A note on optimal domains in a reaction-diffusion problem (Q1066356)

This note deals with the problem \(\partial u/\partial t-Au=\lambda g(u),\) \(u=0\), \(u(x,0)=0\), where A is a uniformly elliptic operator and the function \(g=g(\sigma)\) subject to five conditions specified in the note. Such problems occur in simple models for reaction-diffusion processes in the case where the reactions are endothermic or isothermal. In the treatment D represents the domain where the reaction takes place, and \(1- u=c\) is the concentration of the reactant. By means of the method of upper and lower solutions it is shown that, independent of the volume, sufficiently thin domains have the property that the vessel where the reaction takes place is optimal if no dead core appears and if the effectiveness is high. The proof is based on an estimate for the solution of the Dirichlet problem.