Homogenization of Dirichlet parabolic problems for coefficients and open sets simultaneously variable and applications to optimal design (Q2491745)

The authors consider linear parabolic equations written in variational formulation as \[ \left\langle \partial _{t}y_{n},v\right\rangle +\int_{\Omega }A_{n}\left( x,t\right) \nabla y_{n}\nabla v\,dx+\int_{\Omega }F_{n}\left( x,t\right) y_{n}v\,d\mu _{n}=\left\langle f,v\right\rangle , \] for every \(v\in L^{2}\left( 0,T;H_{0}^{1}\left( \Omega \right) \right) \cap L_{\mu _{n}}^{2}\left( \Omega \right) \). At each time, the solution \(y_{n}\) has to satisfy homogeneous Dirichlet boundary conditions on \(\partial \Omega \times \left( 0,T\right) \) and it starts from zero at \(t=0\). Here the matrix \(A_{n}\) is supposed to be elliptic and continuous uniformly with respect to the time parameter in \(\left( 0,T\right) \), \(F_{n}\) belongs to \( L_{\mu _{n}}^{\infty }\left( \Omega \right) \) and is positive and bounded, \( \left( \mu _{n}\right) \) is a sequence of nonnegative Borel measures which vanish on the subsets of \(\Omega \) having a capacity (with respect to the \(H_{0}^{1}\left( \Omega \right) \)-norm) equal to zero, \(f\) belongs to \( L^{2}\left( 0,T;H^{-1}\left( \Omega \right) \right),\) and \(\left\langle \cdot ,\cdot \right\rangle \) denotes the duality pairing between \( H_{0}^{1}\left( \Omega \right) \) and \(H^{-1}\left( \Omega \right) \). For example, the measure \(\mu _{n}\) can be associated to an open subset \(\Omega _{n}\) of \(\Omega \) through \(\mu _{n}\left( B\right) =+\infty \), if cap\(\left( B\cap \Omega \setminus \Omega _{n},B\right) >0\) and zero otherwise. Thus doing, both the coefficients of the parabolic equation and the domain in which the problem is posed can vary. The subsets \(\Omega _{n}\) can be associated to perforations of the domain \(\Omega \), thus leading to homogenization problems. The purpose of the paper is twofold. In the first part, the authors extend convergence results they proved in a previous paper. They here prove that, for every \(t\in \left( 0,T\right) \), \( y_{n}\left( \cdot ,t\right) _{n}\) converges in the strong topology of \(L^{2}\left( \Omega \right) \) to the solution \(y( \cdot ,t)\) of a linear parabolic problem of the same kind. The matrix \(A\) of this limit problem is the \(H\) -limit of \(\left( A_{n}\right) _{n}\) and the limit measure \(\mu \) is obtained when taking the \(G\)-limit of \(-\Delta +\mu _n\). In the second part of the paper, the authors consider an optimal design problem within this context. They consider a cost functional \(J\) which is lower semicontinuous on \(L^{2}\left( 0,T;H_{0}^{1}\left( \Omega \right) \right) \cap C^{0}\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) \). The quantity \(J\left( y\right) \) has to be minimized over all open subsets \(\widetilde{\Omega }\) of \(\Omega \) and all matrices \(A\in \mathcal{A}\), \(y\) being the solution of the linear parabolic problem \( \partial _{t}y-\text{div}\left( A\left( .,t\right) \nabla y\right) =f\) in \( \widetilde{\Omega }\times \left( 0,T\right) \) with homogeneous Dirichlet boundary conditions on \(\partial \widetilde{\Omega }\times \left( 0,T\right) \) and starting from zero at \(t=0\). The authors first prove an existence result for this optimal design problem. They then apply the previous relaxation convergence result within this context.