Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 38, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu OPTIMIZATION PROBLEMS AND MATHEMATICAL ANALYSIS OF OPTIMAL VALUES IN ORLICZ SPACES ZAHRA DONYARI, MOHSEN ZIVARI-REZAPOUR, BEHROUZ EMAMIZADEH Abstract. This article concerns a minimization problem related to an ellip- tic equation in Orlicz-Sobolev spaces. We prove existence and uniqueness of optimal solutions and show that they are monotone and stable. Furthermore, by employing a characterization of the tangent cones in L∞ spaces, we de- rive some qualitative properties of the optimal solutions. We also derive some results regarding the optimal values. 1. Introduction 1.1. General overview. This article addresses an optimization problem related to the boundary value problem −∇ · (a(|∇u|)∇u) = f(x) in Ω, u = 0 on ∂Ω. (1.1) The conditions that we employ in (1.1) will be described in the next section. As we shall see the imposed restrictions on the function a(·) suggest considering Orlicz- Sobolev space as the underlying function space in which we seek the solution of (1.1). The existence and uniqueness of a solution to the boundary value problem is a straightforward task of implementing the direct method to prove the former and a typical strict convexity argument to guarantee the latter. Denoting the solution by uf , to stress the dependence on the force function f(·), the goal function γ(f) := ∫ Ω (fuf − Φ(|∇uf |)) dx, is minimized relative to f ∈ Aα = { f ∈ L∞(Ω) : 0 ≤ f ≤ 1, ∫ Ω f(x) dx = α } . The function Φ that appears in the definition of γ(·) is an appropriate N -function closely related to the function a(·). In order to appreciate the results reported in this paper a thorough understanding of the admissible set Aα is an advantage. This set can be decomposed as Aα = C+ ∩ B(0, 1) ∩ Λ−1(α). Here C+ denotes the positive cone of L∞(Ω), B(0, 1) the closed unit ball in L∞(Ω), and Λ(f) =∫ Ω f dx the continuous linear functional on L∞(Ω). Identifying L∞(Ω) with the 2010 Mathematics Subject Classification. 35J25, 49K20. Key words and phrases. Existence; uniqueness; Orlicz spaces; minimization; tangent cone; optimal solutions; optimal values. c©2021 Texas State University. Submitted April 12, 2021. Published May 6, 2021. 1 2 Z. DONYARI, M. ZIVARI-REZAPOUR, B. EMAMIZADEH EJDE-2021/38 dual of L1(Ω), it is readily verified that Aα is convex and weak*-compact in L∞(Ω). Unfortunately this decomposition does not reveal more properties of Aα which happen to be core in what follows. To discover other properties of Aα we first recall the definition of a measure preserving transformation from a measure space into another measure space. The mapping ξ : (X,σX , µX) → (Y, σY , µY ) is a measure preserving transformation if and only if (i) ξ is measurable i.e. for every S ∈ σY , ξ−1(S) ∈ σX ; (ii) the equation µX(ξ−1(S)) = µY (S) holds for every S ∈ σY . Here is an example of a measure preserving transformation when X = Y = [0, 1], σX and σY are the Borel sets, and µX = µY = dL, the Lebesgue measure: Let ξ(t) : [0, 1]→ [0, 1] be defined by ξ(t) = kt mod 1, for some k ∈ N. Whence ξ(t) = k−1∑ i=0 k ( t− i k ) χ[i/k,(i+1)/k)(t). Henceforth χE denotes the characteristic function supported on the set E. So χ(x) is equal to 1 when x ∈ E and equal to 0 otherwise. Let us consider the open interval (a, b) ⊆ [0, 1]. Observe that ξ−1(a, b) = ∪k−1 i=0 (a+ i k , b+ i k ) , so L(ξ−1(a, b)) = L(a, b) = b−a. Since the family of open intervals (a, b) generates the open sets of [0, 1], we infer that L(ξ−1(O)) = L(O) for every O, an open subset of [0, 1]. Finally using a well-known extension theorem in Ergodic theory we deduce that L(ξ−1(B)) = L(B), for every B ∈ B, the Borel sets of [0, 1]. So ξ is a measure preserving transformation as desired. Let MΩ→[0,1] = {ξ : Ω→ [0, 1] : ξ is a measure preserving transformation}, and f∆ : [0, |Ω|]→ [0, 1] defined by f∆(t) = χ[0,α)(t). Define R = {f∆ ◦ ξ : ξ ∈MΩ→[0,1]}. The fact that Aα = Rσ(L∞,L1) , the w∗ closure of R in L∞(Ω), and R = extAα, the set of extreme points of Aα in L∞(Ω), belong to the folklore, see for example [4, 5, 16]. Note that functions in R belong to {0, 1}Ω i.e. they are {0, 1}-valued whereas clearly those in Aα belong to [0, 1]Ω. The existence of optimal solutions for the minimization inf f∈Aα γ(f) shall be shown using the w∗ continuity of γ(·) in conjunction with the w∗ com- pactness of Aα, in L∞(Ω). However, similar to many other optimization problems, particularly from the numerical point of view, it would be significantly more efficient to know that the optimal solutions belong to a smaller set than Aα. Indeed, we shall prove that they belong to the extreme points of Aα i.e. R. This milestone will be achieved using a very friendly characterization of the tangent cones of subsets of L∞(Ω). The uniqueness of the optimal solution is another achievement which is an immediate consequence of the strict convexity of γ(·). Since the optimal solutions are of type χΩ̂ ∈ R we can identify them with the shape Ω̂. One then could explore the qualitative properties of Ω̂. We shall see, for example, that Ω̂ behave monoton- ically with respect to the parameter α in the sense that for β ≤ α, Ω̂β ⊆ Ω̂α, where Ω̂β and Ω̂α denote the optimal shapes relative to Aβ and Aα, respectively. It will EJDE-2021/38 OPTIMAL VALUES IN ORLICZ SPACES 3 also be shown that Ω̂, an essentially open set, is connected, thanks to the fact that Ω is simply connected, and also that Ω̂ forms a layer around ∂Ω, the boundary of Ω. In the final part of this note we shall derive some mathematical analysis results about the optimal value: `(α) = inf f∈Aα γ(f). In particular, we shall prove that `(·) is Lipschitz continuous, strictly convex and differentiable. We shall also apply a Lagrange multiplier argument to show the estimate `(α) ≤ Cα for some positive constant C. 1.2. Description of the minimization problem and preliminaries. Let Ω be a bounded smooth domain in RN (N ≥ 2) and a : (0,∞)→ (0,∞) a function such that the map ϕ(t) = { a(|t|)t t 6= 0, 0 t = 0, is an odd strictly increasing homeomorphism from R to R. Thus, the function Φ(t) = ∫ t 0 ϕ(s) ds, t ∈ R, is an N -function, see for example [1] for the definition. The conjugate of Φ, denoted Φ∗, is defined by Φ∗(t) = ∫ t 0 ϕ−1(s) ds, for all t ∈ R. It is known that Φ∗ is also an N -function, and can be reformulated as Φ∗(t) = sup s≥0 (st− Φ(s)). The set KΦ(Ω) = { u : Ω→ R : u is measurable and ∫ Ω Φ(|u(x)|) dx <∞ } , is called the generalized Orlicz class while the generalized Orlicz space is defined by LΦ(Ω) = { u : Ω→ R : u is measurable and lim τ→0+ ∫ Ω Φ(τ |u(x)|) dx = 0 } . LΦ(Ω) is a Banach space endowed with the Luxemburg norm |u|Φ = inf { τ > 0 : ∫ Ω Φ ( |u(x)| τ ) dx ≤ 1 } , or the equivalent Orlicz norm |u|LΦ = sup {∣∣ ∫ Ω uv dx ∣∣ : v ∈ LΦ∗(Ω), ∫ Ω Φ∗(|v(x)|) dx ≤ 1 } . Moreover, the following Hölder type inequality holds, [1],∣∣ ∫ Ω uv dx ∣∣ ≤ 2|u|Φ|v|Φ∗ , ∀u ∈ LΦ(Ω), v ∈ LΦ∗(Ω). (1.2) Henceforth we assume that there exist two positive constants λ and µ such that 1 < λ ≤ tϕ(t) Φ(t) ≤ µ <∞, ∀t > 0. (1.3) The relation (1.3) ensures that the differential equation in (1.1) is uniformly elliptic, see [13], and that Φ satisfies the ∆2-condition: Φ(2t) ≤ CΦ(t), ∀t ≥ 0, (1.4) 4 Z. DONYARI, M. ZIVARI-REZAPOUR, B. EMAMIZADEH EJDE-2021/38 where C is a positive constant, [15, Proposition 2.3]. In turn, the ∆2-condition implies that LΦ(Ω) and KΦ(Ω) are identical, and the dual of LΦ(Ω) coincides with LΦ∗(Ω), see for example [1]. Furthermore, we assume that the function [0,∞) 3 t→ Φ( √ t), (1.5) is convex. Condition (1.5) guarantees LΦ(Ω) is uniformly convex and hence re- flexive, see [15, Proposition 2.2]. The generalized Orlicz-Sobolev space is defined by W 1,Φ(Ω) = { u ∈ LΦ(Ω) : ∂u ∂xi ∈ LΦ(Ω), i = 1, . . . , N } . It is well known that W 1,Φ(Ω) endowed with the norm ‖u‖1,Φ = | |∇u| |Φ + |u|Φ is a reflexive Banach space. The space W 1,Φ 0 (Ω) denotes the closure of C∞0 (Ω) with respect to ‖u‖1,Φ-norm. Using the Poincaré inequality in Orlicz-Sobolev spaces, it follows that ‖u‖ := ‖∇u‖Φ is equivalent to ‖u‖1,Φ. The Orlicz-Sobolev space W 1,Φ 0 (Ω) is also a reflexive Banach space, [15]. In [15] it is shown that for u ∈ LΦ(Ω) the following holds |u|Φ > 1 ⇒ |u|λΦ ≤ ∫ Ω Φ(|u(x)|) dx ≤ |u|µΦ. (1.6) Also, from (1.3), one can prove that the following embeddings are continuous, Lµ(Ω) ↪→ LΦ(Ω) ↪→ Lλ(Ω), and Lλ ′ (Ω) ↪→ LΦ∗(Ω) ↪→ Lµ ′ (Ω), (1.7) where λ′ and µ′ denote the conjugate component of λ and µ respectively, see for example [1]. Definition 1.1. Let f ∈ LΦ∗(Ω). We say that u ∈ X := W 1,Φ 0 (Ω) is a weak solution of (1.1) if ∫ Ω a(|∇u|)∇u · ∇v dx = ∫ Ω fv dx, (1.8) for all v ∈ X. Using the direct method followed with a strict convexity argument one can prove the following basic result. Theorem 1.2. The boundary value problem (1.1) has a unique solution uf ∈ W 1,Φ 0 (Ω). The solution uf is the unique minimizer of the energy functional Ĵf (u) = ∫ Ω (Φ(|∇u|)− fu) dx, relative to u ∈W 1,Φ 0 (Ω). We define the functional Jf : X → R by Jf = −Ĵf i.e. Jf (u) = ∫ Ω (fu− Φ(|∇u|)) dx. We are interested in the minimization problem inf f∈Aα γ(f), (1.9) where γ(f) = Jf (uf ). We note that for f ∈ Aα, uf is positive, see [8, Lemma 3.4], and that uf ∈W 2,Φ(Ω), [3]. EJDE-2021/38 OPTIMAL VALUES IN ORLICZ SPACES 5 It is worth pointing out that when p > 1 and ϕ(t) = |t|p−2t, (1.1) becomes the well-known Dirichlet p-Laplace boundary value problem −∆pu = f(x) in Ω, u = 0 on ∂Ω. (1.10) In [14] the authors investigated the minimization problem (1.9) related to (1.10) for p = 2. We close this section with some physical examples of function Φ. (i) nonlinear elasticity: Φ(t) = (1 + t2)δ − 1, δ > 1 2 ; (ii) plasticity: Φ(t) = tδ(log(1 + t))ε, δ ≥ 1, ε > 0; (iii) generalized Newtonian fluids: Φ(t) = ∫ t 0 s1−δ(sinh−1 s)ε ds, 0 ≤ δ ≤ 1, ε > 0. For details, see [6, 7]. This article is organized as follows. In section 2, existence and uniqueness of optimal solutions to the minimization problem (1.9) are discussed. In section 3, we recite the definition of the tangent cones in L∞(D), and use them to derive the optimality conditions satisfied by the optimal solutions of (1.9). Section 4 is devoted to further properties of the optimal solutions. In particular, we prove the optimal solutions increase as the parameter α increases. Also, when α is close to, say, β, the respective optimal solutions will be close to each other in the Lp-norm. The section is closed by showing that the optimal value grows linearly with respect to the parameter α. 2. Existence and uniqueness of optimal solutions In this section we prove that the minimization problem (1.9) has a unique solu- tion i.e. there is an f̂ ∈ Aα such that γ(f̂) = inff∈Aα γ(f). To this end, we first prove the following result. Lemma 2.1. The functional γ : LΦ∗(Ω)→ R satisfies the following properties: (i) γ is weakly sequentially continuous; (ii) γ is strictly convex; (iii) γ is Fréchet differentiable, and 〈γ′(f), g〉 = ∫ Ω guf dx, for all g ∈ LΦ∗(Ω). Proof. (i) Assume fn ⇀ f , in LΦ∗(Ω). We have γ(f) + ∫ Ω (fn − f)uf dx = ∫ Ω fnuf dx− ∫ Ω Φ(|∇uf |) dx = Jfn(uf ) ≤ Jfn(ufn) = γ(fn) = Jf (ufn) + ∫ Ω (fn − f)ufn dx ≤ Jf (uf ) + ∫ Ω (fn − f)ufn dx = γ(f) + ∫ Ω (fn − f)ufn dx. (2.1) Since fn ⇀ f in LΦ∗(Ω) we deduce that ∫ Ω (fn−f)uf dx→ 0. Whence, to complete the proof of the assertion, it suffices to show∫ Ω (fn − f)ufn dx→ 0. 6 Z. DONYARI, M. ZIVARI-REZAPOUR, B. EMAMIZADEH EJDE-2021/38 The sequence {fn} is bounded in LΦ∗(Ω). If ‖ufn‖ > 1 then by (1.6) we have ‖ufn‖λ ≤ ∫ Ω Φ(|∇ufn |) dx ≤ ‖ufn‖µ. (2.2) From (1.3) and (2.2) we infer that∫ Ω fnufn dx = ∫ Ω a(|∇ufn |)|∇ufn |2 dx = ∫ Ω ϕ(|∇ufn |)|∇ufn |dx ≥ λ ∫ Ω Φ(|∇ufn |) dx ≥ λ‖ufn‖λ. Now, by the Hölder inequality we have λ‖ufn‖λ ≤ ∫ Ω fnufn dx ≤ C|fn|Φ∗ |ufn |Φ ≤ C‖ufn‖. Thus, since λ > 1 we deduce that {ufn} is bounded in X. Hence, up to a subse- quence, there exists w ∈ X such that ufn ⇀ w in X. By the Sobolev’s embedding theorem, X is compactly embedded into LΦ(Ω), [1]. So, ufn → w in LΦ(Ω). So by the Hölder inequality we infer ∫ Ω (fn − f)ufn dx→ 0. Therefore, γ(fn)→ γ(f), as desired. Remark 2.2. We point out that w is equal to uf a.e. in Ω. Indeed, since the functional u 7→ ∫ Ω Φ(|∇u|) dx is weakly lower semi-continuous, see [15, Lemma 4.3], we have that γ(f) = Jf (uf ) ≥ Jf (w) = ∫ Ω fw dx− ∫ Ω Φ(|∇w|) dx ≥ lim sup n→∞ (∫ Ω fnufn dx− ∫ Ω Φ(|∇ufn |) dx ) = lim sup n→∞ γ(fn) = γ(f). Hence γ(f) = Jf (uf ) = Jf (w). Therefore the uniqueness of the maximizer yields w = uf a.e. in Ω. (ii) The proof of this part is similar to that of [2, Lemma 3.2]. So we omit it. (iii) Let f, g ∈ LΦ∗(Ω). For any t ∈ (0, 1) we set ht = f + tg. By (2.1) we have γ(f) + ∫ Ω (ht − f)uf dx ≤ γ(ht) ≤ γ(f) + ∫ Ω (ht − f)uht dx. By Remark 2.2, we infer that uht → uf , as t→ 0+, in LΦ(Ω). So 〈γ′(f), g〉 = lim t→0+ γ(ht)− γ(f) t = ∫ Ω guf dx. Therefore γ is Gâteaux differentiable; moreover, γ′(f) = uf . Next, we show that γ′ is continuous at f ∈ LΦ∗(Ω). Let {fn} be a sequence in LΦ∗(Ω) such that fn → f EJDE-2021/38 OPTIMAL VALUES IN ORLICZ SPACES 7 in LΦ∗(Ω). By part (i) and Remark 2.2, we deduce that ufn → uf in LΦ(Ω). Thus, for all g ∈ LΦ∗(Ω) we have |〈γ′(fn)− γ′(f), g〉| = ∣∣ ∫ Ω g(ufn − uf ) dx ∣∣→ 0, as n→∞. Therefore γ is Fréchet differentiable in LΦ∗(Ω). � The main result of this section reads as follows. Theorem 2.3. The minimization problem (1.9) has a unique solution. Proof. It’s well known that Aα is w* closed in L∞(Ω) in addition to being convex; so it is weak* compact. Since the dual space of LΦ(Ω) is LΦ∗(Ω), by Lemma 2.1(i) and the inclusions LΦ(Ω) ⊂ L1(Ω) and L∞(Ω) ⊂ LΦ∗(Ω) we infer that γ is weak* continuous in L∞(Ω). Therefore the minimization (1.9) has a solution. The uniqueness of the solution is a consequence of strict convexity of γ. � 3. Characterization of the optimal solution and its consequences In this section we use tangent cones to derive the optimality condition satisfied by the optimal solutions, and obtain some qualitative results from this condition. Definition 3.1. Let V be a normed linear space and K a nonempty subset of V . The inner (intermediate or derivable) tangent cone of K at z ∈ K, denoted by T ′K(z), is defined as follows; v ∈ T ′K(z) if and only if for each decreasing real numbers tn ↓ 0 there exists a sequence {vn} in V such that limn→∞ vn = v and z + tnvn ∈ K for all n ≥ 1. The following two lemmas are useful for deriving the minimality conditions as- sociated with problem (1.9). The proof of the following lemma is in [10, Theorem 4.14]. Lemma 3.2. Let K be a nonempty subset of a real normed space V , and let F be a functional defined on an open superset of K. If z is a minimizer of F in K and if F is Fréchet differentiable at z, then 〈F ′(z), v〉 ≥ 0, ∀v ∈ T ′K(z), (3.1) where 〈·, ·〉 denotes the pairing between V and V ′, the dual of V . Here F ′(z) stands for the Gâteaux derivative of F at z. The condition (3.1) is called the first order optimality condition. For the proof of the following Lemma see [14, Lemma 2.2]. Lemma 3.3. Let V be a normed linear space, K a nonempty convex subset of V and F : V → R a convex functional which is Gâteaux differentiable. If 〈F ′(z), v〉 ≥ 0 for all v ∈ T ′K(z), then z is a minimizer of F in K. For f ∈ Aα and n ∈ N we use the following notation: • Ω0 := {x ∈ Ω : f(x) = 0}, • Ω∗ := {x ∈ Ω : 0 < f(x) < 1}, • Ω1 := {x ∈ Ω : f(x) = 1}, • Ω0 n = {x ∈ Ω : f(x) ≤ 1/n}, • Ω1 n = {x ∈ Ω : f(x) ≥ 1− 1/n}. 8 Z. DONYARI, M. ZIVARI-REZAPOUR, B. EMAMIZADEH EJDE-2021/38 To determine the characteristics of tangent cones in Aα, we now state and prove some lemmas that are known but we have not been able to find their proofs. Lemma 3.4. Let f ∈ Aα and h ∈ L∞(Ω). If h ∈ T ′Aα(f) then (i) ∫ Ω hdx = 0, (ii) limn→∞ ‖χΩ0 n h−‖∞ = 0, (iii) limn→∞ ‖χΩ1 n h+‖∞ = 0, where h+ (resp. h−) is the positive (resp. negative) part of h. Proof. (i) The proof of this part is simple. (ii) Let ε > 0. We set tn = 1 ε‖χΩ0 n f‖∞ for n ∈ N. Thus there exists a sequence {hn} in L∞(Ω) such that hn → h in L∞(Ω) and f + tnhn ∈ Aα for all n ∈ N. So we have h ≥ (h−hn)− f tn in Ω. Thus h− ≤ ‖hn−h‖∞+ ε a.e. in Ω0 n for all n ∈ N. Hence lim supn→∞ ‖χΩ0 n h−‖∞ ≤ ε. Since ε > 0 is arbitrary, the result of this part is obtained. (iii) The proof of this part is similar to (ii). � Lemma 3.5. Let f ∈ Aα and h ∈ L∞(Ω) be such that (i) ∫ Ω0 n h− dx = ∫ Ω1 n h+ dx for all n ∈ N. (ii) limn→∞ ‖χΩ0 n h−‖∞ = 0, (iii) limn→∞ ‖χΩ1 n h+‖∞ = 0. Then h ∈ T ′Aα(f). Proof. From (i) for n = 1 we infer ∫ Ω hdx = 0. Assume ‖h‖∞ 6= 0. Let tn ∈ (0, 1 n‖h‖∞ ) for n ≥ 1. For each n we define hn := h+ χΩ0 n h− − χΩ1 n h+ in Ω. From (ii) and (iii) we deduce hn → h in L∞(Ω). Also, for any n, ∫ Ω hn dx = 0 by (i). Thus ∫ Ω (f + tnhn) dx = α for all n ≥ 1. Since Ω0 n ∩ Ω1 n = ∅ for n ≥ 3, it is easy to check that 0 ≤ f + tnhn ≤ 1 in Ω for all n ≥ 3. Therefore h ∈ T ′Aα(f). � Lemma 3.6. Let f ∈ Aα. If h ∈ T ′Aα(f), then h(x) ≥ 0 a.e. in Ω0, h(x) ≤ 0texta.e. inΩ1. Proof. Since Ω0 ⊂ Ω0 n and Ω1 ⊂ Ω1 n for all n ∈ N, the assertion readily follows from Lemma 3.4. � The following Theorems are the main results of this section. Theorem 3.7. f̂ minimizes γ(f) relative to Aα if and only if (i) |Ω∗| = 0, (ii) uf̂ (x0) ≥ uf̂ (x1) for all (x0, x1) ∈ Ω0 × Ω1. Proof. Let f̂ ∈ Aα be the solution of (1.9). We have Ω∗ = ∪∞n=1Ω∗n, where Ω∗n := { x ∈ Ω : 1 n ≤ f̂(x) ≤ 1− 1 n } . Note that Ω∗n ⊂ Ω∗n+1. We show that uf̂ is constant on Ω∗. To derive a contradic- tion, assume not. Hence, uf̂ is not constant on Ω∗n for some n ∈ N. Thus, there exist two measurable sets ω1 and ω2 in Ω∗n such that |ω1| = |ω2| and ∫ ω1 uf̂ dx < ∫ ω2 uf̂ dx. (3.2) EJDE-2021/38 OPTIMAL VALUES IN ORLICZ SPACES 9 Let h(x) :=  1 x ∈ ω1, −1 x ∈ ω2, 0 x ∈ (ω1 ∪ ω2)c. So h ∈ T ′Aα(f̂) by Lemma 3.5. From Lemma 2.1 (iii) and (3.2) we deduce 〈γ′(f̂), h〉 = ∫ Ω huf̂ dx = ∫ ω1 uf̂ dx− ∫ ω2 uf̂ dx < 0, which is a contradiction by Lemma 3.2. Thus, uf̂ is constant on Ω∗. To show that the measure of Ω∗ is zero we proceed as follows. Using the regularity of uf̂ , the differential equation in (1.1) holds almost everywhere. So restricting that equation to the set Ω̂∗ will give a contradiction unless the measure of Ω∗ is zero. (ii) To derive a contradiction, suppose there exist two measurable sets ω0 ⊂ Ω0 and ω1 ⊂ Ω1 such that |ω0| = |ω1| and ∫ ω0 uf̂ dx < ∫ ω1 uf̂ dx. (3.3) Let h(x) :=  1 x ∈ ω0, −1 x ∈ ω1, 0 x ∈ (ω0 ∪ ω1)c which belongs to T ′Aα(f̂). By Lemma 2.1 (iii) and (3.3) we have 〈γ′(f̂), h〉 = ∫ Ω huf̂ dx = ∫ ω0 uf̂ dx− ∫ ω1 uf̂ dx < 0, which is a contradiction by Lemma 3.2. Therefore uf̂ (x0) ≥ uf̂ (x1) for all (x0, x1) ∈ Ω0 × Ω1. Conversely, assume (i) and (ii) hold. Thus c = sup x∈Ω1 uf̂ (x) = inf x∈Ω0 uf̂ (x) > 0. Fix h ∈ T ′Aα(f̂), and apply Lemmas 2.1 (iii), 3.4 and 3.6 to obtain 〈γ′(f̂), h〉 = ∫ Ω0 huf̂ dx+ ∫ Ω1 huf̂ dx ≥ ∫ Ω0 hcdx+ ∫ Ω1 hc dx = c ∫ Ω hdx = 0. Therefore, we deduce from Lemma 3.3 that f̂ is a minimizer. � Henceforth, we suppose that Ω is simply connected. Also, we will make the following assumptions on the functions a(t) and ϕ(t): (A1) a ∈ C1(0,+∞) and there exist positive constant λ1 and µ1 such that 0 < λ1 < tϕ′(t) ϕ(t) ≤ µ1, ∀t > 0. Theorem 3.8. Let f̂ be the minimizer of γ(f) relative to Aα. Then f̂ is a char- acteristic function which is equal to χ{uf̂ 0, see [12] and [13, Theorem 1.7]. From |Ω∗| = 0 we infer that there exists Ω̂ ⊂ Ω1 such that |Ω̂| = α and f̂ = χΩ̂. Note that Ω1 contains a neighborhood of ∂Ω. We set ĉ = sup x∈Ω1 uf̂ (x) = inf x∈Ω0 uf̂ (x) > 0. From the continuity of uf̂ we deduce that uf̂ = ĉ on ∂Ω0. Restricting the differential equation in (1.1) to the set Ω0 we get ∇ · (a(|∇uf̂ |)∇uf̂ ) = 0, in Ω0, and uf̂ = ĉ, on ∂Ω0. Let w = uf̂ − ĉ; so we have ∇ · (a(|∇w|)∇w) = 0, in Ω0, and w = 0, on ∂Ω0. Thus, ∫ Ω0 a(|∇w|)|∇w|2 dx = 0. Since a is a positive function we infer that ∇w = 0 a.e. in Ω0. Therefore, w = 0 on ∂Ω0 implies uf̂ = ĉ in Ω0. Whence, Ω̂ = {x ∈ Ω : uf̂ (x) < ĉ}, where ĉ = maxΩ̄ uf̂ . We know that ∂Ω̂ ⊂ {x ∈ Ω : uf̂ = ĉ} ∩ Ω1. If |{x ∈ Ω : uf̂ = ĉ} ∩ Ω1| > 0, then f̂ = 0 in this set, which leads to a contradiction. Thus |∂Ω̂| = 0. We now prove that Ω̂ is connected. Suppose not, and consider E an open com- ponent of Ω̂ whose boundary does not intersect the boundary of Ω. Since uf̂ = ĉ on ∂E and −∇ · (a(|∇uf̂ |)∇uf̂ ) = 1 in E, we obtain ∫ E (uf̂ − ĉ) dx = ∫ E a(|∇uf̂ |)∇uf̂ · ∇(uf̂ − ĉ) dx = ∫ E a(|∇uf̂ |)|∇uf̂ | 2 dx ≥ 0. This is a contradiction, because uf̂ < ĉ in E. Therefore, Ω̂ is connected. � 4. Monotonicity, stability and regularity Let α, β ∈ (0, |Ω|). Let f̂α ∈ Aα and f̂β ∈ Aβ be the solutions of inf f∈Aα γ(f) and inf f∈Aβ γ(f), respectively. By Theorem 2.3, we know that f̂α = χΩ̂α and f̂β = χΩ̂β . Moreover, we have Ω̂α = {x ∈ Ω : uα(x) < cα} and Ω̂β = {x ∈ Ω : uβ(x) < cβ}, (4.1) where cα = maxΩ̄ uα and cβ = maxΩ̄ uβ . Recall that −∇ · (a(|∇uα|)∇uα) = χΩ̂α in Ω, uα = 0 on ∂Ω, (4.2) and −∇ · (a(|∇uβ |)∇uβ) = χΩ̂β in Ω, uβ = 0 on ∂Ω. (4.3) We now state the monotonicity results. The proof of the following lemma is similar to [14, Theorems 4.1 and 4.2], so we omit it. EJDE-2021/38 OPTIMAL VALUES IN ORLICZ SPACES 11 Lemma 4.1. If 0 < β < α < |Ω|, then Ω̂β ⊂ Ω̂α, and cβ < cα. Now we state a stability result. Let 0 < αn < |Ω|, n ∈ N, and χΩ̂αn denote the unique solution of the minimization problem inf f∈Aαn γ(f). Lemma 4.2. Let χΩ̂α denotes the minimizer of problem (1.9), satisfying |Ω̂α| = α. If αn → α, then χΩ̂αn → χΩ̂α in Lp(Ω) for any p ≥ 1. Moreover, |Ω̂αn 4 Ω̂α| → 0. Here ∆ denotes the symmetric difference of sets. The proof of the above lemma is similar to that of [14, Theorem 5.1]. Next we prove the continuity of the mapping α→ cα, compared with [9, Theorem 2.4 ]. Lemma 4.3. For α ∈ (0, |Ω|), the map α 7→ cα is continuous. Proof. Let α ∈ (0, |Ω|). We only prove continuity from the left at α. The right continuity is proved similarly. To this end, consider {αn}, a sequence in (0, |Ω|) such that αn ↑ α. By Lemma 4.2 we infer χΩ̂αn → χΩ̂α in Lλ ′ (Ω), hence χΩ̂αn → χΩ̂α in LΦ∗(Ω) by (1.7). From Lemma 2.1 (i), Remark 2.2, we deduce uαn → uα in LΦ(Ω), so by (1.7), uαn → uα in Lλ(Ω). Assume cαn does not convergent to cα. In that case, there exists a constant ε > 0 such that for every n ∈ N there is mn > n such that cα − cαmn > ε. From Lemma 4.1 we have Ω̂αmn ⊂ Ω̂α, so uα = cα and uαmn = cαmn in Ω \ Ω̂α. Hence for all n ≥ 1 we deduce∫ Ω |uα − uαmn | λ dx ≥ ∫ Ω\Ω̂α |cα − cαmn | λ dx > ελ(|Ω| − α). This is a contradiction, because uαn → uα in Lλ(Ω). This completes the proof. � We prove now our first results related to the functional `. Theorem 4.4. For α ∈ (0, |Ω|), let `(α) = inff∈Aα γ(f). The mapping α 7→ `(α) is Lipschitz continuous, strictly convex and differentiable, with derivative cα. Proof. Since `(α) = γ(χΩ̂α ), we have `(α) = ∫ Ω (χΩ̂α uα − Φ(|∇uα|)) dx = min |D|=α max v∈X ∫ Ω (χDv − Φ(|∇v|)) dx ≥ min |D|=α ∫ Ω (χDv0 − Φ(|∇v0|)) dx, (4.4) for any positive function v0 ∈ X. By the Bathtub Lemma, see [11], we have min |D|=α ∫ Ω χDv0 dx = ∫ Ω χD̃v0 dx, where D̃ is such that |D̃| = α and {x ∈ Ω : v0(x) < t} ⊂ D̃ ⊂ {x ∈ Ω : v0(x) ≤ t}, for a suitable t > 0. Thus, from (4.4) we deduce `(α) ≥ ∫ Ω (χD̃v0 − Φ(|∇v0|)) dx. (4.5) 12 Z. DONYARI, M. ZIVARI-REZAPOUR, B. EMAMIZADEH EJDE-2021/38 Let 0 < β < α < |Ω|. We know `(β) = ∫ Ω (χΩ̂β uβ − Φ(|∇uβ |)) dx. (4.6) From (4.5) with v0 = uβ we infer `(α) ≥ ∫ Ω (χD̃αuβ − Φ(|∇uβ |)) dx, (4.7) where {x ∈ Ω : uβ(x) < cβ} ⊂ D̃α ⊂ {x ∈ Ω : uβ(x) ≤ cβ}, |D̃α| = α. Since Ω̂β ⊂ D̃α we have |D̃α \ Ω̂β | = α − β. Now, since uβ = cβ outside Ω̂β , from (4.6) and (4.7) we deduce `(α)− `(β) ≥ ∫ D̃α\Ω̂β uβ dx = (α− β)cβ . (4.8) By a similar argument we can derive `(α)− `(β) ≤ (α− β)cα. (4.9) Thus, from (4.8) and (4.9) we obtain cβ ≤ `(α)− `(β) α− β ≤ cα. (4.10) Therefore, ` is Lipschitz continuous and from Lemma 4.3 we deduce that ` is differ- entiable and `′(α) = cα. Since the mapping α 7→ cα is strictly increasing, Lemma 4.1 implies that ` is strictly convex. � Let u1 ∈W 1,Φ 0 (Ω) be the solution of (1.1) for f = 1. Let γ1 := 1 |Ω| γ(χΩ) = 1 |Ω| ∫ Ω (u1 − Φ(|∇u1|)) dx. Our final result is an upper bound for `(α)/α. Theorem 4.5. For each α ∈ (0, |Ω|) we have `(α) ≤ γ1α. Proof. Let K := {f ∈ L∞(Ω) : 0 ≤ f ≤ 1}. Define the linear functional Λ : L∞(Ω) → R by Λ(f) := ∫ Ω f dx. Thus Aα and K ∩ Λ−1({α}) are identical. Let fα ∈ Aα be the solution of `(α) = minf∈Aα γ(f). Hence γ(f) − `(Λ(f)) ≥ 0 for all f ∈ K and γ(fα) − `(Λ(fα)) = 0. Thus by enforcing a standard minimality condition we obtain 0 ∈ ∂(γ − `(Λ))(fα) +NK(fα), (4.11) where NK(fα) denotes the normal cone to K at fα. By (4.11), for g ∈ NK(fα) we have γ′(fα)(fα − f)− `′(α)(α− Λ(f)) = 〈g, f − fα〉 ≤ 0, (4.12) for all f ∈ K. Hence, we obtain γ′(fα)(fα)− ∫ Ω Φ(|∇ufα |) dx− γ′(fα)(f)− α`′(α) + Λ(f)`′(α) ≤ 0, (4.13) for all f ∈ K. Since `(α) = γ(fα), by Lemma 2.1 (iii) and (4.13) we infer that `(α)− γ′(fα)(f)− α`′(α) + Λ(f)`′(α) ≤ 0, ∀f ∈ K. (4.14) EJDE-2021/38 OPTIMAL VALUES IN ORLICZ SPACES 13 In particular, setting f = 0 in (4.14) yields α`′(α)− `(α) ≥ 0. Thus we obtain d dα (`(α) α ) ≥ 0 in (0, |Ω|). Integrating both sides of the last inequality above, on the interval (α, |Ω|), we obtain `(α) α ≤ `(|Ω|) |Ω| in (0, |Ω|). Therefore, we obtain the desired conclusion. � 5. Conclusions In this work, an elliptic partial differential equation with zero Dirichlet bound- ary condition is considered. The differential operator is of elliptic type, and the external force only depends on the space variables. The structure of the equation organically suggests that a suitable function space to find solutions would be the Orlicz-Sobolev space. Next, an energy functional is introduced which depends on the force function that itself belongs to an α-admissible set of measurable functions taking values between 0 and 1 while its integral is equal to a prescribed value. The energy functional is minimized over the admissible set, and existence of opti- mal solutions are verified. Moreover, by proving strict convexity of the functional, uniqueness of optimal solutions are guaranteed. The remaining of the paper fo- cusses on derivation of qualitative properties of the optimal solution. To this end, we have used the tangent cones in order to derive the optimality condition which, in turn, is utilized to show that the optimal solution is indeed classical i.e. it is {0, 1}-valued. We have shown that the optimal solution grows when the parameter α increases. Furthermore, a stability result has been shown in the sense that if α is close to β, then their corresponding optimal solutions are close in the Lp-norm. Our final result concerns the optimal value `(α). More precisely, we have shown that `(α)/α is bounded from above, so the growth of the optimal value is linear with respect to α. Acknowledgments. The authors are grateful to the referee for careful reading of the paper and valuable suggestions and comments. Z. Donyari and , M. Zivari- Rezapour are grateful to the Research Council of Shahid Chamran University of Ahvaz for the financial support (SCU.MM99.441). References [1] R. Adams; Sobolev spaces. Academic Press, New York, 1975. [2] N. Amiri, M. 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Communications in Nonlinear Science and Numerical Simulation 25 (2015), 94-101. [15] M. Mihilescu, V. Rdulescu; Neumann problems associated to non-homogeneous differential operators in Orlicz-Sobolev spaces. Ann. Inst. Fourier 6 (2008), 2087-2111. [16] J. V. Ryff; Majorized functions and measures. Indag. Math., 30 (1968), 431-437. Zahra Donyari Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran Email address: z-donyari@stu.scu.ac.ir Mohsen Zivari-Rezapour (corresponding author) Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran Email address: mzivari@scu.ac.ir Behrouz Emamizadeh School of Mathematical Sciences, University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo 315100, China Email address: Behrouz.Emamizadeh@nottingham.edu.cn 1. Introduction 1.1. General overview 1.2. Description of the minimization problem and preliminaries 2. Existence and uniqueness of optimal solutions 3. Characterization of the optimal solution and its consequences 4. Monotonicity, stability and regularity 5. Conclusions Acknowledgments References