Isotropic realizability of electric fields around critical points

In this paper we study the isotropic realizability of a given regular gradient field $\nabla u$ as an electric field, namely when $\nabla u$ is solution of the equation $\div\left(\si\nabla u\right)=0$ for some isotropic conductivity $\si>0$. The case of a function $u$ without critical point was investigated in \cite{BMT} thanks to a gradient flow approach. The presence of a critical point needs a specific treatment according to the behavior of the dynamical system around the point. The case of a saddle point is the most favorable and leads us to a characterization of the local isotropic realizability through some boundedness condition involving the laplacian of $u$ along the gradient flow. The case of a sink or a source implies a strong maximum principle under the same boundedness condition. However, when the critical point is not hyperbolic the isotropic realizability is not generally satisfied even piecewisely in the neighborhood of the point. The isotropic realizability in the torus for periodic gradient fields is also discussed in particular when the trajectories of the gradient system are bounded.


Introduction
The starting point of the present paper is the following issue: given a gradient field ∇u from R d into R d , under which conditions ∇u is an isotropically realizable electric field, namely there exists an isotropic conductivity σ > 0 such that div (σ∇u) = 0 in R d ? A quite complete answer is given in [5] when u is regular and ∇u is periodic. On the one hand, assuming that u ∈ C 3 (R d ) and inf we may construct a continuous conductivity σ in R d (this is a straightforward extension of Theorem 2.15 in [5]). In dimension two this implies that ∇u can be rectified globally in R 2 into a constant vector (see Theorem A.1 in the Appendix). In contrast the rectification theorem only applies locally for an arbitrary smooth non-vanishing vector field (see, e.g., [4]). On the other hand, when ∇u is periodic, it is not always possible to derive a periodic conductivity under the sole condition (1.1). Actually, the Theorem 2.17 of [5] shows that ∇u is isotropically realizable in the torus under the extra assumption sup x∈R d ˆτ (x) 0 ∆u X(s, x) ds < ∞, (1.2) where X is the gradient flow defined by and τ (x) is the time (unique by (1.1)) for the flow X(·, x) to reach the equipotential {u = 0}. Moreover, the boundedness condition (1.2) is also necessary to derive a periodic conductivity σ in C 1 (R d ). It is then natural to ask what happens when condition (1.1) does not hold due to the existence of critical points. This paper provides partial answers to this question.
We essentially study the realizability of a gradient field in the neighborhood of an isolated critical point. The originality of this local problem lies in the following fact: Surprisingly, the boundedness assumption (1.2) which in [5] is useless for the realizability around any regular point but necessary for the global realizability in the torus, turns out to be crucial in its local version to derive the realizability around an isolated critical point. Consider a function u ∈ C 2 (R d ), for d = 2, 3, and a point x * ∈ R d such that x * is an isolated critical point of u. (1.4) The issue is to know when ∇u is isotropically realizable in the neighborhood of x * . To this end the gradient system (1.3) plays an important role as in [5]. In view of (1.2) we establish a strong connection between the isotropic realizability of ∇u around the critical point x * and the boundedness of the function (t, x) −→ˆt 0 ∆u X(s, x) ds. (1.5) More precisely, we study the local isotropic realizability according to the nature of the critical point x * . The following cases are investigated: • x * is a saddle point, i.e. ∇ 2 u(x * ) is invertible with both positive and negative eigenvalues; • x * is a sink (resp. a source), i.e. ∇ 2 u(x * ) is negative (resp. positive) definite, or more generally, x * is a stable point for the gradient system (1.3) (see definition (3.1) below); • x * is not a hyperbolic point, i.e. det ∇ 2 u(x * ) = 0 and x * is not stable.
When x * is a saddle point, we prove (see Theorem 2.1) that the local isotropic realizability is (in some sense) equivalent to a local version of the bound (1.2) combined with (1.4). In Section 2.3 the two-dimensional example u(x, y) = f (x) + g(y) illustrates the sharpness of the boundedness condition which also implies that ∆u(x * ) = 0 (see Remark 2.2). Moreover, in spite of its simplicity this example shows that the question of realizability around a critical point is rather delicate (see Section 2.1 and Proposition 2.4). In particular, the condition ∆u(x * ) = 0 turns out to be not sufficient to derive the isotropic realizability in the neighborhood of the point x * . Indeed, we construct a function u ∈ C 2 [−1, 1] 2 which admits (0, 0) as a saddle point satisfying ∆u(0, 0) = 0, but the gradient of which is not isotropically realizable around (0, 0) (see Proposition 2.4 iii)).
In the case of a sink or a source the boundedness of the function (1.5) leads us to a strong maximum principle (see Theorem 3.3). When x * is not a hyperbolic point, the situation is much more intricate. We study a two-dimensional example where the isotropic realizability is only satisfied in some regions around x * , again in connection with condition (1.2) (see Proposition 4.1).
We conclude the paper with the isotropic realizability problem in the torus. The natural extension of [5] (Theorem 2.17) is that any regular periodic gradient field which vanishes at isolated points is isotropically realizable provided that the boundedness condition (1.2) holds (see Conjecture 5.1). We prove this result under the additional assumption that the trajectories of (1.3) are bounded (see Theorem 5.2), and we illustrate it by Proposition 5.4. At this level the dimension two is quite particular. Indeed, by virtue of [1] (see also [3] for the non-periodic case) a non-zero periodic gradient field which is isotropically realizable with a smooth periodic conductivity does not vanish in R 2 , and the trajectories of the gradient system (1.3) are then unbounded (see Remark 5.3). This shows that the boundedness of the trajectories together with the presence of critical points is a reasonable assumption in the periodic framework.
2 The case of a saddle point

A preliminary remark
Let u ∈ C 2 (R d ), for d ≥ 2, and let x * be a non-degenerate critical point of u, namely ∇u(x * ) = 0 and the hessian matrix ∇ 2 u(x * ) is invertible. By Morse's lemma (see, e.g., [8] Assume that the gradient field ∇v is isotropically realizable with a smooth conductivity τ > 0 in W * , namely div τ ∇v = 0 in W * . This implies that ∆v Φ(x * ) = 0 by virtue of Remark 2.2 below, and thus d = 2m. Conversely, if d = 2m, the function v is harmonic and ∇v is isotropically realizable with the conductivity τ ≡ 1. Hence, by the change of variables y = Φ(x) we get that div (σ∇u Therefore, when d = 2m, the gradient ∇u is realizable with the smooth conductivity σ of (2.2), but which is a priori anisotropic. We will see that the isotropic realizability of ∇u around the point x * is more delicate to obtain. In particular the equality ∆u(x * ) = 0 is generally not a sufficient condition of isotropic realizability contrary to the case of the quadratic function (2.1).

The general result
Let u ∈ C 2 (R d ), for d = 2, 3. Consider a critical point x * of u, which is also a saddle point of u in the sense that ∇ 2 u(x * ) has a non-zero determinant with both positive and negative eigenvalues.
Without loss of generality we may also assume that u(x * ) = 0.
The Hadamard-Perron Theorem (see, e.g., [2] p. 56, and [7] Section 8.3) claims that in some compact neighborhood K * of x * , containing no extra critical point, there exist two smooth invariant manifolds of the flow (1.3): Γ s of dimension k and Γ u of dimension d − k (which are smooth curves for in dimension two), such that • Γ s contains only stable trajectories of (1.3) (converging to x * as t → ∞) and Γ u contains only unstable trajectories (converging to x * as t → −∞), • for any x ∈ K * \ (Γ s ∪ Γ u ), the trajectory X(t, x) leaves K * as |t| increases.
Note that we have for any (t, (2.4) which implies that the function u X(·, x) is increasing. Hence, u X(·, x) is negative for  On the other hand, if u ∈ C 3 (R d ) the gradient flow X defined by (1.3) is in C 1 (R × R d ). By virtue of the implicit functions theorem, the regularity of X and (2.4) imply that τ belongs to C 1 Q * \ (Γ s ∪ Γ u ) . Then, we may define the function w in Q * \ (Γ s ∪ Γ u ) by which belongs to C 1 Q * \ (Γ s ∪ Γ u ) . If u is only in C 2 (R d ), then w does not belong necessarily to C 1 Q * \ (Γ s ∪ Γ u ) . However, in the sequel we will assume that We have the following result: Theorem 2.1. Let u ∈ C 2 (R d ), for d = 2, 3, and let x * ∈ R d be such that conditions (1.4), (2.3), (2.5) and (2.8) hold with an open neighborhood Q * and a compact neighborhood K * of x * satisfying Q * ⊂ int (K * ).
i) Assume that w belongs to L ∞ (Q * ). Then, ∇u is isotropically realizable in Q * , with a positive conductivity σ such that σ, ii) Conversely, assume that ∇u is isotropically realizable in the interior of K * , with a conductivity σ such that σ, σ −1 ∈ L ∞ int (K * ) ∪ C 1 int (K * ) \ (Γ s ∪ Γ u ) . Then, the function w belongs to L ∞ (Q * ).
Remark 2.2. In the part i) of Theorem 2.1, the regularity of the conductivity σ implies that ∆u(x * ) = 0. Indeed, when σ ∈ C 1 (Q * ) we have More generally, having in mind the part ii) of Theorem 2.1, the boundedness of the function w (2.7) also implies that ∆u(x * ) = 0. Indeed, taking up a heuristic point of view, consider a sequence The sequence |τ (x n )| tends to ∞, since the trajectory X(·, x) does not intersect the equipotential {u = 0}. Then, under some suitable assumption satisfied by ∆u X(s, ·) in the neighborhood of x * and the fact that X(t, x) tends to x * as t → ∞, we have This will be rigorously checked in the particular case of subsection 2.3 (see Proposition 2.4 i)). However, Proposition 2.4 iii) will show that the equality ∆u(x * ) = 0 is not sufficient to get the isotropic realizability of ∇u around the point x * .
Proof of Theorem 2.1.
Proof of i). Assume that w ∈ L ∞ (Q * ). Then, the conductivity defined by σ := e w and σ −1 . Moreover, following the proof of Theorem 2.14 in [5] but with the weaker regularity (2.8), the equation div (σ∇u) = 0 holds in each connected component of Q * \ (Γ s ∪ Γ u ). For the reader's convenience we recall the main steps of the proof: Let Ω be a connected component of Q * \ (Γ s ∪ Γ u ). First note that, since u ∈ C 2 (R d ), the flow X belongs to C 0 (R × R d ). Hence, for any x ∈ Ω and t close to 0 the trajectory X(t, x) remains in Ω. Then, due to the semi-group property satisfied by the gradient flow X (1.3) combined with the uniqueness of τ , we have for any x ∈ Ω and any t close to 0, (2.11) which, using successively the semi-group property and the change of variable r = s + t, yields (2.12) Taking the derivative with respect to t, which is valid since w ∈ C 1 (Ω) by (2.8), it follows that Therefore, for t = 0 we get that which combined with σ = e w implies that It thus remains to prove that the equation is satisfied in Q * . To this end we distinguish the cases d = 2 and d = 3.
First assume that d = 2. Then, Γ s and Γ u are two smooth curves in K * which only intersect at the point x * . Consider ε > 0 such that the open ball B(x * , ε) centered on x * and of radius ε, is contained in Q * . Let ϕ ε be a Lipschitz function in Q * , with compact support in Q * and ϕ ε ≡ 0 in B(x * , ε). Let Ω be a connected component of Q * \ (Γ s ∪ Γ u ). Note that σ∇u is a divergence free vector-valued function in L ∞ (Ω) d , thus has a trace on ∂Ω. Then, integrating by parts we havê where n denotes the normal outside to ∂Ω. However, since Γ s and Γ u are trajectories of the Using that v ε converges strongly to 1 in W 1,1 (Ω) and σ∇u ∈ L ∞ (Ω) d , we deduce from equality (2.17) that In dimension three one of the manifold Γ s or Γ u is a curve Γ, while the other one is a smooth surface Σ = {f = 0} composed of trajectories (1.3). Consider a Lipschitz function ϕ ε with compact support in Q * , which is zero in a tube of radius ε surrounding the curve Γ and containing the ball B(x * , ε). For any x ∈ Σ and t close to 0, the derivative of f X(t, x) = 0 at t = 0 yields ∇f X(0, x) · X ′ (0, x) = ∇f (x) · ∇u(x) = 0. Hence, the normal at each point x of Σ is orthogonal to ∇u(x), which again leads us to equality (2.17). Therefore, passing to the limit as ε → 0 we obtain that σ∇u is divergence free in Q * . (2.20) This combined with (2.5) implies that for any x ∈ Ω ∩ Q * , Remark 2.3. If the function w is not bounded in Q * , the previous proof then shows that the function σ := e w belongs to C 1 Q * \ (Γ s ∪ Γ u ) and blows up near Γ s ∪ Γ u . However, the equation div (σ∇u) = 0 still holds in Q * \ {x}.
The following two-dimensional application shows that the boundedness of the function (2.7) is crucial for deriving the isotropic realizability of ∇u in the neighborhood of x * = (0, 0), and that the sole equality ∆u(x * ) = 0 is not sufficient (see Proposition 2.4).
Proof of i). Assume that x → 0 with x > 0, and y → y 0 > 0. By (2.26) and (2.38) we also have a, b > 0. Then, we have F (x) → −∞, which implies that τ + F (x) → −∞. Indeed, if τ + F (x) ≥ c for some constant c < 0, then a ≥ F −1 (c) together with τ → ∞ and τ + G(y) ≥ τ → ∞. Therefore, a 0 and b → 0, which contradicts (2.40). Similarly, we show that τ + G(y) → ∞. Therefore, we obtain that a, b → 0, and as a consequence This combined with the boundedness of w (2.41) implies that On the other hand, by (2.33) we have Proof of ii). For the sake of simplicity let us assume that f ′′ (0) = −g ′′ (0) = 1. By virtue of Theorem 2.1 i) we have to show that the function w of (2.7) is bounded locally in Q * . However, since w ∈ C 1 Q * \ (Γ s ∪ Γ u ) , it is enough to prove that w(x, y) remains bounded as x → 0 or/and y → 0. Let (x, y) be a point of Q * \ (Γ s ∪ Γ u ). Without loss of generality we can assume that x, y > 0, which by (2.26) and (2.38) also implies that a, b > 0. First, assume that x → 0 and y → y 0 > 0. By the part i) we have a, b → 0. Then, taking into account (2.23) the asymptotics of f, g at the point 0 with f ′′ (0) = −g ′′ (0) = 1, and formula (2.41) imply that Proof of iii). Set α = 1. It is easy to check that there exists a unique one-to-one increasing An easy computation yields (x → 0 + means that x → 0 with x > 0) Putting this asymptotic in (2.56) together with (2.55) we get that for any y 0 ∈ (0, 1), This combined with the fact that w is continuous in (0, 1) 2 , shows that w does not belong to L ∞ (V * ) for any neighborhood V * of (0, 0). Therefore, the part ii) of Theorem 2.1 allows us to conclude.

The case of a sink or a source
Let u ∈ C 2 (R d ), for d ≥ 2. Consider a point x * ∈ R d satisfying (1.4). Assume that x * is stable for the gradient system (1.3), namely there exist a compact neighborhood K * of x * , containing no extra critical point, and a neighborhood Q * of x * , with Q * ⊂ int (K * ), such that In the first case x * is said to be positively stable, while in the second case it is said to be negatively stable.
Remark 3.1. If x * is a sink (resp. source) point of the linearized system, namely ∇ 2 u(x * ) has only negative (resp. positive) eigenvalues, then x * is positively (resp. negatively) stable.
Remark 3.2. If x * is a strict local extremum, then by Lyapunov's stability (see, e.g., [7] Theorem p. 194) x * is stable in the sense of definition (3.1). Conversely, if x * is stable, then it is asymptotically stable, namely each trajectory X(t, x) for x ∈ Q * , converges as t → ∞ (resp. t → −∞) to the isolated critical point x * (see, e.g., [7] Proposition p. 206). Therefore, since the function u X(·, x) is non-decreasing, x * is a local maximum (resp. minimum) of u.
In connection with Remark 3.2, the following strong maximum principle holds: . Let x * be a positively (resp. negatively) stable point for (1.3), satisfying (1.4) with ∆u(x * ) = 0. Assume that there exists a constant C * > 0 such that Then, u is constant in a neighborhood of x * . Using the semigroup property X s, X(t, x) = X(s + t, x), the function w n satisfies for any t ≥ 0 and any x ∈ Q * ,  Taking the derivative of (3.4) with respect to t at the origin, this yields Hence, we get that div (e wn ∇u) = e wn (∇w n · ∇u + ∆u) = e wn ∆u X(n, ·) in Q * .
(3.6) By condition (3.2) the sequence e wn is bounded in L ∞ (Q * ), thus converges weakly- * up to a subsequence to some σ in L ∞ (Q * ).
On the other hand, by virtue of Remark 3.2, for any x ∈ Q * the sequence X(n, x) ∈ K * converges to x * as the unique critical point of K * . This combined with ∆u(x * ) = 0 and the boundedness of e wn , implies that the sequence e wn ∆u X(n, ·) converges to 0 everywhere in Q * . By (3.2) it is also bounded in L ∞ (Q * ). Now, integrating by parts (3.6), then using the weak- * convergence of e wn and Lebesgue's dominated convergence theorem, we get that for any which yields that σ∇u is divergence free in Q * . Therefore, ∇u is realizable with the nonnegative conductivity σ ∈ L ∞ (Q * ). Moreover, condition (3.2) shows that σ is also bounded from below by e −C * , so that σ −1 ∈ L ∞ (Q * ). Hence, u is a weak solution of div (σ∇u) = 0 in Q * , where the positive conductivity σ satisfies σ, σ −1 ∈ L ∞ (Q * ). Moreover, by Remark 3.2 the point x * is a local maximum of u. Therefore, by the strong maximum principle for weak solutions to second-order elliptic pde's (see, e.g., [6] Theorem 8.19), the function u is constant in a neighborhood of x * .

An example with a non-hyperbolic point
Consider a point x * satisfying (1.4), which is not hyperbolic in the sense that ∇u 2 (x * ) has a zero determinant, and which is not stable for (1.3). Generally speaking, a neighborhood of x * can be divided in several regions such that for which of them the gradient system (1.3) mimics either a saddle point, a sink or a source. The coexistence of these different behaviors in the phase portrait prevents ∇u from being isotropically realizable.

A conjecture and a general result
Let Y be the unit cube of R d . In view of the previous results and Theorem 2.17 in [5] we may state the following conjecture on the isotropic realizability in the torus: Assume that the critical points of u are isolated, and that the conditions (5.1) and (5.2) below hold. Then, ∇u is isotropically realizable in the torus with a positive conductivity σ such that σ, σ −1 ∈ L ∞ ♯ (Y ). Conjecture 5.1 is a natural extension, allowing the existence of isolated critical points, of the realizability in the torus which was derived in [5] in the absence of critical point and under the same estimate (5.2). For the moment we have not succeeded to prove this result. However, the next result gives a partial answer to Conjecture 5.1 under the extra assumption that the trajectories are bounded: Theorem 5.2. Let u ∈ C 3 (R d ), for d ≥ 2, be a potential the gradient of which is Y -periodic, and such that for almost every x ∈ R d , the trajectory X [0, ∞), x of (1.3) -defined as the range of the mapping t ≥ 0 → X(t, x) -or the trajectory X (−∞, 0], x is bounded in R d . i) Assume that ∀ x ∈ R d , ∇u(x) = 0 ⇒ ∆u(x) = 0, (5.1) and that there exists a constant C > 0 such that Then, ∇u is isotropically realizable in the torus with a positive conductivity σ such that σ, σ −1 ∈ L ∞ ♯ (Y ). ii) Conversely, assume that ∇u is isotropically realizable in the torus with a positive conductivity σ ∈ C 1 ♯ (Y ). Then, conditions (5.1) and (5.2) hold true. Proof.
ßi) For the sake of simplicity Y = [0, 1] d . Consider the function w n defined by (3.3). Let x ∈ R d be such that the trajectory say X [0, ∞), x is bounded. Consider any subsequence k n of n such that X(k n , x) converges to some x * ∈ R d . The point x * is necessarily a critical point of u (see, e.g., [7] Proposition p. 206). Hence, by (5.1) and (5.2) the sequence e w kn (x) ∆u X(k n , x) converges to 0. Therefore, the whole sequence e wn ∆u X(n, ·) converges to 0 almost everywhere in R d . By (5.2) this sequence is also bounded in L ∞ (R d ). It thus follows from Lebesgue's dominated convergence theorem that e wn ∆u X(n, ·) converges strongly to 0 in L 1 loc (R d ). However, again using estimate (5.2), up to a subsequence e wn converges weakly- * in L ∞ (R d ) to some positive function σ 0 , with σ −1 0 ∈ L ∞ (R d ). Therefore, passing to the limit n → ∞ in (3.6), we get that div (σ 0 ∇u) = 0 in the distributions sense on R d . Finally, up to extract a new subsequence the average σ n defined for any positive integer n by converges weakly- * in L ∞ (R d ) to some Y -periodic function σ which does the job as in the non-critical case. We refer to [5] for further details about this average argument.
ii) The proof is quite similar to the part ii) of Theorem 3.3.
Remark 5.3. The dimension two is quite particular in the framework of Conjecture 5.1 and Theorem 5.2. Indeed, consider a smooth potential u defined in R 2 such that ∇u is Y -periodic and is not identically zero in R 2 . Also assume that there exists a smooth Y -periodic conductivity σ > 0 such that div (σ∇u) = 0 in R 2 . Then, thanks to Proposition 2 in [1] the function u has no critical point in R 2 . Moreover, the trajectories X [0, ∞), x and X (−∞, 0], x are unbounded, since any limit point of a bounded trajectory is a critical point of the gradient (see [7] Proposition p. 206). Therefore, the absence of critical point agrees with the unboundedness of the trajectories in the two-dimensional smooth periodic case.
The following example illustrates the strong connection between the isotropic realizability and estimate (5.2).

Application
Let u i , for i ∈ {1, . . . , d}, be a function in C 2 (R) such that its derivative u ′ i is T i -periodic for some T i > 0 and has only isolated roots t ∈ R also satisfying u ′′ i (t) = 0. Define u ∈ C 2 (R d ) by Acknowledgment. The author is very grateful to G.W. Milton for stimulating discussions on the topic.