Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 116, pp. 1–22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GENERAL p-CURL SYSTEMS AND DUALITY MAPPINGS ON SOBOLEV SPACES FOR MAXWELL EQUATIONS DHRUBA R. ADHIKARI, ERIC STACHURA Abstract. We study a general p-curl system arising from a model of type- II superconductors. We show several trace theorems that hold on either a Lipschitz domain with small Lipschitz constant or on a C1,1 domain. Certain duality mappings on related Sobolev spaces are computed and used to establish surjectivity results for the p-curl system. We also solve a nonlinear boundary value problem for a general p-curl system on a C1,1 domain and provide a variational characterization of the first eigenvalue of the p-curl operator. 1. Introduction We study the following nonlinear system related to the Maxwell system of elec- tromagnetism in Banach spaces: |u|p−2u + curlp(u) + divp(u) = f(x,u), where f : Ω×R3 → R3 is a vector-valued Carathéodory function (see Section 7) and the operators curlp and divp (see their definitions in (3.5) and (5.2)) act on subspaces of the Sobolev space W 1,p(curl,Ω)∩W 1,p(div,Ω), 1 < p < +∞, with Ω a bounded domain in R3. The operators curlp and divp are Banach space generalizations of the classical curl and divergence operators which act on the Hilbert spaces H(curl,Ω) and H(div,Ω) [24]. The p-curl system we study arises from a model of magnetic induction in a high temperature superconductor [9]. However, the system we study here is more general than the one in Bean’s critical state model for type-II superconductors [32], as we allow for vector fields with nonzero divergence. Recently there has been growing interest in various properties of the p-curl sys- tem; see in particular [31] and the references therein. Frequently the roughness of the underlying domain plays a crucial role in the analysis of, for example, well- posedness of the system. Our interest is keeping the domain as rough as possible, i.e. Lipschitz. However, this is not always possible due to various embedding failures and in particular, a lack of simple Poincaré inequality; see Section 4. Frequently, the smoothness of the domain can be relaxed to Lipschitz by restricting the range of p-values for which the corresponding results hold. 2010 Mathematics Subject Classification. 49J40, 46E35, 49J50. Key words and phrases. p-curl operator; duality mappings; trace theorems; Nemytskii operator. c©2020 Texas State University. Submitted January 19, 2020. Published November 24, 2020. 1 2 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 We establish a framework suited for variational methods and calculating duality mappings on various Sobolev spaces associated to the p-curl system; see Sections 3–5. We prove that the p-curl operator can be expressed in terms of a duality mapping. It is worth mentioning that the geometry of Banach spaces is closely related to analytical properties of their duality mappings. We begin by proving a number of trace results for the Banach spaces under consideration. In many cases, we take the domain to be Lipschitz with small Lip- schitz constant. This is needed in order to obtain an Lp-estimate for the gradient of solutions to a certain elliptic boundary value problem. We generalize the duality mapping procedure to general Banach spaces having dual norm which is uniformly Frechét differentiable on the unit sphere; see Section 6. For further details on duality mappings and their applications to the solvability of nonlinear operator equations in Banach spaces, the reader is referred to [2, 5, 8, 26] and the references therein. In Section 7, we consider the nonlinear p-curl system on a C1,1 or convex domain. Under a particular growth assumption (similar to one commonly employed for the p-Laplace equation), we obtain existence of solutions to the nonlinear boundary value problem (7.1) by using the Nemytskii operator. Section 8 details the one-dimensional version of the eigenvalue problem consid- ered in Section 7, and we obtain a formula for the first eigenvalue of the p-curl operator explicitly; see equation (8.2). This result closely resembles the result for the first eigenvalue of the p-Laplace operator on W 1,p 0 (Ω). This is perhaps not surprising due to the similarities between the p-curl and p-Laplace operator; see in particular Theorem 4.5. Such one-dimensional eigenvalue problems have been studied by Drábek and Manásevich in [15] and Cringanu in [11]. We should mention that we have said “p-curl operator”, but the operator we consider in (7.6) also has a divergence term. This is due to the fact that a basic Poincaré inequality does not hold in this setting, and so we must also consider vector fields with well-defined divergence. Equation (4.3) provides the general Friedrichs type inequality for the Lp-norm of the gradient that holds in this setting. 2. Function spaces and trace theorems In this section, we prove trace theorems with respect to the spaces W 1,p(curl,Ω) and W 1,p(div,Ω) and obtain Green’s theorems corresponding to the trace results. We begin with the following definition. Definition 2.1. A bounded domain Ω ⊂ R3 is called a Lipschitz domain if for each point p ∈ ∂Ω there exists an open set O ⊂ R3 such that p ∈ O, and an orthogonal coordinate system with coordinates ξ = (ξ1, ξ2, ξ3) having the following property: there exists a vector b ∈ R3 so that O = {ξ : −bj < ξj < bj , 1 ≤ j ≤ 3} and a Lipschitz continuous function φ defined on the set O′ = {ξ′ ∈ R2 : −bj < ξ′j < bj , 1 ≤ j ≤ 2} such that Ω ∩ O = {ξ : ξ3 < φ(ξ′), ξ′ ∈ O′}, ∂Ω ∩ O = {ξ : ξ3 = φ(ξ′), ξ′ ∈ O′}. EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 3 The domain is said to be of class Cm,1 for an integer m ≥ 1 if the map φ can be chosen to be m-times differentiable with Lipschitz continuous partial derivatives of order m. We also need the notion of a Lipschitz domain with small Lipschitz constant. We say a domain Ω ⊂ R3 is a Lipschitz domain with small Lipschitz constant if it is a Lipschitz domain as in Definition 2.1 and there exists θ ∈ (0, 1] such that sup ξ′,η′∈O′, ξ′ 6=η′ |φ(ξ′)− φ(η′)| |ξ′ − η′| ≤ θ. We work with the general spaces W k,p(curl,Ω) = { u ∈ ( W k−1,p(Ω) )3 : ∇× u ∈ ( W k−1,p(Ω) )3} , 1 < p < +∞, with the norm ‖u‖Wk,p(curl,Ω) = ( ‖u‖p Wk−1,p + ‖∇ × u‖p Wk−1,p )1/p . Additionally, we define W k,p(div,Ω) = { u ∈ ( W k−1,p(Ω) )3 : ∇ · u ∈W k−1,p(Ω) } with the norm ‖u‖Wk,p(div,Ω) = ( ‖u‖p Wk−1,p + ‖∇ · u‖p Wk−1,p )1/p , where the last norm on the right hand side above is a scalar Sobolev norm. As in the case of Hilbert spaces, one can prove the denseness of smooth functions( C∞(Ω) )3 in these Sobolev spaces. We further define W 1,p 0 (curl,Ω) as the comple- tion of (C∞0 (Ω)) 3 in the W 1,p(curl,Ω) norm, and W 1,p 0 (div,Ω) as the completion of (C∞0 (Ω)) 3 in the W 1,p(div,Ω) norm. We also need the spaces Wp = W 1,p(curl,Ω) ∩W 1,p(div,Ω), WN = {u ∈Wp : γt(u) = 0}. We endow these spaces with the obvious graph norm. The map γt above is the tangential trace map, and it is defined classically for a smooth vector function u ∈ ( C∞(Ω) )3 by γt(u) = ν × u ∣∣ ∂Ω , where ν denotes the outer unit normal on ∂Ω. Remark 2.2. It is known that Wp does not compactly embed into Lp. However, we do have compact embedding of WN into Lp; see [3, Lemma 3.3], and this requires that the domain Ω have C1,1 regularity. Furthermore, we need the Besov spaces Bqs,p on the boundary of a Lipschitz do- main. In what follows, S denotes the Schwartz space of rapidly decreasing functions. Additionally, for f ∈ S, we denote by f̂ the Fourier transform of f . Moreover, we set Mj = { ξ ∈ R3 : 2j−1 ≤ |ξ| ≤ 2j+1 } for j = 1, 2, . . . and M0 = {ξ ∈ R3 : |ξ| ≤ 2}. Definition 2.3. For −∞ < q < ∞, 1 < s < ∞, 1 ≤ p < ∞, the Besov space Bqs,p is defined by Bqs,p = { f ∈ S ′ : f = ∞∑ j=0 aj(x), supp (âj) ⊂M0; ‖aj‖ <∞ } , 4 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 where the equality of f above is in the sense of tempered distributions, and the norm of aj is given by ‖aj‖ = [ ∞∑ j=0 ( 2qj‖aj‖Ls )p ]1/p , where ‖ · ‖Ls is the usual norm on the Lebesgue space. For a complete definition of Besov spaces on domains, we refer the reader to [29]. Definition 2.4. We say a distribution u on ∂Ω belongs to Bqs,p(∂Ω) if the compo- sition u ◦ φ ∈ Bqs,p ( O′ ∩ φ−1 (∂Ω ∩ O) ) for all possible O, φ as in Definition 2.1. We now prove trace results and Green’s theorems as their consequences. Theorem 2.5. Suppose Ω is a Lipschitz domain with small Lipschitz constant. The mapping γt(u) = ν×u ∣∣ ∂Ω defined on ( C∞(Ω) )3 can be extended by continuity to a continuous linear map γt from W 1,p(curl,Ω) to ( B −1/p′ p′,p′ (∂Ω) )3 . Moreover, the following Green’s theorem holds for any u ∈W 1,p(curl,Ω) and φ ∈W 1,p′(curl,Ω): 〈γt(u),φ〉∂Ω = ∫ Ω u · ∇ × φ dx− ∫ Ω ∇× u · φ dx. (2.1) The angle brackets above denote the duality pairing between ( B 1− 1 p p,p (∂Ω) )3 and( B −1/p′ p′,p′ (∂Ω) )3 . Remark 2.6. Assume p = 2. Then W 1,p(curl,Ω) is identified with the space H(curl,Ω). Additionally, the Besov space becomes B − 1 2 2,2 ≈ W− 1 2 ,2 which we can identify with the dual of H− 1 2 . Thus Theorem 2.5 is consistent with the well-known trace theorem for H(curl,Ω) functions, see [24, Theorem 3.29]. Theorem 2.7. If Ω is a Lipschitz domain with small Lipschitz constant, then W 1,p 0 (curl,Ω) = { u ∈W 1,p(curl,Ω) : γt(u) = 0 } = { u ∈W 1,p(curl,Ω) : ∫ Ω u · ∇ × φ dx = ∫ Ω ∇× u · φ dx ∀φ ∈ ( C∞(Ω) )3 } . (2.2) 2.1. Proof of Theorem 2.5. We adapt the techniques from the proof of [28, Lemma 6.2 ]; now to the Banach space setting. Our starting point is the formula∫ Ω ∇× u · φ dx = ∫ Ω u · ∇ × φ dx+ 〈γt(u),φ〉∂Ω (2.3) which holds for any u, φ ∈ ( C∞(Ω) )3 . This follows directly from the divergence theorem. By the standard density argument, (2.3) holds for φ ∈ W 1,p′(curl,Ω). The Cauchy-Schwarz inequality and Hölder’s inequality then yield |〈γt(u),φ〉∂Ω| ≤ ‖u‖W 1,p(curl,Ω)‖φ‖W 1,p′ (curl,Ω) EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 5 for all u ∈ ( C∞(Ω) )3 and φ ∈ W 1,p′(curl,Ω). Let µ ∈ B1− 1 p p,p (∂Ω). Consider the Neumann problem ∆v = 0 in Ω, ∂v ∂ν = µ on ∂Ω, v ∈W 1,p(Ω). (2.4) Take φ = ∇v where v solves (2.4). Then ‖φ‖W 1,p′ (curl,Ω) = ‖∇v‖Lp′ (Ω) since any gradient is in the kernel of the curl (as viewed as operators on Lq). Thus, we need to estimate the Lp ′ -norm of the gradient of the solution of (2.4). This highly depends on the geometry of the domain, which is why we restrict the domain to be Lipschitz with small Lipschitz constant. (Note that any C1 domain satisfies this assumption.) Then, by [14, Theorem 5], we can find a constant C > 0, depending on the Lipschitz nature of ∂Ω, such that ‖∇v‖Lp′ (Ω) ≤ C‖µ‖ B 1− 1 p p,p (∂Ω) . For a general Lipschitz domain, the entire range of p′s for such an estimate to hold is not expected; thus, if we wish to weaken the smoothness of the boundary we also have to decrease the range of allowed p’s. This is due to restrictions on solvability of the Neumann problem (2.4). Indeed, there is a (sharp) range of p values for solvability together with an Lp estimate for the gradient, see in particular [16], [19], or [33]. Note also that the result in [14] does not characterize the trace estimates using Besov spaces, but by adapting the ideas of [16] one can easily obtain the above estimate. Indeed, this can be done by using the fact that the trace of W 1,p(Ω) is the Besov space B 1− 1 p p,p (∂Ω), see [19]. We then have that ‖γt(u)‖( B −1/p′ p′,p′ (∂Ω) )3 = sup φ∈ ( B 1− 1 p p,p (∂Ω) )3 , ‖φ‖=1 |〈γt(u),φ〉| ≤ ‖u‖W 1,p(curl,Ω)‖φ‖W 1,p′ (curl,Ω) ≤ C‖u‖W 1,p(curl,Ω)‖µ‖ B 1− 1 p p,p (∂Ω) = C‖u‖W 1,p(curl,Ω), where C = C(θ), i.e. the constant depends on the Lipschitz character of the domain. For more on the dual of Besov spaces, see [25]. Additionally, for this characterization of the Besov space norm on the boundary, see [21]. 2.2. Proof of Theorem 2.7. We need the following lemma to prove the theorem. Lemma 2.8. Suppose that u ∈W 1,p(curl,Ω) is such that for each φ ∈ ( C∞(Ω) )3 , it holds ∫ Ω ∇× u · φ dx− ∫ Ω u · ∇ × φ dx = 0. Then u ∈W 1,p 0 (curl,Ω). 6 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 Assuming Lemma 2.8 for now; it in particular implies that the set{ u ∈W 1,p(curl,Ω) : ∫ Ω ∇× u · φ dx− ∫ Ω u · ∇ × φ dx = 0, ∀φ ∈ ( C∞(Ω) )3 } is a subset of W 1,p 0 (curl,Ω). Then we apply Theorem 2.5 to u such that γt(u) = 0 to obtain{ u ∈W 1,p(curl,Ω) : γt(u) = 0 } ⊂ { u ∈W 1,p(curl,Ω) : ∫ Ω ∇× u · φ dx− ∫ Ω u · ∇ × φ dx = 0, ∀φ ∈ ( C∞(Ω) )3 } . Since ( C∞0 (Ω) )3 ⊂ {u ∈W 1,p(curl,Ω) : γt(u) = 0} and the set {u ∈W 1,p(curl,Ω) : γt(u) = 0} is closed due to continuity of the trace map, we conclude that W 1,p 0 (curl,Ω) ⊂ { u ∈W 1,p(curl,Ω) : γt(u) = 0 } . Proof of Lemma 2.8. The proof is similar to the proof of [24, Lemma 3.27] with a few adjustments, and so we shall provide a sketch of the proof with the necessary adjustments. Since Ω is a bounded Lipschitz domain, we can find a collection of open sets Uj such that Ω ⊂ ∪Mj=1Uj and such that each Ωj := Uj ∩Ω is a bounded and starlike Lipschitz domain. Then there is a partition of unity subordinate to this open cover; that is, there exist functions {αj}Mj=1 such that each αj ∈ C∞0 (Uj), as well as 0 ≤ αj(x) ≤ 1 and ∑M j=1 αj = 1 for all x ∈ Ω. Let ũ denote the extension of u by zero outside of Ω. Clearly, ũ ∈ W 1,p(curl,R3). By the construction of αj , we have ũ(x) = M∑ j=1 αjũ(x), x ∈ Ω, and ũj := αjũ ∈ W 1,p(curl,R3). Let ũj,t(x) := ũj(x/t) for 0 < t < 1. Then ũj,t → ũj in W 1,p(curl,R3) as t→ 1. Let Mε = ρε ∗ v for v ∈ (Lp(R3))3 denote the mollification of v for a mollifier ρε. Then Mε → v in (Lp(R)3)3 as ε→ 0, and by differentiability properties of the convolution, we have ∇ ×Mε = ρε ∗ (∇ × v). Thus, ρε ∗ ũj,t → ũj,t as ε → 0 in W 1,p(curl,R3). We can then find sequences {tk}, {εk}, with 0 < tk, εk < 1, such that εk → 0, tk → 1 and ρεk ∗ ũj,tk → ũj in W 1,p(curl,Ωj). The function ũ(k) := M∑ j=1 ρεk ∗ ũj,tk → u in W 1,p(curl,Ω). Thus, we conclude that u ∈ W 1,p 0 (curl,Ω) (note that ũ(k) ∈ (C∞0 (Ω)) 3 for each k). � 2.3. Traces of W 1,p(div,Ω) functions. We can similarly analyze traces of func- tions belonging to W 1,p(div,Ω). First, we define for a smooth vector u the normal trace γn(u) = u ∣∣ ∂Ω · ν, where ν is the outer unit normal vector on ∂Ω. EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 7 Theorem 2.9. Suppose Ω is a C1,1 domain. The mapping γn(u) defined on( C∞(Ω) )3 can be extended by continuity to a continuous linear map γn from W 1,p(div,Ω) to B −1/p p,p (∂Ω). Moreover, the following Green’s theorem holds for any u ∈W 1,p(div,Ω) and φ ∈W 1,p′(Ω): 〈γn(u), φ〉∂Ω = (u,∇φ)L2 + (∇ · u, φ)L2 . (2.5) Note that Ω is C1,1 is needed here to obtain the estimate (2.6) below. Again, if Ω were merely Lipschitz, then further restrictions of p would need to be imposed; see [19]. Proof. Similar to the proof of Theorem 2.5, we start with the Green’s formula for smooth functions φ ∈ C∞(Ω) (v,∇φ)L2 + (∇ · v, φ)L2 = 〈φ, γn(v)〉∂Ω which by density argument can be extended to hold for φ ∈ W 1,p′(Ω). By the Cauchy-Schwarz inequality, we have |〈γn(v), φ〉| ≤ ‖v‖W 1,p(div,Ω)‖φ‖W 1,p′ (Ω) ∀φ ∈W 1,p′(Ω), v ∈ (C∞(Ω))3. Let g ∈ B 1− 1 p′ p′,p′ (∂Ω). Take φ = u where u solves ∆u = 0 in Ω, with boundary condition u ∣∣ ∂Ω = g. Such a solution exists, and one can find a constant c > 0 such that ‖u‖W 1,p′ (Ω) ≤ c‖g‖ B 1− 1 p′ p′,p′ (∂Ω) , (2.6) see [19]. Then, as in the proof of Theorem 2.5, we see that ‖γn(v)‖ B −1/p p,p (∂Ω) ≤ c‖v‖W 1,p(div,Ω) ‖g‖ B 1− 1 p′ p′,p′ (∂Ω) , and therefore the continuity is established. � Remark 2.10. Just as in Remark 2.6, when p = 2, the space W 1,p(div,Ω) is iden- tified with the space H(div,Ω). We recover the normal trace result [24, Theorem 3.24] in this case as well. Later on, we will need the space W 0 N := {u ∈WN : γn(u) = 0} . Similar to Theorem 2.7, it is straightforward to see that the following result holds. Theorem 2.11. If Ω is a Lipschitz domain with small Lipschitz constant, then W 1,p 0 (div,Ω) = { u ∈W 1,p(div,Ω) : γn(u) = 0 } . (2.7) Thus, we see that W 0 N = W 1,p 0 (curl,Ω) ∩W 1,p 0 (div,Ω). (2.8) 8 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 3. Duality mapping on W 1,p(curl,Ω) Let X be a real Banach space and X∗ its dual. Let 〈·, ·〉 denote the duality pairing. Given an operator T : X → 2X ∗ , define the range of T by R(T ) = ∪x∈D(T )Tx, where as usual D(T ) := {x ∈ X : Tx 6= ∅} is the effective domain of T . The graph of T is the set G(T ) := {(x, y) ∈ X ×X∗ : y ∈ Tx, x ∈ D(T )}. The operator T is said to be monotone if 〈x∗1 − x∗2, x1 − x2〉 ≥ 0 (3.1) for all (x1, x ∗ 1), (x2, x ∗ 2) ∈ G(T ). The operator T is strictly monotone if it is mono- tone and the equality in (3.1) implies x1 = x2. We say that a continuous function φ : R+ → R+ is a normalization function if it is strictly increasing, φ(0) = 0, and φ(t) → ∞ as t → ∞. The duality mapping corresponding to φ is the set valued mapping Jφ : X → 2X ∗ defined by Jφx = {x∗ ∈ X∗ : 〈x∗, x〉 = φ(‖x‖)‖x‖, ‖x∗‖ = φ(‖x‖)} , x ∈ X. We note that D(Jφ) = X by the Hahn-Banach theorem. Some main properties of Jφ are collected in the following theorem (cf. [20]). Theorem 3.1. If φ is as above, then (1) for all x ∈ X, Jφx is a bounded, closed, and convex subset of X∗; (2) Jφ is monotone, i.e. 〈x∗1 − x∗2, x1 − x2〉 ≥ (φ(‖x1‖)− φ(‖x2‖)) (‖x1‖ − ‖x2‖) ≥ 0 for all (x1, x ∗ 1), (x2, x ∗ 2) ∈ G(Jφ); and (3) for every x ∈ X, there holds Jφx = ∂ψ(x), where ψ(x) = ∫ ‖x‖ 0 φ(t) dt (3.2) and ∂ψ : X → 2X ∗ is the subdifferential of ψ defined by ∂ψ(x) = {x∗ ∈ X∗ : ψ(y)− ψ(x) ≥ 〈x∗, y − x〉 ∀y ∈ X} . Further, recall that a functional f : X → R is called Gâteaux differentiable at x ∈ X if there exists f ′(x) ∈ X∗ such that lim t→0 f(x+ th)− f(x) t = 〈f ′(x), h〉 for all h ∈ X. Additionally, we need the following definition. Definition 3.2. A real Banach space X is said to be (1) uniformly convex if for each ε ∈ (0, 2], there exists δ = δ(ε) > 0 such that if ‖x‖ = ‖y‖ = 1 and ‖x− y‖ ≥ ε, then ‖x+ y‖ ≤ 2(1− δ); (2) locally uniformly convex if for ‖x‖ = ‖xn‖ = 1 and ‖xn+x‖ → 2 as n→∞, then xn → x strongly in X; and (3) strictly convex if for every x, y ∈ X with ‖x‖ = ‖y‖ = 1, x 6= y and λ ∈ (0, 1), there holds ‖λx+ (1− λ)y‖ < 1. Remark 3.3. It is well-known that if X is reflexive with both X and X∗ locally uniformly convex, the duality mapping Jφ is a single-valued homeomorphism of X onto X∗. For these and further properties of duality mappings, the reader is referred to [8, 10]. EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 9 Example 3.4. For X = W 1,p 0 (Ω) with 1 < p < ∞, and φ(t) = tp−1, it is shown in [13] by applying the Poincaré inequality that Jφ in this context is precisely the negative of the p-Laplacian ∆p: Jφ : W 1,p 0 (Ω)→W−1,p′(Ω), Jφ(u) = −∆pu := −div ( |∇u|p−2∇u ) , u ∈W 1,p 0 (Ω). When X = W 1,p(Ω), it is shown in [11] that Jφ : W 1,p(Ω)→ ( W 1,p(Ω) )∗ , Jφu = −∆pu+ |u|p−2u, u ∈W 1,p(Ω). Throughout this section, unless otherwise noted, we assume that p ≥ 2 and that Ω is a bounded Lipschitz domain. We next compute the duality mapping on W 1,p(curl,Ω) with respect to its norm given by ‖u‖pW 1,p(curl,Ω) = ‖u‖pLp + ‖∇ × u‖pLp and corresponding to the normalization function φ(t) = tp−1. Recall now that if a convex functional f : W 1,p(curl,Ω) → R is Gâteaux differentiable at u, then ∂f(u) = f ′(u), where ∂f is the subdifferential of f . By Theorem 3.1, part (3), we know that Jφ = ∂ψ, where ψ(u) = 1 p (‖u‖pLp + ‖∇ × u‖pLp) := ψ1(u) + ψ2(u). It is well-known that the functional F : (Lp(Ω)) 3 → R given by u 7→ p−1‖u‖pLp is Gâteaux differentiable and 〈F ′(v),h〉 = 3∑ i=1 ∫ Ω |vi|p−2vihi dx ∀v,h ∈ (Lp(Ω)) 3 . (3.3) Thus, it remains to compute the Gâteaux derivative of ψ2. Now, we write ψ2 = FG, where F is the functional above and G : W 1,p(curl,Ω) → (Lp(Ω)) 3 is defined by G(v) = |∇ × v|. We need to check differentiability of the functional G. But, the derivative is easily computed to be G′(u) · v = ∇× u · ∇ × v |∇ × u| . We obtain 〈ψ′(u),v〉 = 〈ψ′1(u),v〉+ 〈ψ′2(u),v〉 = 〈|u|p−2u + curlp(u),v〉, (3.4) where we define curlp : W 1,p(curl,Ω)→ ( W 1,p(curl,Ω) )∗ by 〈curlp(u),v〉 = ∫ Ω |∇ × u|p−2∇× u · ∇ × v dx ∀u,v ∈W 1,p(curl,Ω). In view of Theorem 2.7, we see that curlp : W 1,p 0 (curl,Ω) → (W 1,p 0 (curl,Ω))∗ is given by curlp(u) := ∇× (|∇ × u|p−2∇× u) ∀u,v ∈W 1,p 0 (curl,Ω). (3.5) Thus, we have shown the following result. 10 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 Theorem 3.5. The duality mapping Jφ : W 1,p(curl,Ω)→ ( W 1,p(curl,Ω) )∗ corre- sponding to the normalization function φ(t) = tp−1 is given by Jφu = |u|p−2u + curlp(u) for each u ∈ W 1,p(curl,Ω). In particular, it is coercive and a strictly monotone homeomorphism. As a result of the surjectivity of the duality mapping, we obtain the following results. Corollary 3.6. For each f ∈ ( W 1,p(curl,Ω) )∗ , the equation |u|p−2u+curlp(u) = f has a unique solution in W 1,p(curl,Ω). Theorem 3.7. The operator ∇p × u := |u|p−2u + curlp(u) satisfies the (S+)- condition; i.e., if un ⇀ u0 weakly in W 1,p(curl,Ω) and lim sup n→∞ 〈−∇p × un,un − u0〉 ≤ 0, then un → u0 strongly in W 1,p(curl,Ω). This follows immediately from the previous results coupled with [13, Prop. 2]. Our next goal is to show that the functional ψ(u) = 1 p (‖u‖pLp + ‖∇ × u‖pLp) is continuously Fréchet differentiable onW 1,p(curl,Ω). To do so, we need the following lemma from [17]. Lemma 3.8. If p ≥ 2, then for all x, y, z ∈ Rn, there exists a constant C1 > 0 such that ∣∣|z|p−2z − |y|p−2y ∣∣ ≤ C1|z − y| (|z|+ |y|)p−2 . Using the above lemma we prove the next theorem. Theorem 3.9. The functional ψ(u) = 1 p (‖u‖pLp + ‖∇ × u‖pLp) is continuously Fré- chet differentiable on W 1,p(curl,Ω). Proof. Let u,v,w ∈ W 1,p(curl,Ω). Then we have from our previous calculations that |〈ψ′(u)− ψ′(v),w〉| = ∣∣〈|u|p−2u− |v|p−2v,w〉+ 〈curlp(u)− curlp(v),w〉 ∣∣ = ∣∣∣ ∫ Ω ( |u|p−2u− |v|p−2v ) ·w dx + ∫ Ω ( |∇ × u|p−2∇× u− |∇ × v|p−2∇× v ) · ∇ ×w dx ∣∣∣ ≤ ‖ |u|p−2u− |v|p−2v‖Lp′‖w‖Lp + ‖ |∇ × u|p−2∇× u − |∇ × v|p−2∇× v‖Lp′‖∇ ×w‖Lp , (3.6) where we have used the Hölder’s inequality. We start by estimating the Lp ′ -norm of the first term on the right hand side of (3.6). Using Hölder’s inequality coupled with Lemma 3.8, we obtain ‖ |u|p−2u− |v|p−2v‖p ′ Lp′ = ∫ Ω ∣∣|u|p−2u− |v|p−2v ∣∣p′ dx ≤ C ∫ Ω |u− v|p ′ (|u|+ |v|)(p−2)p′ dx EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 11 ≤ C‖ |u− v|p ′ ‖Lp−1‖ (|u|+ |v|)(p−2)p′ ‖ L p−1 p−2 = C‖u− v‖ p p−1 Lp ‖ |u|+ |v|‖(p−2)p′ Lp , which implies ‖ |u|p−2u− |v|p−2v‖Lp′ ≤ C‖u− v‖Lp‖ |u|+ |v| ‖(p−2) Lp . (3.7) In a similar fashion, we can find a constant C ′ > 0 such that ‖ |∇ × u|p−2∇× u− |∇ × v|p−2∇× v‖Lp′ ≤ C ′‖∇ × (u− v)‖Lp‖ |∇ × u|+ |∇ × v| ‖p−2 Lp . (3.8) Combining estimates (3.7) and (3.8) allows us to conclude that there exists some constant C (after renaming) such that |〈ψ′(u)− ψ′(v),w〉| ≤ C‖u− v‖W 1,p(curl,Ω)‖w‖W 1,p(curl,Ω) for all u,v,w ∈W 1,p(curl,Ω), which establishes the desired result since a functional is continuously Fréchet differentiable if and only if it is continuously Gâteaux dif- ferentiable. � 4. Duality mappings on WN : part I We would like to compute Jφ for W 1,p 0 (curl,Ω) with φ(t) = tp−1. In the case of the standard Sobolev spaces W 1,p 0 (Ω), a key result that was used in [11, 13] to compute the duality mapping was the Poincaré inequality ‖u‖Lp ≤ C(Ω, n)‖∇u‖Lp ∀ u ∈W 1,p 0 (Ω). It was shown in [3] that for u ∈W 1,p 0 (curl,Ω) on a C1,1 domain, ‖u‖Lp ≤ C (‖∇ × u‖Lp + ‖∇ · u‖Lp + |〈u · ν, 1〉∂Ω|) . Thus, we can generally estimate the Lp-norm of a function u not only in terms of its curl, but also its divergence and a certain boundary trace. So, an equivalent norm on W 1,p 0 (curl,Ω) is given by w 7→ ‖∇ ×w‖Lp + ‖∇ ·w‖Lp + |〈w · ν, 1〉∂Ω|. In any case, one must have a well-defined divergence, and hence if we are interested in computing a duality mapping on a “trace-zero” space, it must be WN . If we assume that Ω has a C1,1 boundary, then WN in the case when p = 2 can be identified with the Sobolev space (H1(Ω))3 with equivalent norms; see [12, Theorem 3, p. 209]. This relies on the vector having zero tangential trace (a similar analysis works if the normal trace vanishes). We want to extend this result for more general p. For this we use Peetre’s lemma from [27]. Lemma 4.1. Let E0, E1, E2 be three Banach spaces, and let A1 : E0 → E1, A2 : E0 → E2 be continuous linear maps, such that (1) A2 is compact and (2) there exists C > 0 such that ‖v‖E0 ≤ C (‖A1v‖E1 + ‖A2v‖E2 ) ∀v ∈ E0. (4.1) 12 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 Then ker(A2) has finite dimension and Im(A1) is closed, and there exists C0 > 0 such that inf w∈ker(A1) ‖v + w‖E0 ≤ C0‖A1v‖E1 . (4.2) Now we show the equivalence of WN with the space WN,0 := {u ∈W 1,p(Ω) : γt(u) = 0}, where the norm here is ‖u‖WN,0 := ‖u‖W 1,p(Ω). Theorem 4.2. Let Ω be a C1,1 domain. Then the spaces WN and WN,0 can be identified and have equivalent norms. Proof. We will apply Lemma 4.1 with E0 = WN,0, E1 = (Lp(Ω)) 3 × Lp(Ω) × (Lp(Ω)) 3 , and E2 = B −1/p′ p′,p′ (∂Ω). The operators we take are given by A1(v) = (∇× v,∇ · v,v), A2(v) ≡ 0. We need to prove an estimate of the form (4.1), which boils down to an Lp-estimate for ∇v. It is precisely here that we require the domain to be C1,1. Given v ∈ E0, by [3, Theorem 3.1], we can find C > 0 such that ‖∇v‖pLp ≤ C (‖v‖pLp + ‖∇ × v‖pLp + ‖∇ · v‖pLp) , (4.3) By using (4.3), we have ‖v‖pWN,0 = ‖v‖pLp + ‖∇v‖pLp ≤ C (‖v‖pLp + ‖∇ × v‖pLp + ‖∇ · v‖pLp) = C‖v‖WN . Thus, since ker(A1) = {0}, (4.2) implies the equivalence of the norms ‖ · ‖WN,0 and ‖ · ‖WN . � Remark 4.3. Let us briefly discuss the assumptions needed in Theorem 4.2. We have used the Lp-estimate from [3], which actually gives ‖∇v‖Lp ≤ C (‖∇ · v‖Lp + ‖∇ × v‖Lp + |〈v · ν, 1〉∂Ω|) . (4.4) This coupled with the following estimate∣∣ ∫ ∂Ω (Tr(B)) (v · ν)2dσ ∣∣ ≤ C ∫ ∂Ω |v|2dσ ≤ 1 2 ‖∇v‖2L2(Ω) + C‖v‖2L2(Ω) yields (4.3). Above, B denotes the curvature tensor of ∂Ω and Tr denotes the trace. It is unclear if these estimates would hold if the domain were merely Lipschitz. Our thought is likely it is not possible, since to get solvability in W 1,p, the domain should be at least C1,1. In [18], particularly Lemma 3.1.1.2, the domain is Lipschitz with the additional assumption that it is piecewise C2. Thus, the Banach space E2 we have taken is somewhat arbitrary due to A2 ≡ 0. However, if we were to use the estimate (4.4), then the boundary term would have to be incorporated into A2 on E2, and in order for this to be compact, one would need compact embedding of certain Besov spaces. Finally, instead of C1,1, one could take the domain to be convex. The idea here is that one can approximate a convex domain by an increasing sequence of convex, C1,1 open sets [18, Lemma 3.2.1.1]. Now that we have the equivalence of norms, we can appeal to the known result [11] for the duality mapping on W 1,p(Ω). First, we need the following theorem. Theorem 4.4. Let Ω be a C1,1 domain. Then the space (WN , ‖ · ‖W 1,p) is uni- formly convex, reflexive, and separable. EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 13 For completeness, we prove the Theorem for 1 < p <∞. Proof. It is well-known that if X is uniformly convex, then it is reflexive. To show uniform convexity, first let p ≥ 2. Take u,v ∈WN with ‖u‖W 1,p = ‖v‖W 1,p = 1, and ‖u−v‖W 1,p ≥ ε ∈ (0, 2]. For z, w ∈ Rn we know that (see [1])∣∣z + w 2 ∣∣p + ∣∣z − w 2 ∣∣p ≤ 1 2 (|z|p + |w|p) . Then we have ‖u + v 2 ‖pW 1,p + ‖u− v 2 ‖pW 1,p = ∫ Ω (∣∣u + v 2 ∣∣p + ∣∣u− v 2 ∣∣p)dx+ ∫ Ω (∣∣∇u +∇v 2 ∣∣p + ∣∣∇u−∇v 2 ∣∣p)dx ≤ 1 2 ∫ Ω (|u|p + |v|p) dx+ 1 2 ∫ Ω (|∇u|p + |∇v|p) dx = 1 2 (‖u‖W 1,p + ‖v‖W 1,p) = 1, and hence ‖u + v 2 ‖pW 1,p ≤ 1− ( ε 2 )p . (4.5) When 1 < p < 2, then it is also known [1] that for z, w ∈ Rn,∣∣z + w 2 ∣∣p′ + ∣∣z − w w ∣∣p′ ≤ [1 2 (|z|p + |w|p) ] 1 p−1 . Take u,v as above. First, notice that ‖ · ‖pLp = ‖ | · |p′‖p−1 Lp−1 . We then have that ‖u + v 2 ‖pW 1,p + ‖u− v 2 ‖pW 1,p = ‖u + v 2 ‖pLp + ‖u− v 2 ‖pLp + ‖∇u +∇v 2 ‖pLp + ‖∇u−∇v 2 ‖pLp = ‖ ∣∣u + v 2 ∣∣p′‖p−1 Lp−1 + ‖ ∣∣u− v 2 ∣∣p′‖p−1 Lp−1 + ‖ ∣∣∇u +∇v 2 ∣∣p′‖p−1 Lp−1 + ‖ ∣∣∇u +∇v 2 ∣∣p′‖p−1 Lp−1 ≤ ‖ ∣∣u + v 2 ∣∣p′ + ∣∣u− v 2 ∣∣p′‖p−1 Lp−1 + ‖ ∣∣∇u +∇v 2 ∣∣p′ + ∣∣∇u−∇v 2 ∣∣p′‖p−1 Lp−1 ≤ 1 2 (‖u‖pLp + ‖∇u‖pLp + ‖v‖pLp + ‖∇v‖pLp) = 1 2 (1 + 1) = 1, where in the first inequality we have used that 0 < p − 1 < 1. Therefore, since ‖u− v‖W 1,p ≥ ε, again we obtain that ‖u + v 2 ‖pW 1,p ≤ 1− ( ε 2 )p (4.6) and uniform convexity is proved. Finally, for separability, we require p <∞, since then Lp is separable. Then it’s easy to construct an isometry from W 1,p onto a subspace of Lp, so separability follows immediately. � 14 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 Theorem 4.5. Let Ω be a C1,1 domain and p ≥ 2. Then the duality mapping Jφ : (WN , ‖ · ‖W 1,p) → (WN , ‖ · ‖W 1,p) ∗ corresponding to the normalization func- tion φ(t) = tp−1 is given by Jφu = −∆pu + |u|p−2u for each u ∈WN . Above, ∆p denotes the p-Laplace operator on (WN , ‖ · ‖W 1,p). Proof. Letting Jφ : (WN , ‖ · ‖W 1,p)→ (WN , ‖ · ‖W 1,p) ∗ , with φ(t) = tp−1, we know that Jφ(u) = ∂Φ(‖u‖) = ∂ (∫ ‖u‖W1,p 0 tp−1 dt ) = Φ1(u) + Φ2(u), where Φ1(u) = 1 p ‖u‖pLp and Φ2(u) = 1 p ‖∇u‖pLp and ∂ denotes the subdifferential. We calculate Φ′2(u). Simple computations using (3.3) imply that 〈Φ′2(u),v〉 = 3∑ j=1 〈|∇uj |p−2∇uj ,∇vj〉. Thus, we conclude that 〈Φ′(u),v〉 = 3∑ j=1 ( 〈|uj |p−2uj , vj〉+ 〈|∇uj |p−2∇uj ,∇vj〉 ) = 3∑ j=1 [ ∫ Ω |uj |p−2ujvj dx+ ∫ Ω |∇uj |p−2∇uj · ∇vj dx ] . � We now make precise the way the p-Laplacian ∆p acts on (WN , ‖ · ‖W 1,p). If u ∈WN and ∇ · ( |∇uj |p−2∇uj ) ∈ (Lp(Ω)) 3 , j = 1, 2, 3, then the traces γn(u) and γn(|∇uj |p−2∇uj) make sense. Setting γn(|∇u|p−2∇u) := ( γn(|∇u1|p−2∇u1), γn(|∇u2|p−2∇u2), γn(|∇u3|p−2∇u3) ) , Theorem 2.9 then implies that 〈γn(|∇u|p−2∇u),φ〉 = 3∑ j=1 [ ∫ Ω ( ∇ · |∇uj |p−2∇uj ) φj dx+ ∫ Ω |∇uj |p−2∇uj · ∇φj dx ] for all u,φ ∈WN . If each |∇uj |p−2∇uj ∈ ker(γn), j = 1, 2, 3, then 3∑ j=1 ∫ Ω − ( ∇ · |∇uj |p−2∇uj ) φj dx = 3∑ j=1 ∫ Ω |∇uj |p−2∇uj · ∇φj dx ∀φ ∈WN . Note that the integral on the right-hand side above exists for all u,φ ∈WN . Thus, we denote ∆pu := ∇ · ( |∇u|p−2∇u ) , u ∈WN . This should be understood componentwise, so that (∆pu)j = ∇ · ( |∇uj |p−2∇uj ) , j = 1, 2, 3. EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 15 Remark 4.6. Suppose u(x1, x2, x3) = (u(x1, x2, x3) 0 0)t. Then the previous calculations for the derivative of Φ(u) reduce to 〈Φ′(u), v〉 = ∫ Ω |u|p−2uv dx+ ∫ Ω |∇u|p−2∇u · ∇v dx ∀u, v ∈W 1,p(Ω). Thus, the p-Laplacian ∆p in this case becomes the usual p-Laplacian on the space W 1,p(Ω) because the space WN reduces to the (scalar) space W 1,p(Ω). The duality mapping from Theorem 4.5 then agrees with that in [11, Theorem 3.1]. In the next section, we consider duality mappings on WN endowed with its graph norm. 5. Duality mappings on WN : part 2 In this section we can take Ω to be a bounded C1,1 domain; the reason for this is to ensure the definition (5.2) below makes sense. We have ‖u‖pWN = ‖u‖pLp + ‖∇ × u‖pLp + ‖∇ · u‖pLp . To compute the duality mapping on WN , define ψ : WN → R by ψ(u) = 1 p (‖u‖pLp + ‖∇ × u‖pLp + ‖∇ · u‖pLp) := ψ1(u) + ψ2(u) + ψ3(u). By Theorem 3.1, part (3), we know that Jφ = ∂ψ(x). We have previously computed ψ′1(u) and ψ′2(u). Thus, it remains to compute the Gâteaux derivative of ψ3. We can write ψ3 = FH, where H(v) = |∇ · v| and F (v) = p−1‖v‖pLp , to obtain H ′(u) · v = (∇ · u)(∇ · v) |∇ · u| . Thus, we get 〈ψ′(u),v〉 = 〈ψ′1(u),v〉+ 〈ψ′2(u),v〉+ 〈ψ′3(u),v〉 = 〈|u|p−2u + curlp(u) + divp(u),v〉, (5.1) where we have defined divp : W 0 N → ( W 0 N )∗ , divp(u) := −∇(|∇ · u|p−2∇ · u), (5.2) in the sense that divp acts, in view of Theorems 2.9 and 2.11, by 〈divp(u),v〉 = ∫ Ω |∇ · u|p−2 (∇ · u)(∇ · v) dx ∀u,v ∈W 0 N . Hence we have shown the following theorem. Theorem 5.1. Let Ω be a bounded C1,1 domain. Then, the duality mapping Jφ : W 0 N → ( W 0 N )∗ corresponding to the normalization function φ(t) = tp−1 is given by Jφu = |u|p−2u + curlp(u) + divp(u) for each u ∈ W 0 N . In particular, it is coercive and a strictly monotone homeomor- phism. As a result of the surjectivity of the duality mapping, we obtain the following result. Corollary 5.2. For each f ∈ ( W 0 N )∗ , the equation |u|p−2u+curlp(u)+divp(u) = f has a unique solution in W 0 N . 16 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 6. Generalizations This method appears to be generalizable as follows. Let now X be an arbitrary Banach space with norm ‖·‖X , such that there exists a ∈ [1,∞) and a C1 functional F so that ‖u‖aX = ∫ Ω F (u(x))dx. (6.1) We require that (X, ‖·‖X) be uniformly convex, which is equivalent to the following which we further assume that the norm on X∗ is uniformly Fréchet differentiable on {x ∈ X : ‖x‖X = 1}. (6.2) Recall that the norm on a Banach space is said to be uniformly Fréchet differentiable on the unit sphere if lim h→0 ∣∣‖x+ hy‖ − ‖x‖ h − fx(y) ∣∣ exists uniformly in x and y on the unit sphere in X. Above fx(y) denotes a support functional; see [23] for more details. As before, we let ψ(u) = a−1‖u‖aX . Then our previous calculations show that 〈Jφ(u), v〉 = 〈ψ′(u), v〉 = ∫ Ω F ′(u(x))v(x) dx. (6.3) This formula agrees with the well-known derivative of the Lp norm by taking a = p and F (u) = |u|p in X = Lp(Ω), as well as the result from [13] by taking a = p, F (u) = |∇u|p and X = W 1,p 0 (Ω). Theorem 6.1. Let (X, ‖ · ‖X) be a uniformly convex Banach space such that (6.1) and (6.2) hold. Then the duality mapping corresponding to the normalization func- tion φ(t) = ta−1 is the single-valued function Jφ : (X, ‖ · ‖X)→ (X, ‖ · ‖X) ∗ satisfying (6.3). 7. On the problem Lp(u) = f(x,u) In this section we assume that Ω has a C1,1 boundary or is convex, so that WN ↪→ Lp compactly. We first define a vector valued variant of Carathéodory functions. Definition 7.1. A vector-valued function f : Ω×R3 → R3 is called Carathéodory provided (i) for each s ∈ R3, the function x 7→ f(x, s) is measurable in Ω; and (ii) for a.e. x ∈ Ω, the function s 7→ f(x, s) is continuous in R3. For a vector-valued Carathéodory function f , for each measurable vector-valued function u = (u1, u2, u3), the function (Nfu)(x) = f(x,u(x)) is measurable. The operator Nf from the set of measurable functions to itself is called the Nemytskii operator. We will consider what conditions on f are required in order to obtain existence of a u ∈W 0 N of the nonlinear boundary value problem |u|p−2u + curlp(u) + divp(u) = f(x,u) in Ω, γt(u) = 0 on ∂Ω. (7.1) EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 17 Note that since we are seeking u ∈W 0 N we also have γn(u) = 0 on ∂Ω. Remark 7.2. Consider the diagram W 1,p 0 (curl,Ω) Id ↪→ (Lq(Ω)) 3 Nf ↪→ ( Lq ′ (Ω) )3 Id∗ ↪→ ( W 1,p 0 (curl,Ω) )∗ . If W 1,p 0 (curl,Ω) ↪→ (Lq(Ω)) 3 compactly, then we would be able to conclude that Nf is a compact operator. Given the results in [22, 6, 7], it is unlikely to expect any compactness without imposing a divergence condition as well. For this reason we seek solutions of (7.1) in WN . Note that (7.1) is understood in the sense of ( W 0 N )∗ : 〈|u|p−2u + curlp(u) + divp(u),v〉 = 〈Nf (u),v〉 ∀v ∈W 0 N . (7.2) The following result will be useful to establish the compactness of Nf , see [30, Theorem 19.1]. Proposition 7.3. Let f : Ω× R3 → R3 be Carathéodory such that |fi(x, u1, u2, u3)| ≤ C 3∑ k=1 |uk|r + bi(x), x ∈ Ω, i = 1, 2, 3, where r > 0, f(x,u(x)) = (f1(x,u(x)), f2(x,u(x)), f3(x,u(x))) with u = (u1, u2, u3), and each bi ∈ Lq(Ω), 1 ≤ q < ∞. Then Nf ( (Lqr(Ω)) 3 ) ↪→ (Lq(Ω)) 3 continuously and maps bounded sets into bounded sets. Note that if r = q − 1 and b ∈ ( Lq ′ (Ω) )3 , then from Proposition 7.3 we have Nf ( (Lq(Ω)) 3 ) ↪→ ( Lq ′ (Ω) )3 and Nf ( (Lq(Ω)) 3 ) ↪→ ( L1(Ω) )3 continuously. Thus, we will assume that the right hand side f in (7.1) is Carathéodory as well as satisfies the growth condition |fi(x, u1, u2, u3)| ≤ C 3∑ k=1 |uk|p−1 + bi(x), x ∈ Ω, i = 1, 2, 3 (7.3) for some C ≥ 0, with bi ∈ Lp ′ (Ω). Thus, by considering W 0 N Id ↪→ (Lp(Ω)) 3 Nf ↪→ ( Lp ′ (Ω) )3 Id∗ ↪→ ( W 0 N )∗ , under the previous assumptions we have that Nf is compact. For u ∈ W 0 N , let ψ(u) = 1 p (‖u‖pLp + ‖∇ × u‖pLp + ‖∇ · u‖pLp) as in Section 5. Then, using the method of proof from Theorem 3.9, it can be shown that ψ is continuously Fréchet differentiable on W 0 N . Next we are interested in seeing when the Nemytskii operator Nf can be written as the gradient of some functional. From [30, Theorem 21.1], we know that if for some real-valued F (x, u1, u2, u3), fi(x, u1, u2, u3) = ∂ ∂ui F (x, u1, u2, u3), F (x, 0, 0, 0) = 0, i = 1, 2, 3 (7.4) with each fi satisfying (7.3), then the functional Φ : W 0 N → R given by Φ(u) = ∫ Ω F (x,u(x)) dx (7.5) satisfies ∇Φ = Nf . 18 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 Thus, the functional F : W 0 N → R given by F (u) = ψ(u)− Φ(u) = 1 p (‖u‖pLp + ‖∇ × u‖pLp + ‖∇ · u‖pLp)− ∫ Ω F (x,u) dx is continuously differentiable on W 0 N and F ′(u) = Lp(u)−Nf (u), where we have defined Lp(u) = |u|p−2u + curlp(u) + divp(u). (7.6) Thus, problem (7.1) is reduced to finding critical points of F on W 0 N . Note that since WN ↪→ (Lp(Ω)) 3 , it holds that F is weakly lower semicontinuous, i.e. when- ever uj → u strongly in WN , it holds that lim infj→∞F (uj) ≥ F (u). It is enough now to prove that F is coercive. One way to show this is via the method of Anane and Gossez for the p-Laplace operator [4]. To this end, we proceed to understand the first eigenvalue of Lp. We see that the equation Lp(u) = 0 arises as the Euler-Lagrange equation for the integral I(u) = ∫ Ω (|u|p + |∇ × u|p + |∇ · u|p) dx, 1 < p <∞. (7.7) Consider now the Rayleigh quotient R(u) = ∫ Ω (|u|p + |∇ × u|p + |∇ · u|p) dx∫ Ω |u|p dx . (7.8) The minimization of this quotient in W 0 N leads to a nonlinear eigenvalue problem. The corresponding Euler-Lagrange equation is Lp(u)− λ|u|p−2u = 0. (7.9) This is obtained by minimizing the functional I(u) subject to the constraint G(u) =∫ Ω |u|p dx = 1. Definition 7.4. A function u ∈ W 0 N ∩ ( C(Ω) )3 will be called a p-eigenfunction if there exists λ ∈ R so that∫ Ω |u|p−2u · v dx+ ∫ Ω |∇ × u|p−2∇× u · ∇ × v dx + ∫ Ω |∇ · u|p−2∇ · u∇ · v dx = λ ∫ Ω |u|p−2u · v dx (7.10) for all v ∈W 0 N . The associated λ will be called a p-eigenvalue. Now, notice that if u is a solution to (7.9), then λ = R(u) as expected. Thus, we see that λ > 0. The smallest eigenvalue of (7.9) is λ1 =λ1(Ω) := inf {∫ Ω (|u|p + |∇ × u|p + |∇ · u|p) dx : u ∈W 0 N , ‖u‖Lp = 1 } . (7.11) We assume that the infimum in (7.11) is attained when u is a multiple of some u1 > 0. (7.12) EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 19 Assume, further, that there exists a function α(x) ∈ L∞(Ω) with α(x) < λ1 on a set of positive measure, such that lim sup |s|→±∞ pF (x, s) |s|p ≤ α(x) ≤ λ1 uniformly in Ω. (7.13) Under these two assumptions, the following theorem holds. Theorem 7.5. Let f : Ω × R3 → R3 be Carathéodory and satisfy (7.3) as well as (7.4). Suppose there exists a function α(x) ∈ L∞(Ω) with α(x) < λ1 on a set of positive measure such that (7.13) holds. Finally, assume the infimum condition (7.12) holds. Then F is coercive, and so (7.1) has solutions in W 0 N . Our next goal is to show that the infimum in (7.11) is attained by u0 which is a weak solution of the Euler-Lagrange equation (7.9). Since WN ↪→ (Lp(Ω)) 3 is compact and WN is reflexive, this is enough to guarantee the existence of a minimizer of (7.11). Indeed, let m ≥ 0 denote the infimum, and suppose {un} is a minimizing sequence of λ1, so that∫ Ω (|un|p + |∇ × un|p + |∇ · un|p) dx −→ λ1, ‖un‖Lp = 1. Then clearly {un} is bounded in W 0 N , and since W 0 N is reflexive, there exists a subsequence unk converging weakly to some u0 in W 0 N . Since W 0 N ↪→ (Lp(Ω)) 3 is compact, it follows that ‖unk − u0‖Lp → 0, and so ‖u0‖Lp = 1. Thus, we have by lower semicontinuity of the Lp-norm that m ≤ ∫ Ω (|u|p + |∇ × u|p + |∇ · u|p) dx ≤ lim inf k→∞ ∫ Ω (|unk |p + |∇ × unk |p + |∇ · unk |p) dx = m, and so u0 ∈W 0 N is a minimizer. As such it satisfies (7.10) with λ = λ1. With this in hand, inspired by the proof of [13, Theorem 13], we turn to the proof of Theorem 7.5. Proof of Theorem 7.5. Define N : W 0 N → R by N(v) = ‖v‖p W 0 N − ∫ Ω α(x)|v(x)|p dx. From (7.11) and the assumption (7.13), we have that N(v) ≥ 0 for all v ∈W 0 N . Suppose now that there exists a sequence {vn} ∈ W 0 N such that ‖vn‖W 0 N = 1 and N(vn) → 0. Since W 0 N is reflexive, we can find a subsequence of {vn} (still denoted {vn}) and some v0 ∈W 0 N such that vn ⇀ v0 weakly in W 0 N and vn → v0 strongly in Lp(Ω). Now, it is clear that the functional v 7→ ∫ Ω α(x)|v(x)|p dx is weakly continuous on W 0 N , which implies that 0 ≤ ‖v0‖pW 0 N − ∫ Ω α(x)|v0(x)|pdx ≤ lim inf n→∞ N(vn) = 0 , 20 D. R. ADHIKARI, E. STACHURA EJDE-2020/116 and so ‖v0‖pW 0 N = ∫ Ω α(x)|v0(x)|pdx. Since N(vn) → 1 − ∫ Ω α(x)|v0(x)|p dx, we have ‖v0‖pW 0 N = ∫ Ω α(x)|v0(x)|pdx = 1 so that v0 6= 0. Again by (7.11) and (7.13) we see that λ1‖v0‖pLp ≤ ‖v0‖pW 0 N ≤ λ1‖v0‖pLp (7.14) since ‖v0‖pW 0 N = ∫ Ω α(x)|v0(x)|p dx. Thus λ1 = ‖v0‖pW 0 N ‖v0‖pLp . By (7.12), we have that v0 is a multiple of u1 > 0. Hence |v0(x)| > 0 a.e. in Ω. Let Ω1 := {x ∈ Ω : α(x) < λ1}, which we have assumed to have positive measure. This means that ∫ Ω α(x)|v0(x)|p dx < λ1‖v0‖pLp by splitting Ω into Ω1 and Ω \ Ω1. This directly contradicts (7.14). Thus, we conclude that there must exist some ε > 0 such that N(v) ≥ ε for all v ∈W 0 N with ‖v‖p W 0 N = 1. (7.15) This implies ‖v‖p W 0 N − ∫ Ω α(x)|v0(x)|pdx ≥ ε‖v‖p W 0 N for all v ∈W 0 N . (7.16) Take ε1 < λ1ε. From (7.13) and (7.3), we can find a constant k such that F (x, s) ≤ α+ ε1 p |s|p + k + c(x), x ∈ Ω, s ∈ R3 (7.17) for some function c(x) ∈ L1(Ω). Then (7.16) together with (7.17) imply that F (v) ≥ λ1ε− ε1 p ‖v‖p W 0 N − k0 −→∞, k0 a constant, as ‖v‖p W 0 N →∞. � Remark 7.6. The condition (7.12) used in Theorem 7.5 could be removed if we had a Harnack inequality for non-negative p-eigenfunctions. We have not yet pur- sued this direction, but believe it would be of independent interest. Moreover, the conservative condition (7.4) is not required in one dimension, as one can simply take F (x, u) = ∫ u 0 f(x, s) ds. 8. The one-dimensional case In the one-dimensional case, we can find the first eigenvalue explicitly. To this end, let u(x1, x2, x3) = (u(x1) 0 0)t only depend on x1. Then the curl of such u vanishes and the divergence of u becomes u′ where ′ = d dx1 . Indeed, the eigenvalue problem in one dimension becomes( |u′|p−2u′ )′ = (1− λ)|u|p−2u (8.1) EJDE-2020/116 p-CURL SYSTEMS AND DUALITY MAPPINGS 21 subject to a Neumann boundary condition (due to the definition of WN ). For simplicity we take the domain Ω = (0, 1). Such problems have been studied in [15] for the p-Laplacian, but a similar analysis carries over here as well. Indeed, the eigenvalue problem (8.1) is the same as the eigenvalue problem for the p-Laplacian with λ− 1 as an eigenvalue instead. Thus, letting α1 = λ− 1, in [15, Theorem 3.2] implies that α1 = ( 4π sin(π/p) )p p− 1 pp . Hence the first eigenvalue of (8.1) in one dimension is λ1 = ( 4π sin(π/p) )p p− 1 pp + 1, (8.2) and more generally, we have λn(p) = npλ1(p). Thus, since λ1 > 0 (for p > 1), we also conclude that each λn > 0 as well (again for p > 1). Conclusion. We have proved a number of trace results for Banach spaces related to the p-curl operator. These were used in turn to calculate duality mappings on various Sobolev spaces related to the p-curl system. We have solved a nonlinear p-curl system in a C1,1 or convex domain by proving that weak solutions are critical points of a coercive and lower semicontinuous functional. We also have a variational characterization of the first eigenvalue of the p-curl operator, and in one dimension, we have obtained an explicit expression for the first eigenvalue. It was necessary to assume positivity of the first eigenfunction of the p-curl operator in order to prove coercivity of the associated functional. It is expected that this can be removed by proving a Harnack inequality for the p-curl operator, which we leave to future work. Acknowledgements. Dhruba R. 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Bian; Global boundeness of the curl for a p-curl system in convex domains, ArXiv preprint arXiv: 1909.00159 (2019). [32] H. M. Yin, B. Q. Li, J. Zou; A degenerate evolution system modeling Bean’s critical-state type-II superconductors, Discrete & Continuous Dynamical Systems-A 8 (2002), no. 3, 781– 794. [33] D. Z. Zanger; The inhomogeneous Neumann problem in Lipschitz domains, Comm. Partial Differential Equations 25 (2000), no. 9-10, 1771–1808. Dhruba R. Adhikari Department of Mathematics, Kennesaw State University, Kennesaw, Georgia 30060, USA Email address: dadhikar@kennesaw.edu Eric Stachura Department of Mathematics, Kennesaw State University, Kennesaw, Georgia 30060, USA Email address: eric.stachura@kennesaw.edu 1. Introduction 2. Function spaces and trace theorems 2.1. Proof of Theorem 2.5 2.2. Proof of Theorem 2.7 2.3. Traces of W1,p(div,) functions 3. Duality mapping on W1,p(curl,) 4. Duality mappings on WN: part I 5. Duality mappings on WN: part 2 6. Generalizations 7. On the problem Lp (u) = f(x,u) 8. The one-dimensional case Conclusion Acknowledgements References