ON A POISSON’S EQUATION ARISING FROM MAGNETISM

We review the proof of existence and uniqueness of the Poisson’s equation ∆u+divm = 0 whenever m is a unit L-vector field on R with compact support; by standard linear potential theory we deduce also the H-regularity of the unique weak solution.


Introduction
In the standard theory of ferromagnetic materials is usually considered an energy, called magnetostatic, which is the energy of the magnetostatic field set up by the magnetization vector field m. It turns out that the magnetostatic energy takes the form where the scalar potential u : R 3 → R satisfies the following equation arising from Maxwell's equations: div(∇u + m χ Ω ) = 0, on R 3 , (1.1) being Ω an open and bounded domain in R 3 , which represents the region occupied by a ferromagnetic material, and χ Ω its characteristic function, that is χ Ω = 1 on Ω and 0 otherwise in R 3 ; for more details on equation (1.1) see [2], [5] and [6]. Without loss of generality, since we will not vary the temperature, which is related with the variation of |m|, we will consider vector fields m : Ω → S 2 , being S 2 the boundary of the unit ball in R 3 . Replacing m χ Ω with There is a huge literature on the Poisson's type equation (1.2); we just mention a very recent application in the context of micromagnetics materials: such an equation has been considered in [3] and [4] where an homogenization procedure of a more complete energy functional for polycrystalline magnetic materials has been investigated. In order to solve equation (1.2) we have to introduce its weak formulation, that is In this short note we will explain how the proof of existence and uniqueness of the solution of equation (1.3) in a suitable space of Sobolev-type works; moreover, we will find, exploiting the standard tools coming from the linear potential theory, the explicit form of the solution from which, in particular, it will descends more regularity of such a solution: more precisely, the unique weak solution turns out to be H 1 (R 3 ), and such a regularity has been stated in [5], but without proof.

Some preliminaries of potential theory
We now recall some well known results coming from potential theory; for details we refer to [7]. Let n ≥ 1 be an integer and, for each f : R n → R measurable and for each α > 0, let I α be the Riesz potential given by for a suitable positive constant c(n, α). It turns out that if α, β > 0 and α + β < n then for Let α ∈ (0, n) and 1 ≤ p < +∞ with First of all, it turns out that if, more generally, f ∈ L p (R n ) then the integral on the right hand-side of (2.1) converges absolutely for almost any x ∈ R n . Moreover, if in particular is linear and continuous. Strictly related with Riesz potentials is the notion of Riesz transform: for any f ∈ L p (R n ), with 1 ≤ p < +∞, and for any j = 1, . . . , n, we let whenever the limit exists; c(n) is a suitable positive constant. It turns out that is linear and continuous; furthermore, we have the following fundamental relation between the first order Riesz potential I 1 and the Riesz transform: for any j = 1, . . . , n.

Existence and uniqueness
Let f ∈ L 2 (R 3 ; R 3 ). We first investigate existence and uniqueness of weak solutions of ∆u + div f = 0 on R 3 , following, for instance, [1]. Let E : R 3 \ {0} → R be the fundamental solution of the Laplace operator on R 3 , i.e. .

Moreover, let
Proof. We divide the proof in some steps.
Step 1. First of all we claim that P(f ) ∈ H. For, let g ∈ L 2 (R 3 ) and, for any i = 1, 2, 3 we Since and P i (g) ∈ L 6 (R 3 ) from (2.3), being g ∈ L 2 (R 3 ). In order to prove that ∂ j P i (g) ∈ L 2 (R 3 ), let . Using the explicit form of E, we immediately get Hence, by the continuity of I 1 : L 2 (R 3 ) → L 6 (R 3 ), we get P i (g h ) → P i (g), strongly in L 6 (R 3 ). Now, for any h ∈ N we have Therefore, using (2.4) and (2.2) we easily get By the continuity of the Riesz transform we deduce that ||∂ j P i (g h )|| 2 ≤c||g h || 2 . Thus ∂ j P i (g h ) u ij , for some u ij ∈ L 2 (R 3 ). Passing to the limit as h → +∞ in we deduce that u ij = ∂ j P i (g) which means that ∇P i (g) ∈ L 2 (R 3 ; R 3 ). In order to conclude it is sufficient to notice that being f (i) , for i = 1, 2, 3, the components of f .
Step 2. Now we prove that P(f ) is a weak solution of the equation ∆u + div f = 0, that is . First of all we have, integrating by parts, and therefore, since E is the fundamental solution of the Laplace operator on R 3 , Passing to the limit as h → +∞ we obtain Therefore we get (3.1).
Step 3. In order to conclude the proof, we have to show that the weak solution in H is then w := u 2 − u 1 satisfies Now, if we choose ϕ h → w in H then passing to the limit as h → +∞, 0 = ∇w · ∇ϕ h dx → |∇w| 2 dx from which we get w constant, and since w ∈ H, we deduce that w = 0, and thus u 1 = u 2 , which yields the conclusion.
We are ready to prove the existence and uniqueness result for the equation (1.3). Proof. Taking into account Theorem 3.1, it is sufficient to prove that P(m) ∈ L 2 (R 3 ). Using the very definition of P(m) and E, we have, since |m| = χ Ω , |P(m)| ≤ cI 1 ( χ Ω ) and I 1 ( χ Ω ) ∈ L 2 (R 3 ) since χ Ω ∈ L ∞ (Ω) and Ω is bounded; this yields the conclusion.