Regularity of Weak Solution to an p curl-system

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2 Regularity of Weak Solution to an p curl-system Hong-Ming Yin Department of Mathematics Washington State University Pullman, WA USA. hyin@wsu.edu Abstract: In this note we study the regularity of weak solution to a nonlinear steady-state Maxwell s equation in conductive media: [ H p 2 H] = F(x), p > 1, where H(x) represents the magnetic field while F(x) is the internal magnetic current. It is shown that the weak solution is of class C 1+α, which is optimal regularity for the weak solution. The basic idea is to transform the system into an p Laplacian equation subject to appropriate boundary condition and then to prove the desired regularity by using the known theory for the scalar equation. AMS Mathematics Subject Classifications: 35Q60 Key Words and Phases: Nonlinear Maxwell s Equations; Optimal regularity of weak solution. 1

3 1. Introduction Let p > 1 and F(x) be a three-dimensional vector function defined in a bounded domain in R 3 with divf(x) = 0 on. In this note a bold letter represents a threedimensional vector or vector function. We are interested in finding the minimum over the Banach space H p 0(curl, div0, ) for the following functional: where I[H] = 1 p H p dx + F Hdx, H p 0(curl, div0, ) := {H L p () : H L p (), H = 0, x }. By using Dirichlet principle, we see that the minimum of I[H] is the solution of the following p curl-system: [ H p 2 H ] = F(x), H = 0, x, (1.1) N H(x) = 0, x, (1.2) where N is the outward unit normal on. Note that if H(x) = {0, 0, h(x 1, x 2 )}, we see that the p curl-system (1.1) becomes the well-known p Laplacian equation which has been studied extensively by many authors (see monographs Choe [4], DiBenedetto [6] and the references therein). On the other hand, Eq.(1.1) is the steady-state approximation for Bean s criticalstate model for type-ii superconductors ([3, 8, 11, 12]) where the magnetic field H is approximated by the solution of the p curl-evolution system: H t + [ H p 2 H] = F(x, t). (1.3) The interested reader may consult [1, 3] for further physical background. The existence of a unique weak solution for the time-dependent p curl-system (1.3) as well as the steady-state case (1.1)-(1.2) is established in [11, 12]. Moreover, some regularity of weak solution is also discussed in [12]. In this note we use the idea of [10] to show the optimal regularity for the weak solution of the steady-state system (1.1). Throughout this paper the following hypothesis is assumed to hold. H(1.1): (a) Let be a bounded and simply-connected domain in R 3 with boundary 2

4 C 2. (b) Let F(x) H(div0, ) C α ( ). The main result in this note is the following theorem. Theorem: Under the assumption H(1.1) the weak solution H(x) of the problem (1.1)-(1.2) belongs to C 1+α ( ) for some α (0, 1). Remark 1.1: The C 1+α -regularity of weak solution to time-dependent p curlsystem (1.3) is still open. 2. Proof We recall some function spaces associated with curl and div-operators ([5]). Other Sobolev spaces such as H 1 () are the same as usual. Let p > 1. H p (curl, ) := {U L p () : U L p ()}; H p (div, ) := {U L p () : U L p ()}; H(curl0, ) := {U H p (curl, ) : U(x) = 0, x }; H(div0, ) := {U H p (div, ) : U = 0, x }. The norm of H p (curl, ) is defined as [ U H p (curl,) = ] 1 U p + U p p ]dx. The norm of H p (div, ) is defined similarly. Under these norms, H p (curl, ) and H p (div, ) are Banach spaces. Moreover, when p = 2 we simply denote by H(curl, ) and H(div, ). They are Hilbert spaces with the following inner products, respectively, (U, K) = (U, K) = We first show some elementary properties. [U K + ( U) ( K)]dx [U K + ( U) ( K)]dx. 3

5 Lemma 2.1: For a vector function F(x) H(div0, ) there exists a vector function G(x) H(curl, div0, ) C α ( ) such that F(x) = G(x), x. Proof: We begin with the classical case by assuming F(x) C 1 () with F(x) = 0 on. Since is a simply-connected domain, we know from the elementary calculus that there exists a vector function, denoted by G 0 (x), such that G 0 (x) = F(x), x. It is clear that for any ψ(x) C 2 ( ) the vector function G 0 (x) + ψ also satisfies the above system (called Gauge invariance in the electromagnetic theory). From the theory of elliptic equations we see that there exists at least one solution for the following equation: ψ + G 0 (x) = 0, x. By choosing ψ 0 (x) to be the solution of the above equation, we find that G(x) = G 0 (x) + ψ 0 (x) satisfies the div-condition. From the result of [9] we further know that G(x) C α ( ). Finally, since C 1 is dense in H 1 () by using the standard approximation, we see the conclusion of Lemma 2.1 holds for any F(x) H(div0, ). Q.E.D. Lemma 2.2: For a vector function K(x) H(curl0, ) there exists a scalar function ψ(x) H 1 () such that K(x) = ψ(x), x. Proof: Again we may assume that K(x) C 1 (). Since the domain is simplyconnected and K(x) = 0, x, then K(x) must be a conservative field. Hence there exists a potential function ψ(x) C 1 () such that K(x) = ψ(x), x. Since C 1 ( ) is dense in H 1 (), we see the conclusion of Lemma 2.2 holds for any K(x) H(curl0, ). Q.E.D. 4

6 Lemma 2.3: For H(x) H p (curl, ) with N H(x) = 0 on in the sense of trace. Then N ( H(x)) = 0, x in the sense of trace. Proof: For any ψ(x) H 1 (), = = [N ( H)ψ]ds = [( H) ψ]dx [(N H) ( ψ)]ds = 0. It follows that N ( H) = 0 in the sense of trace. Proof of the Theorem: From Lemma 2.1, we choose such that Now we can rewrite Eq. (1.1) as follows: [( H)ψ]dx G(x) H(curl, div0, ) C α ( ) G = F(x), G(x) = 0, x. [ H p 2 H G] = 0, x. (2.1) On the other hand, Lemma 2.2 implies that there exists a scalar function ψ(x) H 1 () such that It follows that From Eq.(2.2)-(2.3) we find H p 2 H G = ψ, x. (2.2) H = ψ + G 1 p 1, x. (2.3) H = H (p 2) [ ψ + G] = ψ + G p 2 p 1 [ ψ + G], x. (2.4) 5

7 Using the identity div(curlk) = 0 in the weak sense for any vector K H(curl, ) with N K = 0 on we have [ ψ + G p 2 p 1 ( ψ + G)] = 0, x. (2.5) On the boundary, we use Lemma 2.3 to obtain N [ ψ + G p 2 p 1 ( ψ + G)] = 0, x, which is equivalent to n ψ = N G, x. (2.6) Note that p 2 p 1 = p 2 := q 2, p 1 p where q = > 1 for any p > 1. p 1 To apply the regularity results for the scalar p Laplacian equation, we have to verify that Eq.(2.5) satisfies the structure conditions as those in [4, 6, 7]. Let Then A(x, t, ψ) := ψ + G q 2 [ ψ + G]. A(x, t, V) V = V + G q 2 [V + G] V = V + G q V + G q 2 [V + G] G V + G q ε V + G q C(ε) G q 1 ε V q C(ε) G q, 2 q where at the final step we have used the following inequality: B 1 + B 2 q 2 q [ B 1 q + B 2 q ], where B 1 and B 2 are two vectors and q > 1. By choosing ε = 1, we see 2 A(x, t, V) V a 0 V q C, 6

8 where a 0 > 0 and C depends only on G q. Moreover, for any vectors B 1, B 2 and β (0, 1) the following inequality also holds: B 1 + B 2 β B 1 β + B 2 β. It follows that A(x, t, V) C V q 1 + C G q 1, where C = max{1, 2 q 1 }. Now we can use the regularity result for scalar p-laplacian equation subject to the Neumann boundary condition( see [4, 6, 7]) to conclude that ψ C 1+α ( ). Moreover, from [7] there exists a constant C such that ψ C 1+α ( ) C, where C depends only on C 1+α -norm of G(x), p, and. From Eq. (2.4), we know that H(x) satisfies H = ψ + G q 2 [ ψ + G], x, H(x) = 0, x, N H(x) = 0, x. Since is simply-connected and hence the second Betti number of is zero. From the result of [2] (Theorem 2.1) we find that H(x) C 1+α ( ) and H C 1+α ( ) C 6 [ H C α (), where C 6 depends only on, and C 5. Q.E.D. Remark 2.1: The simply-connectedness of the domain is necessary not only for Lemma 2.1 and Lemma 2.2 but also for the result of [2] since the C 1+α -estimate of U depends on divh, curlh, N H and the topology of the domain (the first and second Betti numbers, see [2] for details). 7

9 Remark 2.2: Unlike steady-state case, for the time-dependent p curl-system (1.3) there must be a restriction on the exponent p in order to derive C 1+α -regularity (see [6]). Acknowledgement: The author would like to thank Professor G. Lieberman for helpful discussions. References [1] C.P. Bean, Magnetization of high-field superconductors, Rev. Mod. Phys., 36(1964), [2] J. Bolik and W. von Wahl, Estimating u in terms of divu, curlu, either (ν, U) or ν U and the topology, Mathematical Methods in the Applied Sciences, Vol.20 (1997), [3] S.J. Chapman, A hierarchy of models for type-ii superconductors, SIAM Review 42(2000), [4] H.J. Choe, Degenerate Elliptic and Parabolic Equations and Variational Inequality, Lectures Notes Series, Number 16, Seoul National University, Korean, [5] P. Ciarlet, Jr. and E. Sonnedrücker, A decomposition of the electromagnetic field- Application to the Darwin Model, Mathematical Models and Methods in Applied Sciences, 7(1997), [6] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, [7] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis 12 (1988), [8] L. Prigozhin, On the Bean critical-state model in superconductivity, European J. Appl. Math. 7 (1996), no. 3, [9] W. von Wahl, Estimating u by divu and curlu, Mathematical Methods in Applied Sciences, 15(1992), [10] H.M. Yin, Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields, Comm. on Pure and Applied Analysis, 1(2002),

10 [11] H. M. Yin, On a p Laplacian type of evolution system and applications to the Bean model in the type-ii superconductivity theory, Quarterly of Applied Mathematics, vol. LIX(2001), [12] H.M. Yin, B.Q. Li and J. Zou, A degenerate evolution system modeling Bean s criticalstate type-ii superconductors, Discrete and Continuous Dynamical Systems, 8(2002),