A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS

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1 A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS A BRESSAN, G M COCLITE, AND W SHEN Abstract 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 analysis also provides results on the existence and uniqueness of strictly positive solutions to an elliptic equation with a measure-valued source and 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 R N, N 2, with smooth boundary Denote by ϕ ϕ(t, x) the density of fish at time t at the point x In absence of fishing activity, we assume that the fish population evolves according to the parabolic equation with source term with Neumann boundary conditions ϕ t ϕ + g(x, ϕ) x, (11) ϕ n 0 x Here n n(x) denotes the unit outer normal to the set at the point x 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 ϕ t ϕ + g(x, ϕ) ϕu Assuming that the harvesting rate remains constant in time, the fish population will reach an equilibrium described by (12) ϕ + g(x, ϕ) ϕu x, together with the Neumann boundary conditions (11) In order to define an optimization problem for the steady state solution (12), we consider the cost (13) c(x) u(x) dx Here c(x) is the cost for a unit of fishing effort at the location x Of course, the simplest choice here is c(x) constant However, one may have a cost c(x) which increases with the distance of the Date: October 26, Mathematics Subject Classification 34B15, 34B18, 49J20, 49N25, 49N90 Key words and phrases Optimal control, measured-valued solutions, fish harvest The second author was partially supported by GNAPA with grant Sistemi di leggi di conservazione con sorgente non-locale singolare The work of the third author is partially supported by an NSF grant no DMS This work was initiated while G M Coclite visited the Department of Mathematics at the Penn State University He is grateful for Department s financial support and excellent working conditions 1

2 2 A BRESSAN, G M COCLITE, AND W SHEN point x from the coastal city where fishing company has its base In addition, if a region 0 is set aside as a marine park where no fishing is permitted, this can be modeled by setting c(x) + for every x 0 More generally, we consider a net profit which depends on the total fish caught, minus the harvesting cost: (14) J(ϕ, u) ϕ(x)u(x)dx c(x) u(x) dx The function u u(x) describes the harvesting strategy It is reasonable to assume that it satisfies constraints of the form (15) u(x) 0, b(x) u(x) dx 1, for some non-negative function b( ) The second constraint determines the maximum amount of harvesting power within the capabilities of the company In practice, this may depend on the number of fishermen and on the size of fishing boats available With minor modifications, all of our analysis remains valid for somewhat more general cost functionals of the type ( ) J(ϕ, u) ϕ(x)u(x)dx Ψ c(x) u(x) dx, assuming that Ψ is a non decreasing, convex, and lower semicontinuous function such that Ψ(0) 0 and Ψ (0) 1 Our main interest is in the existence and the qualitative properties of optimal solutions This problem has two distinct features: (i) In general, a given strategy u( ) does not determine a unique solution to the nonlinear elliptic problem (12), (11) In particular, one always has the trivial solution ϕ 0 For this reason, it is convenient to consider optimal pairs (u, ϕ), where u is an admissible harvesting strategy satisfying the constraints (15), while ϕ is a positive solution of the corresponding elliptic problem (12), (11) (ii) Since the cost functional (14) has only linear growth wrt u, there is no guarantee that the optimal strategy u( ) will lie in the space L 1 () Indeed, existence of optimal solutions will be proved within the larger space of bounded Radon measures supported on the closure of the domain By deriving suitable necessary conditions for optimality, one can then understand whether the optimal measure µ is absolutely continuous wrt Lebesgue measure In the one-dimensional case, necessary conditions were obtained in [7] by means of the Pontryagin maximum principle In the multidimensional case, we expect that optimality conditions can be derived by similar techniques as in [8] When a marine park is present, so that c(x) + for x 0, the optimal measure µ may concentrate a positive amount of mass along the boundary 0 An example where this holds was constructed in [7] The paper is organized as follows In Section 2 we show the existence of a positive solution for the problem (12), (11), assuming that u Cc () Section 3 introduces a suitable definition of solution in the case where u is replaced by a measure µ supported on the closure Finally, in Section 4 the existence of an optimal measure-valued control is established Problems of optimal harvesting of a marine resource, governed by a semilinear elliptic equation, have been the subject of several investigations [2, 9, 11, 12, 15] We remark that a quadratic harvesting cost such as c(x)u 2 (x) dx is entirely natural from a mathematical point of view, and guarantees that the optimal strategy is described by a function u opt L 2 () However, the linear cost (13) provides a more realistic model We also remark that most of the theory of elliptic equations with measure-valued right hand side is

3 A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS 3 concerned with Dirichlet boundary conditions In our fishery model, setting ϕ(x) 0 for x would mean that the fish is instantly killed as soon as it touches the shore Of course this is not the case The Neumann boundary conditions (11) yield a more appropriate model Finally, we remark that in these optimization problems governed by a linear (as in [8]) or by a semilinear elliptic equation it is natural to assume that the measure µ vanishes on every set of capacity zero In connection with our harvesting problem, this can be explained as follows Assume µ µ 0 +µ 1, where µ 0 is a positive measure concentrated on a set of capacity zero As shown in [10], one should have ϕ on the support of the measure ϕµ 0 Since in our case we know a priori that the fish density is ϕ 0, we conclude that ϕµ 0 must be the zero measure, hence ϕ 0 on the support of µ 0 If µ 0 0, then the harvesting strategy µ 1 yields a strictly better payoff than µ, because fishing at a location where the fish density is zero increases the cost without increasing the payoff For this reason, all of our analysis will be concerned with measures which vanish on sets of zero capacity In the one-dimensional case, the existence and a characterization of measure-valued optimal controls were proved in [6, 7] The multi-dimensional case, studied in the present paper, requires a more careful analysis, relying on Sobolev space theory For the basic theory of elliptic equations with measure-valued right hand side we refer to [3, 4, 10] 2 An elliptic problem with smooth coefficients As a first step in the analysis, we derive some estimates on the solution to an elliptic boundary value problem with smooth coefficients Consider the semilinear elliptic problem with Neumann boundary conditions ϕ + β(x)ϕ g(x, ϕ), in, (21) n ϕ 0, on, where n denotes the derivative in the direction of the outer normal to the boundary The following assumptions will be used (A1) The domain R N is bounded, open, connected, and has a smooth boundary (A2) The nonlinear source term can be written as g(x, s) f(x, s) s, where f : R R is a smooth function satisfying (22) (23) s f(x, s) < 0, for all (x, s) R, f(x, s) > 0, if and only if s < h(x), and h : R is a smooth nonnegative function (A3) The function β Cc () is non-negative and satisfies (24) β(x) dx < f(0, x) ds By (A2), there exists a constant M such that (25) g(x, s) M, h(x) M for all x, s 0 Lemma 21 Under the assumptions (A1) (A3), the boundary value problem (21) has a positive smooth bounded solution ϕ ϕ(x), such that (26) 0 < δ ϕ(x) M for all x, where M is the constant in (25)

4 4 A BRESSAN, G M COCLITE, AND W SHEN Proof By (23), g(x, M) f(x, M) M 0 for all x Hence the constant function identically equal to M is a supersolution of (21) In order to construct a subsolution, we consider the functional I(u) 1 2 and the manifold of codimension 1 u 2 dx B u H 1 (); [ β(x) f(x, 0) Since β and f(x, ) are smooth, the following facts are clear: ] u 2 dx, } u 2 dx 1 (i) I n is a continuous, quadratic functional on H 1 () (ii) I n is coercive on B Namely, if u k } k B and lim u k H k 1 () (iii) I is weakly lower semicontinuous in H 1 () Namely u k u weakly in H 1 () I(u) lim inf k u H 1 (),, then lim k I(u k) I(u k) Therefore, restricted to B, the functional I is bounded from below Since B H 1 () is weakly closed, by taking the limit of a minimizing sequence we obtain a function u B such that l inf I(w) I(u) w B The necessary conditions for optimality yield the existence of a scalar Lagrange multiplier η such that the Gateaux derivative of I at u satisfies DI(u) ηu 0 Observing that I is positively homogeneous of degree 2, we obtain η d dλ I(λu) d λ1 dλ [λ2 I(u)] 2I(u) λ1 This means that u and η solve the linear eigenvalue problem u + β(x) u f(x, 0) u + ηu, on, n u 0, on, where 1 Since the map with constant value mn () ( (27) η 2 min I(w) 2I w B 1 mn () η 2I(u) belongs to B, by (24) we obtain ) ( ) 1 βdx f(x, 0)dx m N () < 0 Since the domain is connected, classical results on linear elliptic eigenvalue problems [13] yield (28) u L (), 0 < c u, for some positive constant c As subsolution for the problem (21) we now choose the function εu, where ε > 0 is a sufficiently small constant Choosing ε small enough, we achieve (29) 0 < εu M Next, observe that (εu) + β εu g(x, εu) ε ( u + βu ) εuf(x, εu) ε ( uf(x, 0) + ηu ) ( (f(x, ) ) εuf(x, εu) εu 0) f(x, εu) + η

5 A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS 5 Due to the boundedness of u, the continuity of f(x, ), and (27), we can choose ε > 0 so small that ( ) η f(x, 0) f(x, εu) + η < 0, x 2 Therefore εu is a subsolution of (21) By (29) there exists a solution ϕ of (21) such that 0 < cε εu ϕ M Since all coefficients of the equation are smooth, it is clear that the solution is smooth as well For future use, we derive here an inequality valid for solutions ϕ of (21) Let G G(t, x; y) be the Green function for the linear parabolic equation with Neumann boundary conditions w t w, t > 0, x, (210) n w 0, t > 0, x As it is well known [14], for each fixed y the function G(, ; y) provides a solution to (210) such that (211) G(t, x; y) dx 1, lim G(t, x; y) φ(x) dx φ(y) t 0 for every φ C() The solution of (210) with a continuous initial data w(0, x) φ(x) is thus given by w(t, x) G(t, x; y) φ(y) dy t > 0, x Let now ϕ C 2 () be any function such that ϕ M, in, (212) n ϕ 0, on In particular, ϕ could be the solution to the elliptic problem (21) constructed in Lemma 21 For any t > 0, consider the averaged function (213) ϕ (t) (y) G(t, x; y) ϕ(x) dx Using the boundary conditions in (210) and (212) to integrate by parts, by the first equations in (211) and (212) one obtains d dt ϕ(t) (y) d G(t, x; y) ϕ(x) dx G t (t, x; y) ϕ(x) dy dt (214) G(t, x; y) ϕ(x) dx G(t, x; y) ϕ(x) dx M By (211), for every y we have (215) lim t 0 ϕ (t) (y) ϕ(y) Together with (214), this implies (216) ϕ(y) ϕ (t) (y) + Mt, for all y, t > 0

6 6 A BRESSAN, G M COCLITE, AND W SHEN 3 An elliptic problem with measure-valued coefficients In general, an optimal strategy for the harvesting problem may be a measure, not necessarily absolutely continuous wrt Lebesgue measure We thus need to study the semilinear elliptic problem with Neumann boundary conditions ϕ + ϕµ g(x, ϕ) x, (31) n ϕ 0 x, where µ is a nonnegative, bounded Radon measure on the closure A solution to (31) need not be smooth, or even continuous In the following, we shall consider solutions of (31) in distributional sense Definition 31 Let ϕ : R be an upper semicontinuous, non-negative function whose pointwise values are determined by (32) ϕ(y) lim ϕ (t) (y) lim G(t, x; y) ϕ(x) dx, t 0 t 0 with G as in (211) (213) We say that ϕ is a solution of (31) if ϕ L () H 1 (), and for every test function φ C 2 (R N ) one has (33) φ ϕ dx + φϕ dµ φ g(x, ϕ) dx Under the assumptions (A1)-(A2) on the domain and on the source term g, made in Section 2, our next result provides the existence and uniqueness of a nontrivial solution to the measure-valued elliptic problem (31) Theorem 31 Let the assumptions (A1)-(A2) hold Assume that (34) µ is a bounded nonnegative Radon measure on, µ(a) 0, µ() < for every A with zero capacity, f(x, 0) dx Then the boundary value problem (31) has a unique positive bounded solution such that (35) 0 < ϕ(x) M for ae x, where M is the constant in (25) Moreover, one has (36) ϕ dµ g(x, ϕ)dx M m N (), (37) ϕ 2 dx + ϕ 2 dµ ϕ g(x, ϕ) dx M 2 m N (), where m N denotes the N-dimensional Lebesgue measure

7 A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS 7 Proof 1 By assumption, the boundary is smooth Hence, by choosing ε > 0 small enough, for every point x R N with distance d(x, ) ε, the perpendicular projection π(x) is well defined For r 0, consider the open sets } r x R N ; d(x, ) < ε, } r x ; d(x, ) > ε We can choose ε > 0 sufficiently small so that the projection x π(x) is smooth on a neighborhood of the closed set ε \ ε Let χ C c ( ε ) be a smooth function such that χ(x) [0, 1] for all x ε, χ(x) 1 for all x By a reflection across the boundary of, we now define a bounded, linear extension operator E : H 1 () H0 1( ε) by setting φ(x) if x, (38) Eφ(x) χ(x) φ(2π(x) x) if x ε \ 2 We shall approximate the measure µ by a sequence of measures µ n having smooth density wrt Lebesgue measure, and study the regularized problems Notice that we can extend µ to a positive measure on ε simply by setting µ( ε \ ) 0 Since µ is zero on all sets with zero capacity, following [10] we can find measures µ and µ such that (39) µ µ + µ, µ, µ 0, µ L 1 ( ε ), µ H 1 ( ε ) More precisely, there exists functions Φ 0, Φ 1,, Φ N such that Φ 0 L 1 ( ε ), Φ i L 2 ( ε ), i 1,, N, and, for every bounded function φ H 1 ( ε ), one has (310) ε φ dµ ε Φ 0 φ dx N i1 ε Φ i φ xi dx By shifting the measure µ strictly inside the domain ε/n and taking a mollification, we obtain a sequence of smooth approximating measures µ n This procedure yields sequences of functions Φ 0,n,, Φ N,n C c () such that (311) β n Φ0,n + (312) β n 0, N (Φ i,n ) xi Cc (), i1 β n (x) dx f(0, x) dx δ for some δ > 0 and all n 1 Moreover, as n, one has the convergence (313) Φ 0,n Φ 0 L 1 ( ε) 0, Φ i,n Φ i L 2 ( ε) 0 i 1,, N

8 8 A BRESSAN, G M COCLITE, AND W SHEN 3 Applying Lemma 21, for every n 1 we obtain the existence of a strictly positive smooth solution ϕ n : ]0, M] to the problem ϕ + β n (x)ϕ g(x, ϕ), in, (314) n ϕ 0, on We claim that (315) ϕ n 2 dx M 2 m N (), for all n 1 Indeed, since ϕ n M and g M, multiplying the first equation in (314) by ϕ n and integrating over one obtains 0 ϕ n ϕ n dx + ϕ 2 nβ n dx ϕ n g(x, ϕ n )dx }} 0 ϕ n 2 dx M 2 m N () 4 Let Eϕ n H 1 0 ( ε) be the extension on ϕ n, defined as in (38) Thanks to the previous estimates, by possibly taking a subsequence and relabeling, we can assume the strong convergence (316) Eϕ n ϕ L 2 ( ε) 0 and the weak convergence (317) Eϕ n ϕ in H 1 0 ( ε ), for some function ϕ H 1 0 ( ε) In addition, we can assume the pointwise convergence (318) lim n Eϕ n(x) ϕ(x) and the convergence of the averages (319) lim n ϕ(t) n (x) ϕ (t) (x) for ae x ε for every fixed t > 0, uniformly for x Indeed, for every fixed time t, the function G(t, ; ) admits a Lipschitz continuous extension defined on Hence the maps x ϕ (t) n (x) are uniformly Lipschitz continuous as well Applying Ascoli s compactness theorem, by taking a subsequence and relabeling, we can achieve (319) for every rational t > 0 By continuity, the convergence holds for every (possibly irrational) t > 0 as well In the remainder of the proof we will show that the restriction of ϕ to the closed domain provides a solution of (31), in the sense of Definition 31 5 We claim that the limit (32) exists, for every x, and defines an upper semicontinuous function Indeed, for every n 1 and x the function t ϕ (t) n (x) + Mt is nondecreasing Taking the limit as n, this implies that the map t ϕ (t) (x) + Mt is nondecreasing as well Hence the function ( ) (320) ϕ(x) inf ϕ (t) (x) + Mt lim ϕ (t) (x) t>0 t 0 is well defined for every x Moreover, ϕ(x) ϕ(x) at every Lebesgue point of ϕ, hence almost everywhere The representation (320) of the function ϕ as the pointwise infimum of a decreasing family of Lipschitz continuous functions shows that ϕ is upper semicontinuous

9 A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS 9 6 To prove that ϕ solves (31), let φ C 2 (R N ) be a test function From (314) it follows 0 ϕ n φ dx + ϕ n β n φ dx g(x, ϕ n )φ dx (321) ( ϕ n φ dx + Eϕ n Φ 0,n + ε N ) (Φ i,n ) xi φ dx i1 ϕ n φ dx + Eϕ n Φ 0,n φ dx ε ε g(x, ϕ n )φ dx N (Eϕ n ) xi Φ i,n φ dx i1 g(x, ϕ n )φ dx Letting n, using the weak convergence Eϕ n ϕ in H 1 ( ε ), the uniform bounds ϕ n L () M, and the strong convergence Φ 0,n Φ 0 L 1 () 0, Φ i,n Φ i L 2 () 0, we conclude (322) 0 ϕ φ dx + ϕ Φ 0 φ dx ε ε N i1 ϕ φ dx + ϕφ dµ g(x, ϕ)φ dx ϕ xi Φ i φ dx g(x, ϕ)φ dx Indeed, since µ is supported on, integrating wrt the measure µ over ε or is the same This shows that ϕ is a solution of (31) 7 Clearly, from (26) it follows 0 ϕ M We now prove that ϕ is strictly positive As a first step we show that (323) ϕ 0 Assume by contradiction that ϕ(x) 0 for all x Then (324) ϕ n 0 weakly in H 1 () By (314) we have ϕ n f(x, ϕ n ) β n ϕ n Integrating over and using the Neumann boundary conditions one obtains ϕ n 2 (325) dx β n dx f(x, ϕ n )dx ϕ 2 n Letting n, thanks to (34), (26), and (324), one obtains ϕ n 2 dx µ() f(x, 0)dx < 0, lim n ϕ 2 n reaching to a contradiction Therefore (323) is proved To prove that ϕ is strictly positive, we observe that ϕ ( ) n ϕn div ϕ n 2 (log(ϕ n )) ϕ n 2 ϕ n ϕ n yields ϕ 2 n (log(ϕ n )) f(x, ϕ n ) β n + ϕ n 2 Thanks to (311), (313), (26), and (325) the right hand side is bounded in L 1 (), hence ϕ 2 n (log(ϕ n ))} n N is bounded in L 1 () ϕ 2 n

10 10 A BRESSAN, G M COCLITE, AND W SHEN By the analysis in [16], this implies log(ϕ n )} n N is bounded in W 1,q (), 1 q < Therefore there exists a function ψ W 1,q (), 1 q < We now have proving our claim N N 1 log(ϕ n ) ψ weakly in W 1,q (), 1 q < N N 1, log(ϕ n ) ψ strongly in L q (), 1 q < N N 1 N N 1 such that, passing to a subsequence, and ae in ψ(x) > and ϕ(x) e ψ(x) > 0 for ae x 8 We now prove the uniqueness of the positive solutions of (31) satisfying (35) The main idea is taken from [5] Assume that there exist two positive solutions ϕ and ϕ of (31), so that ϕ ϕ + µ f(x, ϕ), ϕ + µ f(x, ϕ), ϕ and hence ϕ ϕ + ϕ f(x, ϕ) f(x, ϕ) ϕ Multiplying by ϕ 2 ϕ 2 and integrating over, using (A2), (35), and the Neumann boundary conditions one obtains ( 0 ϕ ϕ + ϕ ) (ϕ 2 ϕ 2) dx (f(x, ϕ) f(x, ϕ)) ( ϕ 2 ϕ 2) dx ϕ ) ( ϕ ϕ + ϕ2 ϕ2 ϕ + ϕ ϕ ϕ dx ϕ ϕ (f(x, ϕ) f(x, ϕ)) (ϕ ϕ) (ϕ + ϕ) dx ) ( ϕ 2 2 ϕϕ ϕ2 ϕ ϕ + ϕ 2 ϕ 2 ϕ2 ϕ ϕ + 2ϕ ϕ ϕ 2 ϕ 2 + ϕ 2 dx + ( ϕ n ϕ ϕ2 ϕ nϕ + ϕ ϕ ) n ϕ ϕ n ϕ ds }} 1 0 ϕ f (x, θϕ + (1 θ) ϕ) (ϕ ϕ) 2 (ϕ + ϕ) dθdx 0 }}}} <0 >0 ae ] ([ ϕ 2 ϕ2 ϕ ϕ + ϕ 2 + [ ϕ 2ϕ ϕ 2 ϕ 2 + inf ϕf (ϕ ϕ) 2 (ϕ + ϕ) dx [0,M] ( [ ϕ ϕ ] 2 ϕ ϕ + [ ϕ ϕϕ ] ) 2 ϕ dx }} 0 inf ϕf (ϕ ϕ) 2 (ϕ + ϕ) dx 0 [0,M] }}}} >0 0 ]) 2 ϕϕ ϕ2 ϕ ϕ + ϕ 2 ϕ 2 dx + inf ϕf (ϕ ϕ) 2 (ϕ + ϕ) dx [0,M]

11 A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS 11 Therefore ϕ ϕ for ae x, proving uniqueness of positive solutions 9 We conclude by proving the estimates (36) and (37) Let ϕ L () H 1 () be the positive bounded solution of (31) Using the function φ 1 as test function in (33) we obtain ϕ dµ g(x, ϕ) dx, therefore (36) follows from (35) Due to the density of C 2 () in L () H 1 (), we can find a sequence of test functions φ ν } ν N C 2 (R N ) such that By (33) one has Letting ν we obtain Therefore (37) follows from (35) φ ν ϕ ae in and in H 1 (), as ν φ ν ϕ dx + φ ν ϕ dµ ϕ 2 dx + ϕ 2 dµ φ ν g(x, ϕ) dx ϕg(x, ϕ) dx 4 Existence of an optimal measure-valued harvesting strategy In this final section we study the existence of an optimal pair (ϕ, µ ) for the problem (41) maximize: J(ϕ, µ) ϕ(x) dµ(x) c(x) dµ(x), where ϕ is the unique strictly positive solution of ϕ(x) + g(x, ϕ) ϕ µ, x, (42) n ϕ 0, x, and µ is a non-negative Radon measure on which satisfies (43) b dµ 1 Remark 41 Integrating the equation (42) over the domain and using the Neumann boundary conditions, one obtains the balance law (44) ϕ dx + g(x, ϕ) dx ϕ dµ } } 0 According to (44), since no fish can enter or escape through the boundary of, at an equilibrium configuration the total growth rate of the fish population must be exactly balanced by the total harvesting rate In the following, instead of (41), we thus consider the equivalent problem (45) maximize: J(ϕ, µ) g(x, ϕ) dx c(x) dµ(x)

12 12 A BRESSAN, G M COCLITE, AND W SHEN In addition to the hypotheses (A1)-(A2) made in Section 2, we now assume (A4) The functions b : R and c : R are both lower semi-continuous, and satisfy (46) b(x) 0, c(x) c 0 > 0, for all x, for some positive constant c 0 Theorem 41 Under the assumptions (A1), (A2), and (A4), the constrained maximization problem (45), (42) admits an optimal solution Proof 1 Let (ϕ n, µ n )} n N be a maximizing sequence, where µ n is a positive measure on and ϕ n is the corresponding positive solution of (31), according to Definition 31 Taking a sequence of test functions φ ν C 2 () such that ϕ ν ϕ n H 1 () 0 as ν, from (33) we obtain ( ) (47) ϕ n 2 dx + ϕ n 2 dµ lim ν φ ν ϕ n dx + φ ν ϕ n dµ lim φ ν g(x, ϕ n ) dx ϕ n g(x, ϕ n ) dx M 2 m N () ν This shows that the sequence (ϕ n ) n 1 is uniformly bounded in H 1 () 2 By the assumptions, there exists a constant c 0 > 0 small enough so that (48) c(x) > c 0, h(x) > c 0 for all x We claim that is not restrictive to assume that (49) supp(µ n ) A n x ; ϕn (x) c 0 } Otherwise we could consider the measure µ n χ An µ n Clearly, µ n satisfies (43) and the function ϕ n provides a subsolution to the equation (31), with µ n replaced by µ n Therefore, we can find a solution ϕ n ϕ n of the same problem We now have J(ϕ n, µ n ) J(ϕ n, µ n) ϕ n dµ n ϕ ndµ n cdµ n + cdµ n ϕ n dµ n ϕ ndµ n cdµ n + cdµ n A n A n ϕ n dµ n cdµ n (ϕ n c)dµ n 0 \A n \A n \A n We can thus replace (ϕ n, µ n ) with the pair (ϕ n, µ n), which satisfies the additional condition (49) 3 For each n 1, from (25) and (48)-(49) it follows (410) 0 < c 0 ϕ n (x) M for all x Indeed, the upper bound in (410) is a consequence of (35) If the lower bound does not hold, consider the set 0 x ; ϕ n (x) < c 0 } Restricted to this set, the function ϕ n provides a smooth solution to the elliptic equation ϕ n g(x, ϕ n (x)) < 0, with boundary conditions ϕ n (x) c 0 on 0, n ϕ(x) 0 on

13 A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS 13 Therefore, the minimum principle for superharmonic functions yields min ϕ n (x) x 0 min ϕ n (x) c 0 x 0 4 We now show that the total mass of the measures µ n remains uniformly bounded Indeed, from (44) and (410) it follows c 0 µ n () ϕ n dµ n g(x, ϕ n ) dx M m N () Consider the positive measures ν n ϕn µ n Clearly, all these measures are uniformly bounded Thanks to the bounds (47), by possibly taking a subsequence and relabeling we can assume (411) ν n ν, µ n µ in the sense of weak convergence of measures on, ϕ n ϕ strongly in L 2 (), and ae in, ϕ n ϕ weakly in H 1 (), for some non-negative Radon measures ν, µ, and some function ϕ H 1 () By (320), the pointwise values of each ϕ n are determined by ( ) (412) ϕ n (x) inf ϕ (t) n (x) + Mt lim ϕ (t) n (x), t>0 t 0 where ϕ (t) n are the averaged functions, defined as in (213) As in step 5 of the proof of Theorem 31, by taking a further subsequence and relabeling, we can assume that the Lipschitz continuous functions converge to ϕ (t) uniformly on, for every fixed t > 0 For each x, the map t ϕ (t) n (x) + Mt is non-decreasing, for every n 1 Hence the same holds for the map t ϕ (t) (x) + Mt As in (32), ϕ (t) n (320), the pointwise values of ϕ are determined by ( ) (413) ϕ(x) inf ϕ (t) (x) + Mt t>0 lim ϕ (t) (x) lim t 0 t 0 G(t, y; x) ϕ(y) dy Being the pointwise infimum of a decreasing family of continuous maps, ϕ is upper semicontinuous By (411), ϕ provides a weak solution to ϕ + g(x, ϕ) ν, x, n ϕ 0, x From the uniform bounds (410) it follows (414) 0 < c 0 ϕ(x) M for all x Defining (415) ϕ ϕ, µ ν ϕ, in the following two steps we will show that the pair (ϕ, µ ) provides an optimal solution to our harvesting problem 5 Since ϕ is an upper semicontinuous function satisfying ϕ(x) c 0 > 0, the definition (415) is meaningful We now establish the key inequality (416) µ µ To prove that (416) holds, thanks to the upper semicontinuity of ϕ it suffices to show that φ (417) ψ dν φ dµ, for every φ, ψ C(), φ 0, ψ > ϕ

14 14 A BRESSAN, G M COCLITE, AND W SHEN Since ψ is continuous on the compact set, we can choose t, δ > 0 small enough so that (418) ϕ(x) ϕ (t) (x) + Mt < ψ(x) δ for all x Since ϕ n ϕ in L 2 (), as n the corresponding functions ϕ (t) n converge to ϕ (t) uniformly on Hence for all n large enough we have ϕ n (x) ϕ (t) n (x) + Mt ϕ (t) (x) + Mt + δ < ψ(x) for all x This yields φ ψ dν lim φ 1 n ψ dν n lim φ ϕ n n ψ dµ n lim φ dµ n n φ dµ 6 We conclude by proving that the pair (ϕ, µ ) optimal Since (ϕ n, µ n ) n 1 is a maximizing sequence, using (411), the lower semicontinuity of the cost function c( ), and then (415), we obtain ( ) sup J(ϕ, µ) lim J(ϕ n, µ n ) lim g(x, ϕ n ) dx c dµ n (ϕ,µ) n n g(x, ϕ ) dx c dµ g(x, ϕ ) dx c dµ J(ϕ, µ ) Finally, the lower semicontinuity of b( ) yields b dµ b dµ lim inf n b dµ n 1 This shows that the pair (ϕ, µ ) is admissible, completing the proof References [1] D H Armitage and S J Gardiner Classical potential theory Springer Monographs in Mathematics Springer-Verlag, London, 2001 [2] O Arino and J A Montero Optimal control of a nonlinear elliptic population system Proc Edinburgh Math Soc 43 (2000), [3] L Boccardo and T Gallouët Nonlinear elliptic equations with right hand side measures Comm Partial Differential Equations 17 (1992), [4] L Boccardo, T Gallouët, and L Orsina Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data Ann Inst H Poincaré Anal Non Linéaire 13 (1996), [5] H Brezis and L Oswald Remarks on sublinear elliptic equations Nonlinear Anal 10 (1986), [6] A Bressan and W Shen Measure-valued solutions for a differential game related to fish harvesting SIAM J Control Optim 47 (2009), [7] A Bressan and W Shen Measure valued solutions to a harvesting game with several players Advances in Dynamic Games 11 (2011), [8] G Buttazzo, N Varchon, and H Zoubairi, Optimal measures for elliptic problems Ann Mat Pura Appl 185 (2006), [9] A Cañada, J L Gámez, and J A Montero Study of an optimal control problem for diffusive nonlinear elliptic equations of logistic type SIAM J Control Optim 36 (1998), [10] G Dal Maso, F Murat, L Orsina, and A Prignet Renormalized solutions of elliptic equations with general measure data Ann Scuola Norm Sup Pisa Cl Sci 28 (1999), [11] M Delgado, J A Montero, and A Suárez Optimal control for the degenerate elliptic logistic equation Appl Math Optim 45 (2002), [12] M Delgado, J A Montero, and A Suárez Study of the optimal harvesting control and the optimality system for an elliptic problem SIAM J Control Optim 42 (2003), [13] E DiBenedetto Partial differential equations Second edition Cornerstones Birkhäuser, Boston, MA, 2010 [14] A Friedman, Partial Differential Equations of Parabolic Type Prentice-Hall, Englewood Cliffs, NJ, 1964 [15] S M Lenhart and J A Montero Optimal control of harvesting in a parabolic system modeling two subpopulations Math Models Methods Appl Sci 11 (2001), [16] G Stampacchia Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus Ann Inst Fourier (Grenoble) 15 (1965),

15 A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS 15 (Alberto Bressan) Department of Mathematics, Penn State University, University Park, PA 16802, USA address: URL: (Giuseppe Maria Coclite) Dipartimento di Matematica, Università di Bari, Via E Orabona 4,I Bari, Italy address: coclitegm@dmunibait URL: (Wen Shen) Department of Mathematics, Penn State University, University Park, PA 16802, USA address: shen w@mathpsuedu URL: w/