Maximizing_Population_Elliptic

Maximization of the population size

We consider an elliptic equation $$ - \mu\Delta \theta=\theta(m(x)-\theta)$$ in dimension 1 and 2. The goal is to maximize the following functional $$\int_{[0,1]^d} \theta dx$$ with respect to $m$ that belong to a class of functions $\mathcal{M}$ $$ \mathcal{M}:=\left{ m\in L^\infty([0,1]^d): 0\leq m\leq 1,\quad \int_{[0,1]^d}mdx=m_0\right}$$