Density estimates for vector minimizers and applications

Abstract

We extend the Caffarelli-Córdoba estimates to the vector case (L. Caffarelli and A. Córdoba, Uniform Convergence of a singular perturbation problem, Comm. Pure Appl. Math. 48 (1995)). In particular, we establish lower codimension density estimates. These are useful for studying the hierarchical structure of minimal solutions. We also give applications.2010 Mathematics Subject Classification. 35J20, 35J47, 35J50.(c) the phase separation system ∆u i − j =i u i u j = 0, i = 1, . . . , m (Caffarelli and Lin [15]) and its variants. All these systems are included in the general framework of Rubinstein, Sternberg and Keller [27], [28].We assumewhere · denotes the Euclidean inner product in R m and | · | the associated norm.In the present paper we limit ourselves to uniformly bounded solutions to (1.1) that are minimal with respect to their boundary conditions (called minimizers by some authors),2 0 (Ω; R m ) (1.6) for every open bounded set Ω ⊂ D. Moreover we assume |u − a| < M, |∇u| < M, on R n .(1.7)The basic estimate for minimal solutions (cr. Lemma 2.1) isWe generally omit the center and write it B R . In this paper, we give two different extensions of the Caffarelli-Cordoba density estimates [14] to the vector case, together with a few applications. In the scalar case, among other things, these estimates refine the linking of the phase transition model to minimal surfaces and have played a major role in the resolution of the De Giorgi conjecture in higher dimensions (Savin [29]). Other extensions of the density estimates in different contexts have been provided by Farina and Valdinoci [20], Savin and Valdinoci [30],[31], Sire and Valdinoci [32], and very recently by Cesaroni, Muratov and Novaga [16].We denote L k the k-dimensional Lebesgue measure and let

Optimal potential energy growth lower bounds for minimizing solutions to the vectorial Allen–Cahn equation in two space dimensions

“…In this note, we will establish in a simple and elementary way the corresponding optimal growth lower bound to for just the potential energy term when n = 2, that is in the case of bounded and minimal solutions. In particular, our result strengthens the above lower bound of when n = 2, even under weaker assumptions on W , and at the same time refines it (Remark ).…”

“…We emphasize that the latter estimates have played a pivotal role in some of the recent constructions of equivariant solutions that were mentioned previously in relation with (see in particular ). This Liouville property was shown originally in , in the case where W is strictly convex near its global minimum (see also for a more recent proof, which rests upon the density estimates developed therein, in the case where that minimum is non‐degenerate). In this regard, we highlight that our main result implies that, at least when n = 2, this Liouville property holds when W is merely monotone near its global minimum, in the sense of below, and with less regularity in fact.…”

“…In view of the aforementioned result of and our theorem, it is natural to conjecture that the assertion of the latter holds in all dimensions, at least when the global minima of W are non‐degenerate. In this regard, we note that the presence of the gradient in the energy was crucial in the corresponding proofs (there were two) of the latter reference.…”

“…If the latter are assumed to be non‐degenerate, by adapting a lemma of from the study of the Ginzburg–Landau system, we showed in that, for any k > 0 and , bounded, nonconstant solutions of satisfy (see also for related estimates for just the potential energy term; there it was also noted that the above estimate remains valid even if the global minima of W have a finite degree of degeneracy). If, in addition, the solution is minimal, it was shown in , as a consequence of the density estimates proven therein (a natural but nontrivial extension of those in for the scalar problem), that which in light of is optimal.…”

“…(see also [30] for related estimates for just the potential energy term; there it was also noted that the above estimate remains valid even if the global minima of W have a finite degree of degeneracy). If, in addition, the solution is minimal, it was shown in [31], as a consequence of the density estimates proven therein (a natural but nontrivial extension of those in [17] for the scalar problem), that…”