A Multidimensional Optimal-Harvesting Problem with Measure-Valued Solutions

The paper is concerned with the optimal harvesting of a marine resource, described by an elliptic equation with Neumann boundary conditions and a nonlinear source term. Since the cost function has linear growth, an optimal solution is found within the class of measure-valued control strategies. The paper also provides results on the existence and uniqueness of strictly positive solutions to the elliptic equation, and an averaging inequality valid for subharmonic functions with Neumann boundary data. 1. The basic model In this paper we study an optimal harvesting problem in a multi-dimensional domain. Consider a bounded connected open set Ω ⊂ RN , N ≥ 2, with smooth boundary. Denote by φ = φ(t, x) the density of fish at time t at the point x ∈ Ω. In absence of fishing activity, assume that the fish population evolves according to the parabolic equation with source term φt = ∆φ+ g(x, φ) x ∈ Ω , with Neumann boundary conditions (1.1) ∇φ · n = 0 x ∈ ∂Ω . Here n = n(x) denotes the unit outer normal to the set Ω at the point x ∈ ∂Ω. A typical choice for the source term is g(x, φ) = α(x) ( h(x)− φ ) φ. Here h(x) denotes the maximum fish population that can be supported by the habitat at x, while α is a reproduction speed. We denote by u = u(t, x) the intensity of harvesting conducted by a fishing company. In the presence of this harvesting activity, the population evolves according to φt = ∆φ+ g(x, φ)− φu . Assuming that the harvesting rate remains constant in time, the fish population will reach an equilibrium described by (1.2) ∆φ+ g(x, φ) = φu x ∈ Ω, together with the Neumann boundary conditions (1.1). In order to define an optimization problem for the steady state solution (1.2), we consider the cost (1.3) ˆ