André Guerra

We study maps u\in W^{1,2}(\Omega;\overline M) that minimize the Alt–Caffarelli energy functional \int_{\Omega} (|Du|^{2} + q^{2} \chi_{u^{-1}(M)})\,dx under the condition that the image u(\Omega) is confined within \overline M . Here, \Omega denotes a bounded domain in the ambient space \mathbb{R}^{n} (with n\geq 1 ), and M represents a smooth domain in the target space \mathbb{R}^{m} (where m\geq 2 ). Since our minimizing constraint maps coincide with harmonic maps in the interior of the coincidence set, {\rm int}(u^{-1}(\partial M)) , such maps are prone to developing discontinuities due to their inherent nature. This research marks the commencement of an in-depth analysis of potential singularities that might arise within and around the free boundary. Our first significant contribution is an \varepsilon -regularity theorem, founded on a novel method of Lipschitz approximation near points exhibiting low energy. Utilizing this approximation and extending the analysis through a bootstrapping approach, we show Lipschitz continuity of our maps whenever the energy is small. Our subsequent key finding reveals that, whenever the complement of M is uniformly convex and of class C^{3} , the maps minimizing the Alt–Caffarelli energy with a positive parameter q exhibit Lipschitz continuity within a universally defined neighborhood of the noncoincidence set u^{-1}(M) . In particular, this Lipschitz continuity extends to the free boundary. A noteworthy consequence of our findings is the smoothness of flat free boundaries and of the resulting image maps.

In this work we prove a sharp quantitative form of Liouville’s theorem, which asserts that, for all n ≥ 3 n\geq 3 , the weakly conformal maps of S n − 1 \mathbb S^{n-1} with degree ± 1 \pm 1 are Möbius transformations. In the case n = 3 n=3 this estimate was first obtained by Bernand-Mantel, Muratov and Simon [Arch. Ration. Mech. Anal. 239 (2021), pp. 219-299], with different proofs given later on by Topping, and by Hirsch and the third author. The higher-dimensional case n ≥ 4 n\geq 4 requires new arguments because it is genuinely nonlinear: the linearized version of the estimate involves quantities which cannot control the distance to Möbius transformations in the conformally invariant Sobolev norm. Our main tool to circumvent this difficulty is an inequality introduced by Figalli and Zhang in their proof of a sharp stability estimate for the Sobolev inequality.

In 1989, Burnett conjectured that, under appropriate assumptions, the limit of highly oscillatory solutions to the Einstein vacuum equations is a solution of the Einstein–massless Vlasov system. In a recent breakthrough, Huneau–Luk (Ann Sci l’ENS, 2024) gave a proof of the conjecture in U(1)-symmetry and elliptic gauge. They also require control on up to fourth order derivatives of the metric components. In this paper, we give a streamlined proof of a stronger result and, in the spirit of Burnett’s original conjecture, we remove the need for control on higher derivatives. Our methods also apply to general wave map equations.

In this short expository note, we present a selection of classic and recent ideas in free boundary theory, with a focus on the vectorial case, referred to here as constraint maps. The note includes a brief historical perspective and highlights the latest heuristic-level results.

We consider the following question arising in the theory of differential inclusions: Given an elliptic set \Gamma and a Sobolev map u whose gradient lies in the quasiconformal envelope of \Gamma and touches \Gamma on a set of positive measure, must u be affine? We answer this question positively for a suitable notion of ellipticity, which for instance encompasses the case where \Gamma \subset \mathbb{R}^{2\times 2} is an elliptic, smooth, closed curve. More precisely, we prove that the distance of \mathrm{D}u to \Gamma satisfies the strong unique continuation property. As a by-product, we obtain new results for non-linear Beltrami equations and recover known results for the reduced Beltrami equation and the Monge–Ampère equation: concerning the latter, we obtain a new proof of the W^{2,1+\varepsilon} -regularity for two-dimensional solutions.

Energy-minimizing constraint maps are a natural extension of the obstacle problem within a vectorial framework. Due to inherent topological constraints, these maps manifest a diverse structure that includes singularities similar to harmonic maps, branch points reminiscent of minimal surfaces, and the intricate free-boundary behavior of the obstacle problem. The complexity of these maps poses significant challenges to their analysis. In this paper, we first focus on constraint maps with uniformly convex obstacles and establish continuity (and therefore higher-order regularity) within a uniform neighborhood of the free boundary. More precisely, thanks to a new quantitative unique continuation principle near singularities (which is new even in the setting of classical harmonic maps), we prove that, in the uniformly convex setting, topological singularities can only lie in the interior of the contact set. We then establish the optimality of this result. Second, while exploring the structure of the free boundary, we investigate the presence of branch points and show how they lead to completely new types of singularities not present in the scalar case.

We prove a sharp quantitative version of the Faber–Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit $\delta (f;\Omega )$ δ ( f ; Ω ) which measures by how much the STFT of a function $f\in L^{2}(\mathbb{R})$ f ∈ L 2 ( R ) fails to be optimally concentrated on an arbitrary set $\Omega \subset \mathbb{R}^{2}$ Ω ⊂ R 2 of positive, finite measure. We then show that an optimal power of the deficit $\delta (f;\Omega )$ δ ( f ; Ω ) controls both the $L^{2}$ L 2 -distance of $f$ f to an appropriate class of Gaussians and the distance of $\Omega $ Ω to a ball, through the Fraenkel asymmetry of $\Omega $ Ω . Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.