arXiv subjects
Publications and source records attributed to André Guerra.
We classify all nonconstant smooth harmonic maps $\mathbb{S}^2\to \mathbb{CP}^n$ and $\mathbb{S}^3\to \mathbb{CP}^n$ whose 0-homogeneous extensions are minimizing: they are, respectively, isometric embeddings or holomorphic isometric embeddings composed with a Hopf map.
We confirm a conjecture posed by T. Iwaniec in 1982: the $L^p$ norm of the Beurling--Ahlfors transform is $p-1$ if $p\geq 2$, and $(p-1)^{-1}$ if $1<p\leq 2$.
We construct a Lipschitz, non-$C^1$ map $u\colon \mathbb{R}^2\to \mathbb{R}^6$ which is the unique global minimizer of a smooth, strongly polyconvex functional. Such an example is necessarily non-homogeneous.
We prove the sharp quantitative Faber--Krahn inequality with a computable dimensional constant, resolving a problem posed by Brasco and De Philippis~\cite[Open Problem~2]{brasco2017spectral}. In contrast with the known proof of the sharp estimate, our argument avoids indirect compactness arguments and selection principles. It runs through a new level-set proof of the Saint--Venant inequality, using that the torsional deficit controls both the average isoperimetric deficit and the oscillation of the gradient magnitude along the boundaries of level sets. An effective local strong isoperimetric estimate for confined sets then yields the result.
We consider solutions of uniformly elliptic equations with measurable coefficients. We assume that the lowest eigenvalue of the coefficient matrix is at least $K^{-1}$ and the largest eigenvalue is at most $K$. In three and higher dimensions we construct $α$-Hölder continuous solutions with $α= \exp(- c_n K)$. This disproves a long-standing conjecture by showing that, except for the two-dimensional case, the Hölder exponent obtained from the Bombieri--Giusti Harnack inequality has the optimal dependence on the ellipticity constant $K$. Using the same construction as a starting point, we disprove a conjecture of De Giorgi about continuity of solutions to non-uniformly elliptic equations.
We use Morse Theory to prove that the Burkholder function is quasiconvex on symmetric matrices, which implies several sharp inequalities for the Beurling--Ahlfors transform when acting on real-valued functions. More generally, we prove that the separating functions introduced by Székelyhidi are quasiconvex on symmetric matrices.
We find an explicit rank-one convex non-quasiconvex integrand in $\mathbb{R}^{2\times 4}$: to falsify the quasiconvexity inequality, we exhibit a map $\mathbb{T}^4\to \mathbb{R}^2$ with $12$ non-zero Fourier modes. In fact, this map is obtained from a scalar potential, so we also find a rank one convex integrand in $\mathbb{R}^{4\times 4}_\text{sym}$ which is not quasiconvex. These examples are obtained by transpositions and restrictions of Grabovsky's example of a rank-one convex, non-quasiconvex integrand in $\mathbb{R}^{8 \times 2}.$ We also modify Šverák's example to construct a rank-one convex non-quasiconvex integrand in $\mathbb{R}^{3\times 3}_\text{sym}$.
We consider polyconvex integrands that are conformally invariant and frame indifferent. In two dimensions, we prove that the corresponding stationary points are smooth outside a discrete set; this result is new even for minimizers. We further show that every orientation-preserving stationary point is $C^1$. Since such solutions are closely related to Teichmüller-type variational problems, our result also confirms, in the case of integrands with linear growth in the distortion, a conjecture of Astala, Iwaniec, Martin, and Onninen from 2005.
Smooth maps $u\colon\mathbb B^3\to\mathbb S^2$ can be lifted to $\hat u\colon\mathbb B^3\to\mathbb S^3$ using the Hopf fibration $h\colon \mathbb S^3\to\mathbb S^2$ via the factorization $u=h\circ\hat u$. In this note we characterize the $W^{1,2}$-maps which have this lifting property in terms of exactness of the pullback form $u^*ω_{\mathbb S^2}$, and deduce a smooth approximation property preserving the constraint $u^*ω_{\mathbb S^2}=dη$.
We prove that, modulo rigid motions, the Hopf map is the unique minimizer of the Faddeev--Skyrme energy in its homotopy class, provided that the radius of the target 2-sphere is not smaller than the radius of the domain 3-sphere.
We consider a strengthening of the usual quasiconvexity condition of Morrey in two dimensions, which allows us to prove lower semicontinuity for functionals which are unbounded as the determinant vanishes. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and Šverák, plays a prominent role in our approach.
This paper explores recent progress related to constraint maps. Building on the exposition in [14], our goal is to provide a clear and accessible account of some of the more intricate arguments behind the main results in this work. Along the way, we include several new results of independent value. In particular, we give optimal geometric conditions on the target manifold that guarantee a unique continuation result for the projected image map. We also prove that the gradient of a minimizing harmonic map (or, more generally, of a minimizing constraint map) is an $A_\infty$-weight, and therefore satisfies a strong form of the unique continuation principle. In addition, we outline possible directions for future research and highlight several open problems that may interest researchers working on free boundary problems and harmonic maps.
We show that if the Hessian of a $C^{1,1}$ function has uniformly positive determinant almost everywhere then its index is locally constant, as conjectured by Šverák in 1992. We deduce this result as a consequence of a more general theorem valid for quasiregular gradient mappings.
Understanding structure-function relationships is essential to advance the manufacturing of next-gen materials with desired properties and functionalities. Precise and rapid measurement of features like wrinkle size, droplet diameter, and surface roughness is essential to establishing such structure-function relationships. To this end, this work developed feature size and surface morphology characterizations through image analysis in Python and validated them with both synthetic and experimental images. Manual measurements of bio-based surfaces resulted in between 3.3% (N=50, visually simple) and 51.2% error (N=100, visually complex) compared to Python analysis results. This analysis was also used to accurately distinguish multiple feature size populations in a given image (which were missed entirely in manual measurements), and to determine the skewness and kurtosis of biological surfaces in a surface roughness map. This work contributes to a larger goal of developing a robust and computationally cheap platform to analyze complex materials to accelerate structure-function discovery.
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 entire solutions $ω\in\dot H^1(\mathbb R^2;\mathbb R^3)$ of the $H$-system $Δω=2ω_x\wedgeω_y,$ which we refer to as bubbles. Surprisingly, and contrary to conjectures raised in the literature, we find that bubbles with degree at least three can be degenerate: the linearized $H$-system around a bubble can admit solutions that are not tangent to the smooth family of bubbles. We then give a complete algebraic characterization of degenerate bubbles.
Acoustic levitation is a container-free method for examining novel crystallization effects, though liquid-to-solid phase change has seen little investigation for levitated nanofluids. Recent developments have allowed for examining the morphological and temperature evolution of multiple levitated nanofluid droplets freezing simultaneously. The fundamental effect of adding nanoparticles to a levitated crystallization system is crystal growth rate enhancement from improved mass transfer at the growing solid front. Nucleation times are unaffected as freezing is initiated by secondary ice nucleation particles (INPs). Instead, the enhancement produces higher instantaneous nucleation pressures and more cracking in the primary ice shell. In turn, more INPs are ejected, resulting in faster protrusion formation on the droplet surface (hastened further in systems containing adjacent droplets). The crystal matrix also includes more defects, resulting in liquid escaping and forming beads at the droplet base and optical clarity loss. During crystal decomposition, thermal gradients create convective currents dampened by the same transport phenomena that enhance crystal growth. Suspension loss after a crystallization-decomposition cycle reduced opacity and light absorbance such that the droplets were 62% closer in appearance to water. However, the non-isobaric, sample-encompassing cooling process resulted in smaller particle clusters than if the droplets were frozen on a solid surface.
In this manuscript, we delve into the study of maps $u\in W^{1,2}(Ω;\overline M)$ that minimize the Alt-Caffarelli energy functional $$ \int_Ω(|Du|^2 + q^2 χ_{u^{-1}(M)})\,dx, $$ under the condition that the image $u(Ω)$ is confined within $\overline M$. Here, $Ω$ 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 the validity of a $\varepsilon$-regularity theorem. This theorem is 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 energy. 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 non-coincidence 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.