Optimal data-driven solutions for a stationary diffusive model of population growth

Westudyoptimaldata-drivensolutionsforthestationarydiffusivepopulationgrowthmodel $-\Delta u=ru$ in a bounded domain $\Omega\subset\mathbb{R}^N$ with Neumann boundary conditions. Instead of prescribing a functional relation between the position $x$, the net per-capita growth rate $r$ and the population size $u$, we look for a pair $(u, r)\in H^1(\Omega)\times L^{\infty}(\Omega)$ that fits a given data set in an optimal way measured by a cost functional I and an additional penalty term. We characterize the relaxed cost functional $\text{sc}^{-} I$ by showing that its density is given as the partial lower convex envelope with respect to the variable r, and prove the existence of optimal data-driven solutions. Furthermore, we establish a consistency result comparing conventional solutions of $-\Delta u=\varrho(x, u)u$ with optimal data-driven solutions where the data set stems from the functional relation $(x, u) \mapsto \varrho(x, u)$. Finally, as data sets evolve, we prove the convergence of optimal solutions via the $\Gamma$-convergence of the associated cost functionals.