta'(0) = V$. \end{enumerate} \end{hyp} Sometimes we consider higher order derivatives too. We will refer to $ \theta $ or $\Phi $ indistinctively, since they relate by $\Phi (\tau )=I+\theta (\tau )$. Thus, the above theorem leads to: \begin{corollary} Let $\Phi $ be a a curve of deformations of class $\mathcal{C}^{k}$. Then $\tau \mapsto v_{\theta (\tau )}$ is of class $\mathcal{C}^{l}$ with $l=\min\{m,k\}$. \end{corollary} Our second result concerns the characterization of $u'$. \begin{theorem} \label{thm:directional diff} Let $g$ satisfy Hypothesis~\ref{hyp:g} and \ref{hyp:g Nemitskij}. Let $\theta $ be a curve satisfying assumptions~\ref{hyp:Phi}. Then $u$ is differentiable along $\Phi $ at least at $0$. That is, the directional derivative $\frac{d}{d\tau }(u\circ \Phi )$ exists, and it is the solution $u'$ of \begin{equation} \begin{gathered} -\Delta u'+\lambda g'(u_{\Omega _0})u'=0 \quad \text{in }\Omega _0, \\ u'=-\nabla u_{\Omega _0}\cdot V, \quad \text{on }\partial \Omega _0. \end{gathered} \label{eq:shape optimization linear BVP} \end{equation} \end{theorem} We point out that the above result shows, in other terms, that $u'(0)\theta $ is the unique weak solution of \begin{equation} \begin{gathered} -\Delta u'+\lambda g'(u_{\Omega _0})u'=0, \quad \text{in }\Omega _0, \\ u'=-\nabla u_{\Omega _0}\cdot \theta , \quad \text{on }\partial \Omega _0. \end{gathered} \label{eq:u' in terms of theta} \end{equation} As consequence we have the following result. \begin{corollary}\label{cor:continuity u'} The function $u':W^{1,\infty }(\mathbb{R}^n,\mathbb{R}^n)\to H^{1}(\Omega )$ is continuous. In fact, since the solution $u$ of \eqref{eq:model problem u} $u\in W^{2,p}(\Omega )$ for any $p\in [ 1,+\infty )$ then for any $q\in [ 1,p]$, \begin{align*} |u'(0)(\theta )|_{q} & \leq c|\nabla u\cdot \theta |_{L^{p}(\partial \Omega _0)} \\ &\leq c|\theta |_{\infty }|\nabla u_{\Omega_0}|_{L^{p}(\partial \Omega _0)} \\ & \leq c(p)|\theta |_{\infty }|u_{\Omega _0}|_{W^{2,p}(\Omega _0)}. \end{align*} \end{corollary} Concerning the differentiability of the effectiveness factor functional we have the following theorem. \begin{theorem} \label{thm:differentiation effectiveness} Under the assumptions of Theorem \ref{thm:differentiability of solution}, let \begin{equation} \hat \eta(\theta )=\int_{(I+\theta )\Omega _0}g(u_{(I+\theta )\Omega_0})dx. \end{equation} Then ${\eta }$ is of class $\mathcal{C}^{m}$ in a neighbourhood of $0$. It holds that \begin{equation} \hat \eta ^{(m)}(0)(\theta _1,\cdots ,\theta _{m}) =\int_{\Omega _0}\frac{{d^n}}{d\theta _n\cdots d\theta _1}(g(v_{\theta }) J_{\theta })\,dx. \label{eq:higher order derivatives of E} \end{equation} Its first derivative can be expressed in terms of $u$, \begin{equation} \hat \eta '(0)(\theta )=\int_{\Omega _0} \left( g'(u_{\Omega _0})u'\ +\operatorname{div}(g(u_{\Omega _0})\theta ) \right) d x, \\ \end{equation} and if $\partial G$ is Lipschitz \begin{equation} \hat \eta '(0)(\theta )=\int_{\Omega _0} g'(u_{\Omega _0})u'\,dx+g(0)\int_{\partial \Omega _0}\theta \cdot n\,dS, \\ \end{equation} where $u'=u'(0)(\theta )$. \end{theorem} \begin{corollary} \label{cor: non isovolumetric effectiveness derivative} Under the assumptions of Theorem \ref{thm:differentiability of solution} it holds that \begin{equation*} \eta '(\theta )= \frac {1}{|\Omega_0| } \Big( \int_{\Omega _0}g'(u_{\Omega _0})u'\,dx -\eta (0)\int_{\partial \Omega _0}\theta \cdot n \,dS \Big). \end{equation*} \end{corollary} \begin{corollary} \label{cor:isovolumetric effectiveness derivative} Under the assumptions of Theorem \ref{thm:differentiability of solution} if $\Phi$ is a volume preserving curve then \begin{equation*} \eta '(\theta )= \frac{1}{|\Omega_0|} \int_{\Omega _0}g'(u_{\Omega _0})u' \,dx\,. \end{equation*} \end{corollary} We point out that if $g$ is Lipschitz (i.e. $g\in W^{1,\infty }(\mathbb{R))}$ then we obtain \begin{equation*} |\eta (\theta )-\eta (0)|=|\eta '(0)(\lambda \theta )|\leq c|g'|_{\infty }|u|_{W^{2,p}}|\theta |_{\infty }. \end{equation*} This allows to get some generalizations of the last result in cases in which the absorption term $g$ is not so regular, as for instance when $\beta(w)=w^{q}$ and $q\in (0,1)$. Nevertheless, if there is a non-empty dead core (in the literature the dead core is defined as $\{x\in \Omega : w_\Omega (x) = 0 \}$ where $w_\Omega$ is the solution of \eqref{eq:model problem w}) some additional arguments must be developed, in the line of \cite{Diaz+Rakotoson:2010}, where some unbounded potentials are considered. This will the subject of a separated paper by the authors \cite{Diaz+Gomez-Castro:2015veryweak}. We end this paper by presenting, in Section 5, some applications of the above results in terms of the Schwarz and Steiner symmetrization as well as by illustrating them for some special families of domains by means of some numerical experiences. \section{Functional setting: Nemitskij operators and the implicit function theorem} Let us formalize what we mean by a shape functional. At the most fundamental level it should be a function defined over a set of domain, that is defined over $\mathfrak{C}\subset \mathcal{P}(\mathbb{R}^n)$. Since we want to differentiate we, at the very least, need to define proximity, that is a way to define \emph{neighbourhood of a set}. As it is usual in the literature of shape optimization we work over the set of weakly differentiable bounded deformations with bounded derivative, the Sobolev space $W^{1,\infty }(\mathbb{R}^n,\mathbb{R}^n)$. \begin{definition} \label{def2} \rm We say that $J$ is defined on a neighbourhood of $\Omega_0 \subset \mathbb{R} ^n$ if there exists $U$ a neighbourhood of $0$ on $W^{1,\infty} (\mathbb{R} ^n , \mathbb{R}^n)$ such that $J$ is defined over $\{(Id + \theta) (\Omega_0): \theta \in U \}$. We say that $J$ is differentiable at $\Omega_0$ if the application $W^{1,\infty}(\mathbb{R}^n; \mathbb{R}^n) \to \mathbb{R}$, \[ \theta \mapsto J ( (Id + \theta) (\Omega_0) ) \] is differentiable at 0. \end{definition} We present a sufficient condition so that Hypothesis \ref{hyp:g Nemitskij} holds. This is widely used in the context of partial differential equations, but as far as we know no reference is known besides it being an exercise in \cite{Henry:1993}. That being the case we provide the usual proof. Other conditions, mainly on the growth of $g$ can be considered so that assumptions~\ref{hyp:g}.\ref{hyp:g Nemitskij} holds. \begin{lemma}\label{lem:continuity of Nemitskij} Let $g \in W^{2,\infty} (\mathbb{R})$. Then the Nemitskij operator \eqref{eq:Netmiskij operator for g} in the sense $L^p (\Omega) \to L^2 (\Omega)$ is of class $\mathcal{C}^1$ for all $p>2$. In particular, Hypothesis \ref{hyp:g Nemitskij} holds. \end{lemma} \begin{proof} Let us define $G$ the Nemitskij operator defined in \eqref{eq:Netmiskij operator for g}. Consider it $G: L^p (\Omega) \to L^2 (\Omega)$ for $p \ge 2$. We first have that, for $L = \max \{ \|g\|_\infty, \|g'\|_\infty, \|g'' \|_\infty \}$ $$ \|G(u) - G(v) \|_{L^2}^2 = \int_{\Omega} |g(u) - g(v)|^2 dx \le L \int_{\Omega} |u - v|^2 dx 2 $$ so that $F$ is continuous. For $p>2$ let $\varphi \in \mathcal C^\infty (\Omega)$ we compute $$ \|g(u + \varphi ) - g(u) - g'(u) \varphi \|_{L^2}^2 = \int_{\Omega} | g' (\xi (x)) - g' (u(x)) |^2 |\varphi (x) |^2 d x $$ for some function $\xi$ by the intermediate value theorem. We, of course, have that \begin{gather*} |g'(\xi(x)) - g' (u((x)) | \le L | \xi (x) - u(x) | \le L | \varphi (x) | \\ |g'(\xi(x)) - g' (u(x)) | \le 2 L \\ |g'(\xi(x)) - g'(u(x)) | \le L 2^{1-\alpha} |\varphi (x) |^{\alpha} , \quad \forall \alpha \in (0,1)\,. \end{gather*} Therefore, $$ \|g(u + \varphi ) - g(u) - g'(u) \varphi\| _{L^2}^2 \le L^2 2^{2 - 2\alpha} \int_{\Omega} |\varphi(x)|^{2 + 2\alpha} dx\,. $$ Let $2 < p < 4$ then we have that $p = 2 + 2\alpha $ with $0 < \alpha < 1$. We then have that $$ \|g(u + \varphi ) - g(u) - g'(u) \varphi\| _{L^2} \le L 2^{1 - \alpha} \| \varphi(x)\|_{L^p}^{1 + \alpha} $$ which proves the Frechet differentiability. Of course for $p > 4$ we have that $L^p(\Omega) \hookrightarrow L^3 (\Omega)$. Furthermore, for any given dimension $n$ we have Sobolev inclusions $H^1 (\Omega) \hookrightarrow L^p (\Omega)$ with $p > 2$, proving the differentiability. \end{proof} Some other well-known results are quoted now. \begin{theorem} Let $g\in W^{1,p}(\mathbb{R}^n)$. Then the map $\mathfrak{G}:W^{1,\infty }(\mathbb{R}^n,\mathbb{R}^n) \to L^{p}(\mathbb{R}^n)$ given by $\theta \mapsto g\circ (I+\theta )$ is differentiable in a neighbourhood of $0$ and \begin{equation*} \mathfrak{G}'(0)=(\nabla g)\circ (I+\theta )\,. \end{equation*} \end{theorem} \begin{theorem}[{\cite[Lemme 5.3.3.]{Henrot+Pierre:2005}}] \label{thm:composition from the right} Let $g\in W^{1,p}(\mathbb{R}^n)$, \begin{equation*} \Psi :W^{1,\infty }(\mathbb{R}^n,\mathbb{R}^n)\to W^{1,\infty }( \mathbb{R}^n,\mathbb{R}^n) \end{equation*} continuous at $0$ with $\Psi (0)=I$, $W^{1,\infty }(\mathbb{R}^n,\mathbb{R}^n) \to L^{p}(\mathbb{R} ^n)\times L^{\infty }(\Omega )$, \[ \theta \mapsto (g(\theta ),\Psi (\theta )) \] differentiable at $0$, with $g(0)\in W^{1,p}(\mathbb{R}^n)$ and \begin{equation*} g'(0):W^{1,\infty}(\mathbb{R}^n,\mathbb{R}^n)\to W^{1,p}(\mathbb{R}^n) \end{equation*} continuous. Then the application $\mathfrak{G}:W^{1,\infty }(\mathbb{R}^n,\mathbb{R}^n) \to L^{p}(\mathbb{R}^n)$, \[ \mathfrak{G}(\theta) = g(\theta )\circ \Psi (\theta ) \] is differentiable at $0$ and \begin{equation*} \mathfrak{G}'(0)=g'(0)+\nabla g(0)\cdot \Psi '(0). \end{equation*} \end{theorem} To conclude this section we state a classical result. \begin{theorem}[Implicit Function Theorem] \label{thm:implicita sobolev} Let $X,Y$ and $Z$ be Banach spaces and let $U,V $ be neighbourhoods on $X$ and $Y$, respectively. Let $F:U\times V\to Z$ be continuous and differentiable, and assume that $ D_{y}F(0,0)\in \mathcal{L}(Y,Z)$ is bijective. Let assume, further, that $F(0,0)=0$. Then there exists $W$ neighbourhood of $0$ on $X$ and a differentiable map $f:W\to Y$ such that $F(x,f(x))=0$. Furthermore, for $x$ and $y$ small, $f(x)$ is the only solution $y$ of the equation $ F(x,y)=0$. If $F$ is of class $\mathcal{C}^{m}$ then so is $f$. \end{theorem} \section{Differentiation of solutions. Proof of Theorems \ref{thm:differentiability of solution} and \ref{thm:directional diff}} For the reader convenience we repeat here the general result in \cite{Simon:1980differentiation}: \begin{theorem} Let $D$ be a bounded domain such that $\partial D$ be a piecewise $\mathcal{C}^1$ and assume that $D$ is locally on one side of $\partial D$. Let $u_0$ be the solution of \eqref{eq:formulation Simon}. Let us use the notation $\mathcal{C}^k = \mathcal{C}^k (\mathbb{R}^n, \mathbb{R}^n)$ and $k \ge 1$. Assume that \begin{equation} u(\theta) \in W^{m,p} ((I+ \theta) D) \end{equation} and that for every open set $D'$ close to $D$ (for example $D'= (I + \theta) D$ for small $\theta'$ in the norm of $\mathcal{C}^k$), \begin{equation} \begin{gathered} A : W^{m,p} (D') \to \mathcal{D}'(D') \\ B: W^{m,p} (D^{\prime 1,1} (D') \\ C: W^{m,p} (D^{\prime 1 }(D') \\ A,B,C : W^{m-1,p} (D') \to \mathcal{D}'(D) \text{ differentiable } \end{gathered} \end{equation} and $\mathcal{C}^k \to W^{m,p}$: $\theta \mapsto u(\theta) \circ (I + \theta)$ is differentiable at $0$. Then: \begin{enumerate} \item The solution is differentiable in the sense that $u:\mathcal{C}^{k}\to W_{\rm loc}^{m-1,p}(D)$ is differentiable and the derivative the local derivative $u'$ in the direction of $\tau $ satisfies \begin{equation} \dfrac{\partial A}{\partial u}(u_0)u'=0,\quad \text{in }D. \end{equation} \item If $\theta \mapsto B(u(\theta ))\circ (I+\theta )$ is differentiable at $0$, into $W^{1,1}(D)$, $B(u_0)\in W^{2,1}(D)$ and $g\in W^{2,1}(\mathbb{R}^n)$, then $u'$ satisfies \begin{equation} \frac{\partial B}{\partial u}(u_0)u'=-\tau \cdot n\frac{\partial }{\partial n}(B(u_0)-g). \end{equation} \item If $\theta \mapsto C(u(\theta ))\circ (I+\theta )$ is differentiable at $0$ into $L^{1}(D)$, and $C(u_0)\in W^{1,1}(D)$, then $\theta \mapsto J(\theta )$ is differentiable and its directional derivative in the direction of $\tau $ is: \begin{equation} \frac{\partial J}{\partial \theta }(0)\tau =\int_D\frac{\partial C}{ \partial u}u' \,dx +\int_{\partial D}\tau \cdot nC(u_0) \,d S. \end{equation} \end{enumerate} \end{theorem} Let us prove now our first contribution. \begin{proof}[Proof of Theorem~\ref{thm:differentiability of solution}] We take several steps. For simplicity, allow the notation $$ \Omega_\theta = (I + \theta) (\Omega_0). $$ We first check that $v_\theta$ satisfies \begin{equation*} -\operatorname{div}(A(\theta)\nabla v) + \lambda J_\theta g( v_\theta ) = (f\circ (I+\theta)) J_\theta \end{equation*} in $H^{-1} (\Omega)$, where \begin{equation*} A(\theta) = J_\theta (I+D\theta)^{-1}(I + ^tD\theta)^{-1}, \quad J_\theta = \det J ( I + \theta )\,. \end{equation*} For that, consider for a given $\varphi \in H_0^1(\Omega_0)$ the auxiliar function $\varphi_\theta = \varphi \circ (I+\theta)^{-1} \in H_0^1(\Omega_\theta)$ by definition of $u_\theta$ we have \begin{equation*} \int_{\Omega_\theta} \left( \nabla u_\theta \nabla \varphi_\theta + \lambda g(u_\theta) \varphi_\theta \right) \,d x = \int_ {\Omega_\theta} f \varphi_\theta \,dS \quad \forall \varphi \in H_0^1(\Omega_0). \end{equation*} Then by a change of variable, the result follows. Let us define the operator $F: W^{1,\infty} \times H_0^1(\Omega_0) \to H^{-1}(\Omega_0)$, by \[ F(\theta,v) = \operatorname{div}(A(\theta) \nabla v) + \lambda J_\theta g(v) - (f\circ(I+\theta))J_\theta \] of class $\mathcal{C}^1$ (or $\mathcal C^m$) in a neighbourhood of $\theta = 0$. On that direction we check \begin{itemize} \item $\theta \in W^{1,\infty} \mapsto J_\theta = \det (I + D\theta) \in L^\infty $ of class $\mathcal{C}^\infty$ since $\theta \in W^{1,\infty} \to I + D\theta \in L^\infty (\mathbb{R}^n, \mathcal{M}_n)$ and $\det $ is a polynomic operator. \item $\theta \in W^{1,\infty} \mapsto (I + D\theta)^{-1} = \sum_{q \ge 0} (-1)^qD\theta^q \in L^\infty (\mathbb{R}^n, \mathcal{M}_n)$ is $\mathcal{C} ^\infty$, \item $(A,v)\in L^\infty(\mathbb{R}^n, \mathcal{M}_n)\times H_0^1(G) \mapsto - \operatorname{div}(A \nabla v) \in H^{-1} (G)$ is $\mathcal{C}^\infty$ since it is bilinear and continuous. \item Through the lemma $\theta \mapsto k(\theta) = (f\circ (I+\theta))J_\theta \in L^2(\mathbb{R}^n) \subset H^{-1}(\Omega_0)$ is $\mathcal{C}^1$ \end{itemize} so $F \in \mathcal C^1$. Note that, if $f = 0$ then $F \in \mathcal C^m$. It holds that \begin{equation*} D_v F(0,0)\varphi = -\Delta \varphi + \lambda g'(u( \cdot :0)) \varphi \end{equation*} and, since $g' \ge 0$, we have that $D_v (0,v) : H_0^1(G) \to H^{-1}(G)$ is a isomorphism by Lax-Milgram's theorem. Through the implicit function theorem (theorem~\ref{thm:implicita sobolev}) there exists a map $\theta \in W^{1,\infty} \to v(\theta) \in H_0^1(\Omega_0)$ of class $\mathcal{C}^1$ is $f \in H^1 (\mathbb R^n)$ and $\mathcal C^m$ if $f = 0$ such that \begin{equation*} F(\theta, v(\theta)) = 0\,. \end{equation*} If we we consider uniqueness for the elliptic problem we find that \begin{equation*} v(\theta) = v_\theta. \end{equation*} Simple substitution returns $u_\theta$. By Theorem \ref{thm:composition from the right} we have the differentiability of $u$. \end{proof} Once this is done we can make explicit calculations for the directional derivative. \begin{proof}[Proof of Theorem \ref{thm:directional diff}] Let us now characterize the directional derivative. Let $\theta \in W^{1,\infty}$ be fixed, let us call $u ' = u'(0) (\theta)$ and let $\Phi$ a curve of perturbations of the identity with $V = \theta$. We differenciate on the variational formulation \begin{equation*} \int_{\mathbb{R}^n} f \varphi \,\mathrm{d}x = \int_{\mathbb{R}^n} \left( -u_\tau \Delta \varphi + \lambda g(u_\tau) \varphi \right) \,dx \quad \varphi \in \mathcal C_c ^\infty (\Omega) \end{equation*} to obtain \begin{equation} 0 = \int_{\Omega_0} \left( - u'\Delta \varphi + \lambda g'(u_0) u'\varphi \right) dx, \quad \varphi \in \mathcal C_c ^\infty (\Omega) \end{equation} (observe that $h(x) = \lambda g'(u_0(x)) $ is a known function). This means that $u'$ is a very weak solution of the aforementioned equation \eqref{eq:u' in terms of theta}. Since we know that $u' \in L^2 (\mathbb R^n)$ we can apply regularity theory for this equation. For the boundary condition $v_\theta = 0$ on $\partial \Omega_0$, for all $\theta$ and therefore $v' = 0, \partial \Omega_0$. Since $v_\tau = u_\tau\circ \Phi(\tau)$ we have \begin{equation*} u'+ \nabla u_{\Omega_0}\cdot \theta = v'\in H_0^1(\Omega_0) \end{equation*} which provides the boundary condition. Therefore, we have \begin{equation} \label{eq:very weak formulation u' in terms of theta} \int_{\Omega_0} \left( - u'\Delta \varphi + \lambda g'(u_0) u'\varphi \right) \,dx = \int_{\partial \Omega_0} \left( (\nabla u_{\Omega_0} \cdot \theta) \partial_{\textbf n} \varphi \right)\,dS , \quad \varphi \in \mathcal C_0 ^2 (\Omega) \end{equation} we can obtain a Kato type inequality to shows uniqueness of very weak solutions (see \cite{Diaz+Rakotoson:2009}). For the regularity we apply the following classical trick. Since $u'$ is know we can take $\tilde f = -\lambda g'(u_0)u' \in L^2 ( \Omega)$ and $\tilde \eta = - \nabla u \cdot \theta \in L^2 (\partial \Omega)$ and find $z$ the unique solution in $ H^1 (\Omega_0) $ of \begin{gather*} - \Delta z = \tilde f, \quad\text{in } \Omega \\ z = \tilde \eta, \quad\text{on } \partial \Omega \end{gather*} classical theory. Then $z$ is a very weak solution of \eqref{eq:very weak formulation u' in terms of theta} and, by uniqueness, $u'(0) = z \in H^1 (\Omega)$. \end{proof} \begin{remark} \label{rmk1} \rm In the case that further regularity is necessary $v\in H_0^{1}\cap H^{m}$ then deformation must taken in $W^{m,\infty }$. A theory analogous to that on \cite{Henrot+Pierre:2005} for higher differentiability can be obtained for the non-linear case. \end{remark} \section{Differentiation the functional. Proof of Theorem \ref{thm:differentiation effectiveness} and its corollaries} We shall follow a reasoning similar to the one presented in \cite{Henrot+Pierre:2005}. Let us define $G_{t}=\Phi (t,G)$ and consider a function $f$ such that $f(\tau )\in L^{1}(G_{t})$. We take interest on the map $I:\mathbb{R} \to \mathbb{R}$, \begin{equation} I(\tau)= \int_{G_{\tau }}f(\tau ,x)\,dx=\int_{G}f(\tau ,\Phi (\tau ,y))J(\tau ,y)\,dy \end{equation} where $f(\tau ,x)=f(\tau )(x)$, \begin{equation*} J(\tau ,y)=\det (D_{y}\Phi (\tau ,y)). \end{equation*} \begin{theorem} \label{thm:derivation of integral formula} Let $\Phi $ very assumptions~\ref{hyp:Phi}, $f$ such that $f:[0,T)\to L^{1}(\mathbb{R}^n)$ is differentiable at $0$ and \begin{equation*} f(0)\in W^{1,1}(\mathbb{R}^{N})\,. \end{equation*} Then, $\tau \mapsto I(\tau )=\int_{G_{\tau }}f(\tau )$ is differentiable at $0$ and \begin{equation*} I'(0)=\int_{G}f'(0)+\operatorname{div}(f(0)V)\,. \end{equation*} If $G$ is an open set with Lipschitz boundary then \begin{equation*} I'(0)=\int_{G}f'(0)+\int_{\partial G}f(0)n\cdot V. \end{equation*} \end{theorem} In simpler terms, under regularity it holds that \begin{equation} \frac{\partial }{\partial \tau }\Big|_{\tau =0} \Big( \int_{G_{\tau}}f(\tau ,x) dx\Big) =\int_{\Omega _0}\Big\{ \frac{\partial f}{\partial \tau }(0,x)+\operatorname{div}\Big( f(0,x)\frac{\partial \Phi }{\partial \tau } (0,x)\Big) \Big\} dx. \end{equation} We have some immediate consequences of Theorem~\ref{thm:derivation of integral formula} \begin{lemma} Let $g\in W^{1,1}(\mathbb{R}^{N})$ and $\Psi :[0,T)\to W^{1,\infty }$ be continuous at $0$ such that $\Psi :[0,T)\to L^{\infty }$ is differentiable at $0$, and let $Z$ be its derivative. Then the mapping $[ 0,T) \to L^{1}(\mathbb{R}^n)$, \[ \tau \mapsto g\circ \Psi (\tau ) \] is differentiable at $0$ and $G'(0)=\nabla g\cdot Z$. \end{lemma} \begin{lemma}[Differentiation under the integral sign] Let $E$ be a Banach space and $f:E\times \Omega \to \mathbb{R}$ be such that $\tilde{f}:E \to L^{1}(\Omega )$ \[ \tilde{f}(x)= f(x,\cdot ) \] is differentiable at $x_0$. Let $F:E \to \mathbb{R}$, \[ F(x)=\int_{\Omega }f(x,y) dy\,. \] Then $F$ is differentiable at $x_0$ and \begin{equation*} DF(x)=\int_{\Omega }(D_{x}\tilde{f})(x)(y). \end{equation*} \end{lemma} Now we can prove the third of our main results. \begin{proof}[Proof of Theorem \ref{thm:differentiation effectiveness}] It is classical that we can differentiate under the integral sign \begin{equation*} \int_{\Omega }f(t,x)dx \end{equation*} with respect to $t$ as many times as $f$ is differentiable, and that the integral commutates with the derivative. This shows the derivability with $vJ_{\theta }$ under the integral sign. For the remaining equations we have to be a little more subtle and apply the previous theorem. Let $f(\tau)=g\circ u_{\tau }$. From the know formulas we must compute \begin{equation*} f'(\tau )=(g'\circ u_0)u' \end{equation*} Thus \begin{equation*} \frac{\partial }{\partial \tau }\Big|_{\tau =0} \Big( \int_{G_{\tau }}g(u_{\tau })\,dx\Big) =\int_{\Omega _0}\left\{ g'(u_0)u'+\operatorname{div}\left( g(u_0)\Phi '(0)\right) \right\} dx. \end{equation*} If $\Omega _0$ is Lipschitz then \begin{equation} \frac{\partial }{\partial \tau }\Big|_{\tau =0} \Big( \int_{G_{\tau}}g(u_{\tau })\,dx\Big) =\int_{\Omega _0}g'(u_0)u' x+g(0)\int_{\partial \Omega _0}\Phi '(0)\cdot n\,dS. \end{equation} Equation \eqref{eq:higher order derivatives of E} is guaranteed since $ g(v):W^{1,\infty }\to H_0^{1}(\Omega )\to L^{1}(\Omega )$ is $\mathcal{C}^{1}$, and so we can differentiate under the integral sign. \end{proof} To show equation \eqref{eq:higher order derivatives of E} we need a formula of differentiation under the integral sign \begin{proof}[Proof of Corollary \ref{cor: non isovolumetric effectiveness derivative}] Given the functional \begin{equation*} I(\Omega)=\frac{1}{|\Omega|}\int_{\Omega}g\circ u_{\Omega} dx \end{equation*} If we do not impose constant volume we have also to differentiate the volume measure \begin{equation*} I(\Phi )=\frac{\int_{\Phi (G)}g\circ u_{\Phi ({\Omega _0})}dx}{\int_{\Phi (G)}dx} \end{equation*} over a curve of deformations $\Phi (\tau )$ we have, applying the formula of differentiation of fractions \begin{align*} \frac{\mathrm{d}I}{\mathrm{d}\tau }\Big|_{\tau =0} &= \frac 1 {|\Omega _0|^2} \Big( |\Omega_0| \frac{d}{d \tau} \Big( \int_{\Phi (G)}g\circ u_{\Phi ({\Omega _0})}dx \Big) \\ &\quad - \Big( \int_{\Omega_0} \operatorname{div} \Phi ' (0) \cdot n \,dx \Big) \Big( \int_{\Phi (G)}g\circ u_{\Phi ({\Omega _0})}dx \Big) \Big), \end{align*} which, once simplified, gives the result. \end{proof} The proof of Corollary \ref{cor:isovolumetric effectiveness derivative} relies on the following Proposition. \begin{proposition} \label{prop:integral derivada deformaciones isovolumetricas} Let $\Phi (\tau )$ be a volume preserving family of deformations of $\Omega _0$ in the sense of Hypothesis \ref{hyp:Phi}. Then \begin{equation*} \int_{\Omega _0}\operatorname{div}\Phi '(0) \,dx=0. \end{equation*} If $G$ is Lipschitz then \begin{equation*} \int_{\partial \Omega _0}\Phi '(0)\cdot {n}\,dS=0. \end{equation*} \end{proposition} \begin{proof} Define $G_\tau = \Phi(\Omega_0,\tau)$; then \begin{equation*} c = \int_{G_\tau} 1 \,\mathrm{d}x\,. \end{equation*} From this and theorem~\ref{thm:derivation of integral formula} we obtain \begin{equation*} 0 = \int_{\Omega_0} \frac{\partial 1 }{\partial \tau } + \operatorname{div} \left(1 \Phi'(0)\right) \,\mathrm{d}x, \end{equation*} which proves the first part of the result. The second is an immediate consequence of the divergence theorem. \end{proof} \begin{remark} \label{rmk2} \rm Note that the condition $\Phi(0) = I$ is paramount. For example consider the family of deformations \begin{equation*} \Phi (\tau) (x,y) = \Big((1+\tau) x, \frac 1 {1 + \tau} y \Big)\,. \end{equation*} These are isovolumetric deformations of any circle centered at $0$, and of course $\Phi (0) = 0$. We can compute \begin{equation*} \operatorname{div} \Phi '(\tau ) = 1 - \frac 1 { (1 + \tau)^2 }\,. \end{equation*} This is only zero at $\tau = 0$ (that is where $\Phi(\tau) = I$) even though the transformations are isovolumetric at any given $\tau$. \end{remark} \begin{remark} \label{rmk} For generalizing to the case $g=g(x,u)$, we need to assume that the Nemitskij operator $G:W^{1,\infty }(\Omega )\times H^{1}(\Omega ) \to L^{2}(\Omega )$, \[ G(\Phi ,v) =g(\Phi (x),v(x)) \] is $C^{m}$ and that \begin{equation*} \frac{\partial g}{\partial v}(x,v)\geq 0. \end{equation*} In this case the operator on the implicit function theorem will be \begin{equation*} F(\theta ,v)=-\operatorname{div}(A(\theta )v)+g((I+\theta )^{-1},v)J_{\theta }=fJ_{\theta } \end{equation*} with derivative \begin{equation*} D_{v}F(0,v)\varphi =-(\Delta \varphi )(x)+\frac{\partial g}{\partial v} (x,v(x))\varphi (x). \end{equation*} \end{remark} \section{Applications} \subsection*{Rearrangement techniques: Schwarz and Steiner symmetrization} From Schwarz symmetrization we know (see e.g. \cite{Diaz:1985}, \cite{Diaz:1991}) that, if $g$ is either concave or convex and $\theta $ is volume preserving then ${\eta}(\theta )\leq {\eta}(0)$ (that is: the sphere is the least effective reactor). Therefore \begin{equation*} \int_{G}g'(u_0)u'=\tilde{\eta}'(0)=0. \end{equation*} For the Steiner symmetrization we know that, as we have proven in \cite {Diaz+Gomez-Castro:2014aims}, for concave $g,$ and in \cite {Diaz+Gomez-Castro:2014pageoph}, for convex $g$ (note that this is equivalent to concave $\beta $), the following holds: \begin{theorem} Let $g $ be a concave or convex continuous nondecreasing function such that $g (0)=0$. Let $f \in L ^2 (\Omega) $ be nonnegative, i.e. $f\geq 0$, and $|B| = |\Omega''|$ with $B$ a ball. Then \begin{equation} \eta(\Omega'\times \Omega'') \le \eta(\Omega'\times B). \end{equation} \end{theorem} So, for $G=B\times G_{2}\ni (x,y)$, we have for all deformations $\theta =(\theta _1,0)$ with $\theta _1$ volume preserving and $g$ convex or concave, \begin{align*} \int_{G}g'(u_0)u' &= \int_{G_{2}}\int_{B}(g'(u_0)u'+\operatorname{div}(g(u_0)\theta )\\ &=\int_{G_{2}}\int_{B}(g'(u_0)u'+\operatorname{div}_{x}(g(u_0)\theta _1) \\ &= \int_{G_{2}}\Big\{ \int_{B}g'(u_0)u'+g(0)\int_{\partial B}\theta _1\cdot n\Big\} \\ &= \int_{G_{2}}\int_{B}g'(u_0)u' \\ &= \int_{G}g'(u_0)u'\,. \end{align*} Whenever the Nemitskij operator for $g$ is of class $\mathcal{C}^{2}$ we get \begin{equation*} \eta '(0)(\theta )=0,\quad \eta ''(0)(\theta,\theta )\leq 0\,. \end{equation*} Applying the bounds for $\eta '(0)$ we have as consequence an a priori estimate of the effectiveness factor in terms of the value of the functional for a circular cylinder: \begin{proposition} If $B$ is a ball such that $|B|=|\Omega '|$ then \begin{equation*} \eta (B\times \Omega '')-c(p)|g'|_{\infty}|u|_{W^{2,p}} |\theta |_{\infty } \leq \mathcal{\eta }(\Omega '\times \Omega '') \leq \eta (B\times \Omega ''). \end{equation*} \end{proposition} \subsection*{Numerical experiments} \begin{figure}[htb] \begin{center} \includegraphics[width = 0.7 \textwidth]{fig1} % effectiveness-ellipse \end{center} \caption{Effectiveness on isovolumetric ellipses with smaller semiaxes $a$, for the kinetic $g(u) = 1 - (1-u)^{1/q}$.} \label{fig1} \end{figure} \begin{figure}[htb] \begin{center} \includegraphics[width = 0.48\textwidth]{fig2a} % elliptic-cylinder-clear \includegraphics[width = 0.48\textwidth]{fig2b} \\ % effectiveness-elliptic-cylinder Solution in elliptic cylinder \hfil Curve of the effectiveness \end{center} \caption{Effectiveness on elliptic cylinders with smaller semiaxes $a$, for the kinetic $g(u) = 1 - (1-u)^{1/q}$, $0 < q < 1$ (this kinetic corresponds to $\beta(w) = w^q$, which is known in chemistry as the Freundlich isotherm).} \label{fig2} \end{figure} \begin{figure}[htb] \begin{center} \centering \includegraphics[width = 0.48\textwidth, height=55mm]{fig3a} %rectangular-cylinder \includegraphics[width = 0.48\textwidth]{fig3b} \\ % rectangular-cylinder-effectivenessh10 Solution in rectangular cylinder \hfil \parbox[t]{6cm}{Surface of the effectiveness for $h = 10$ and $a,b$ as parameters. Dotted lines represent curves of equal area} \end{center} \caption{Effectiveness on rectangular cylinders $[0,a]\times[0,b]\times[0,h]$, for the kinetic $g(u) = 1 - (1-u)^2$ and $h = 10$.} \label{fig3} \end{figure} \begin{figure}[htb] \begin{center} \includegraphics[width = 0.6\textwidth]{fig4} % triangular-cylinder.png \end{center} \caption{A triangular cylinder.} \label{fig4} \end{figure} The following numerical experiments were performed with COMSOL Multiphysics. \begin{example}[Schwarz symmetrization] \rm Let $g = g_1 + g_2$ where $g_1$ is convex and $g_2$ is concave. It is well known, see \cite{Diaz:1985} and \cite{Diaz:1991}, that a sphere is the least effective reactor for our problem in each isoperimetric family (to be more precise, isovolumetric families). We can see this in terms of derivatives through a family of ellipses \begin{equation*} \Phi(x,y,\tau) = (a(\tau)x, a(-\tau) y) \end{equation*} for $a$ regular such that $a(0) = 1$, even when we have no volume conservation. It turns out that since this is a symmetric curve of linear transformations we have that $$ I (\tau ) = I ( - \tau )\,. $$ Since we have differentiability it must hold that $ I'(0) = 0$. Since we have that this is a minimum and we are able to differentiate twice $ I'' (0) = 0$. \end{example} \begin{example}[Steiner symmetrization] \rm The same computations hold for transformations \begin{equation*} \Phi(x,y,z,\tau) = (a(\tau)x, a(-\tau) y,z) \end{equation*} This is a particular case of the results in \cite{Diaz+Gomez-Castro:2014aims} and \cite{Diaz+Gomez-Castro:2014pageoph}. If we consider a (uniparametric) family of elliptic cylinders of fixed height then we have the analogous result\,. We can even do this analysis on two parametric families, for example in square or triangular cylinder were we consider both dimensions on the basis. This analysis can be repeated over other families, like triangular cylinders with results of the same exact nature. \end{example} \subsection*{Acknowledgments} The authors would like to thank Prof. J. M. Arrieta for the suggestion of reference \cite{Henry:1993} and for the proof of Lemma \ref{lem:continuity of Nemitskij}. Researches partially supported by the projects MTM2011- 26119 and MTM2014-57113 of the DGISPI (Spain) and by the UCM Research Group MOMAT (Ref. 910480). \begin{thebibliography}{99} \bibitem{Allaire:2007} G. Allaire; \emph{Conception Optimale de Structures (Mathematiques et Applications)}, Springer, Berlin, 2007. \bibitem{Aris:1975} R.~Aris; \emph{The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts}, Oxford University Press, Oxford, 1975. \bibitem{Aris+Strieder:1973} R. Aris, W.~Strieder; \emph{Variational Methods Applied to Problems of Diffusion and Reaction}, vol.~24 of Springer Tracts in Natural Philosophy, Springer-Verlag, New York, 1973. \bibitem{Brezis:1999} H. Br\'ezis; \emph{Is there failure of the inverse function theorem?}, Morse theory, Minimax theory and their Applications in Nonlinear Differential Equations, Proc. Workshop held at the Chinese Acad. of Sciences, Beijing, 1999 (2003). \bibitem{Conca+Diaz+Linan+Timofte:2004} C. Conca, J. I. D\'iaz, A.~Li\~n'an, C. Timofte; \emph{Homogenization in chemical reactive flows}, Electronic Journal of Differential Equations, 40 (2004), pp.~1--22. \bibitem{Conca+Diaz+Timofte:2003} C. Conca, J. I. D\'iaz, C. Timofte; \emph{Effective chemical process in porous media}, Mathematical Models and Methods in Applied Sciences, 13 (2003), pp.~1437--1462. \bibitem{Dervieux:1980perturbationplasma} A.~Dervieux; \emph{Perturbation des \'equations d'\'equilibre d'un plasma confin\'e: comportement de la fronti{\`e}re libre, \'etude des branches de solutions}. \newblock INRIA. Research Report RR-0018. https://hal.inria.fr/inria-00076543, 1980. \bibitem{Diaz:1985} J. I. D{\'i}az; \emph{Nonlinear Partial Differential Equations and Free Boundaries, Vol.I.: Elliptic equations}, Research Notes in Mathematics, Pitman, London, 1985. \bibitem{Diaz:1991} J. I. D\'iaz; \emph{Simetrizaci\'on de problemas parab\'olicos no lineales: Aplicaci\'on a ecuaciones de reacci\'on - difusi\'on}, vol.~XXVII of Memorias de la Real Acad. de Ciencias Exactas, F\'isicas y Naturales, Madrid, 1991. \bibitem{Diaz+Gomez-Castro:2014aims} J. I. D\'iaz, D.~G\'omez-Castro; \emph{{Steiner} symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem}, To appear in Discrete and Continuous Dynamical Systems, (2015). \newblock \bibitem{Diaz+Gomez-Castro:2014pageoph} J. I. D\'iaz, D.~G\'omez-Castro; \emph{On the effectiveness of wastewater cylindrical reactors: an analysis through {Steiner} symmetrization}, To appear in Pure and Applied Geophysics, (2015). \newblock DOI 10.1007/s00024-015-1124-8 \bibitem{Diaz+Gomez-Castro:2015veryweak} J. I. D\'iaz, D.~G\'omez-Castro; \emph{Very weak solutions of elliptic equations with an absorption potential unbounded on a subset of the boundary: applications to shape differentiation}. \newblock Submitted, 2015. \bibitem{Diaz+Rakotoson:2009} J. I. D\'iaz, J.-M. Rakotoson; \emph{On the differentiability of very weak solutions with right hand side data integrable with respect to the distance to the boundary}, Journal of Functional Analysis, 257 (2009), pp.~807--831. \bibitem{Diaz+Rakotoson:2010} J. I. D\'iaz, J.-M. Rakotoson; \emph{On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary}, Discrete and Continuous Dynamical Systems, 27 (2010), pp.~1037--1058. \bibitem{Henrot+Pierre:2005} A.~Henrot, M.~Pierre; \emph{Optimization des Formes: Un analyse g\'{e}ometrique}, Springer, Berlin, 2005. \bibitem{Henry:1993} D.~Henry; \emph{Geometric Theory of Semilinear Parabolic Equations}, Lecture Notes in Mathematics, Springer Berlin Heidelberg, 1993. \bibitem{Mignot+Murat+Puel:1979variation} F.~Mignot, J.~Puel, F.~Murat; \emph{Variation d'un point de retournement par rapport au domaine}, Communications in Partial Differential Equations, 4 (1979), pp.~1263--1297. \bibitem{Murat+Simon:1976} F.~Murat, J.~Simon; \emph{Sur le contr \^{o}le par un domaine g\'eom\'etrique}, no.~76015 in Pr\'e-publications du Laboratoire d'Analyse Num\'erique, Universit\'e de Paris VI, 1976. \bibitem{Pironneau:2012optimal} O.~Pironneau; \emph{Optimal shape design for elliptic systems}, Springer Science \& Business Media, New York, 2012. \bibitem{Simon:1980differentiation} J.~Simon; \emph{Differentiation with respect to the domain in boundary value problems}, Numerical Functional Analysis and Optimization, 2 (1980), pp.~649--687. \bibitem{Sokolowski+Zolesio:1991} J.~Sokolowski, J.-P. Zolesio; \emph{Introduction to Shape Optimization: Shape Sensitivity Analysis}, vol.~16 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1991. \end{thebibliography} \end{document}