Analyticity of free boundary in an optimal harvesting problem

Behrouz Emamizadeh,Yichen Liu

doi:10.1007/s11005-020-01259-z

Abstract

This note revisits an optimisation problem pertaining to the optimal harvesting of a marine species. The existence of solutions and the corresponding optimal conditions they satisfy have already been proved. It is known that the optimal solutions can be identified with n-dimensional shapes. We will obtain an interesting result concerning the free boundary of the optimal shapes. Indeed, we will prove that if a parameter in the admissible set is kept sufficiently small then the free boundaries will be real analytic hypersurfaces.

Highlights

This note presents an interesting free boundary result that complements those in [12]
We briefly review the essential parts of [12] that are relevant to our goal, and close the section with our main result
We use DEto denote the region of maximal harvesting effort corresponding to the optimal strategy E, i.e., DE = uE(x) ≤ t

Summary

Introduction

This note presents an interesting free boundary result that complements those in [12]. When the diffusion constant fails to be large enough, the corresponding population uE will be identically zero, Jε(·, E) will vanish at uE Such a strategy cannot be optimal, according to the sign result. We use DEto denote the region of maximal harvesting effort corresponding to the optimal strategy E, i.e., DE = uE(x) ≤ t. In many rearrangement optimisation problems, where the admissible set is co(R), it turns out that the optimal solutions can (and sometimes should be) selected from the extreme set, dramatically reducing the amount of work in relevant numerical simulations. For such optimisation problems the reader is referred to a sample of papers [6, 9, 13]. The paper is organised as follows: In Section 2, we will include all the useful results which assist us to prove Theorem 1.1; The last section is devoted to the proof of Theorem 1.1

Preliminaries

Proof of the Main Result