arXiv subjects

Adolfo Arroyo-Rabasa

Publications and source records attributed to Adolfo Arroyo-Rabasa.

At least 19 recordsLinked to original sources

arXiv2026

Quantified rigidity of $\mathcal{A}$-free Young measure concentrations

We establish quantitative stability of the De Philippis--Rindler rigidity on conical convex regions. By constructing $\mathcal{A}$-quasiconvex functions with a tailored concavity---which additionally yields a new, elementary proof of the original theorem for constant-rank operators---we prove a spreading inequality that strictly limits the angular concentration of $\mathcal{A}$-free measures. This geometric bound establishes that singular concentrations cannot cluster arbitrarily close to a direction outside the Tartar wave cone. As a consequence, we obtain a new $L^1$ compensated compactness result: $\mathcal{A}$-free sequences with uniformly bounded mass are forced to be equi-integrable, thereby preventing the formation of mass concentrations, provided their targets are asymptotically restricted to cones whose aperture is controlled by a power of the distance to the wave cone of $\mathcal A$.

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arXiv2026

Complex-ellipticity, dimensional estimates and plane wave rigidity in $BV^{\mathbb A}$

We characterize complex-elliptic operators $\mathbb A(D)$ through a hierarchy of overdeterminacy ($\ell$-vanishing) quantifying the structural twisting of their symbols. This framework yields the optimal dimensional estimate for $BV^{\mathbb A}$-functions: a measure ${\mathbb A} u$ cannot concentrate on sets of dimension below $n-1$. Consequently, the jump part of ${\mathbb A} u$ is characterized as an $(n-1)$-dimensional surface measure with density given by the symbol and the two-sided traces. Building on this dimensional bound, we prove that measures satisfying $\frac{{\mathbb A} u}{|{\mathbb A} u|} \in \mathrm{span}\{P_0\}$ precisely decompose into finite sums of one-dimensional $BV$ profiles. Ultimately, these results reveal that complex-ellipticity strictly enforces a plane-wave structure on tangent measures.

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arXiv2025

Affine rigidity of functions with additive oscillation

We prove that a locally integrable function $f:(a,b) \to \mathbb R$ must be affine if its mean oscillation, considered as a function of intervals, can be extended to a locally finite Borel measure. In particular, we show that any function $f$ satisfying the integro-differential identity $|Df|(I)=4\text{osc}(f,I)$ for all intervals $I \subset {(a,b)}$ must be affine.

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arXiv2025

Structural properties of one-dimensional metric currents: SBV-representations, connectedness and the flat chain conjecture

A comprehensive study of one-dimensional metric currents and their relationship to the geometry of metric spaces is presented. We resolve the one-dimensional flat chain conjecture in this general setting, by proving that its validity is equivalent to a simple geometric connectedness property. More precisely, we prove that metric currents can be approximated in the mass norm by normal currents if and only if every $1$-rectifiable set can be covered by countably many Lipschitz curves up to an $\mathscr{H}^1$-negligible set. Building on this, we demonstrate that any $1$-current in a Banach space can be completed into a cycle by a rectifiable current, with the added mass controlled by the Kantorovich--Rubinstein norm of its boundary. We further refine our approximation result by showing that these currents can be approximated by polyhedral currents modulo a cycle. Finally, in arbitrary complete metric spaces, we establish a Smirnov-type decomposition for one-dimensional currents. This decomposition expresses such currents as a superposition, without mass cancellation, of currents associated with curves of bounded variation that have a vanishing Cantor part.

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arXiv2024

Local Poincar\'e constants and mean oscillation functionals for $BV$ functions

We introduce the concept of local Poincar\'e constant of a $BV$ function as a tool to understand the relation between its mean oscillation and its total variation at small scales. This enables us to study a variant of the BMO-type seminorms on $\varepsilon$-size cubes introduced by Ambrosio, Bourgain, Brezis, and Figalli. More precisely, we relax the size constraint by considering a family of functionals that allow cubes of sidelength smaller than or equal to $\varepsilon$. These new functionals converge, as $\varepsilon$ tends to zero, to a local functional defined on $BV$, which can be represented by integration in terms of the local Poincar\'e constant and the total variation. This contrasts with the original functionals, whose limit is defined on $SBV$ and may not exist for functions with a non-trivial Cantor part. Moreover, we characterize the local Poincar\'e constant of a function with a cell-formula given by the maximum mean oscillation of its $BV$ blow-ups. As a corollary of this characterization, we show that the new limit functional extends the original one to all $BV$ functions. Finally, we discuss rigidity properties and other challenging questions relating the local Poincar\'e constant of a function to its fine properties.

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arXiv2022

Functional and variational aspects of nonlocal operators associated with linear PDEs

We introduce a general difference quotient representation for non-local operators associated with a first-order linear operator. We establish new local to non-local estimates and strong localization principles in various spaces of functions, measures and distributions, which fully generalize those known for gradients. Under suitable assumptions, we also establish the invariance of quasiconvexity within the proposed local-nonlocal setting. Applications to the fine properties of $\mathcal A$-gradient measures are further discussed.

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arXiv2022

A Lebesgue-Lusin property for linear operators of first and second order

We prove that for a homogeneous linear partial differential operator $\mathcal A$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation satisfying \[ \mathcal A u(x)= f(x) \qquad \text{almost everywhere}. \] This extends a result of G. Alberti for gradients on $\mathbf R^N$. In particular, for $0 \le m < N$, it is shown that every integrable $m$-vector field is the absolutely continuous part of the boundary of a normal $(m+1)$-current.

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arXiv2021

Representation of the total variation as a $\Gamma$-limit of $BMO$-type seminorms

We address a question raised by Ambrosio, Bourgain, Brezis, and Figalli, proving that the $\Gamma$-limit, with respect to the $L^1_{\rm loc}$ topology, of a family of $BMO$-type seminorms is given by $\tfrac14$ times the total variation seminorm. Our method also yields an alternative proof of previously known lower bounds for the pointwise limit and conveys a compactness result in $L^1_{\rm loc}$ in terms of the boundedness of the $BMO$-type seminorms.

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arXiv2021

New projection and Korn estimates for a class of constant-rank operators on domains

Let $1 < p < \infty$ and let $\Omega$ be an open and bounded set of $\mathbb R^n$. We establish classical Korn inequalities \[ \inf_{\substack{v \in L^p(\Omega)\\\mathcal A v = 0}} \|u - v\|_{W^{k,p}(\Omega)} \le C \| \mathcal A u\|_{L^p(\Omega)} \] for all $k$th order operators $\mathcal A$ satisfying the maximal-rank condition. This new condition is satisfied by the divergence, Laplacian, Laplace-Beltrami, and Wirtinger operators, among others. As such, our estimates generalize Fuchs' estimates for the del-bar operator to maximal-rank operators and to arbitrary domains. For domains with sufficiently regular boundary $\partial \Omega$, we are able to construct an $L^p(\Omega)$-bounded projection $P$, onto the kernel of the operator. This projection is shown to satisfy a classical Fonseca-M\"uller projection estimate \[ \|u - Pu\|_{L^p(\Omega)} \le C \| \mathcal A u\|_{W^{-k,p}(\Omega)} \] as well as analogous estimates for higher-order derivatives. As a particular application of our results, we are able to establish a weak Korn inequality for general constant-rank operators (by taking the infimum over all $\mathcal A$-harmonic maps instead of taking it over all $\mathcal A$-free maps). Several examples are discussed.

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arXiv2021

An elementary approach to the homological properties of constant-rank operators

We give a simple and constructive extension of Rai\c{t}\u{a}'s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement on the order of the operators constructed by Rai\c{t}\u{a}, as well as the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we establish a generalized Poincar\'e lemma for constant-rank operators and homogeneous spaces on $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.

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arXiv2021

A Bourgain-Brezis-Mironescu representation for functions with bounded deformation

We establish a non-local integral difference quotient representation for symmetric gradient semi-norms in $BD(\Omega)$ and $LD(\Omega)$, which does not require the manipulation of distributional derivatives. Our representation extends the formulas for the symmetric gradient established by Mengesha for vector-fields in $W^{1,p}(\Omega;\mathbb R^d)$, which are inspired by the gradient semi-norm formulas introduced by Bourgain, Brezis and Mironescu in $W^{1,p}(\Omega)$ and by D\'avila in $BV(\Omega)$.

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arXiv2021

Higher integrability for measures satisfying a PDE constraint

We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[ \mathcal{A} \mu = \sigma, \] where $\mu \in \mathcal{M}(\Omega;V)$ and $\sigma\in \mathcal{M}(\Omega;W)$ are vector measures and the polar $\frac{\mathrm{d} \mu}{\mathrm{d} |\mu|}$ is uniformly close to a convex cone of $V$ intersecting the wave cone of $\mathcal{A}$ only at the origin. More precisely, we prove local compensated compactness estimates of the form \[ \|\mu\|_{\mathrm{L}^p(\Omega')} \lesssim |\mu|(\Omega) + |\sigma|(\Omega), \qquad \Omega' \Subset \Omega. \] Here, the exponent $p$ belongs to the (optimal) range $1 \leq p < d/(d-k)$, $d$ is the dimension of $\Omega$, and $k$ is the order of $\mathcal{A}$. We also obtain the limiting case $p = d/(d-k)$ for canceling constant-rank operators. We consider applications to compensated compactness and {applications to the theory of} functions of bounded variation and bounded deformation.

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arXiv2020

Slicing and fine properties for functions with bounded $\mathcal A$-variation

We study the slicing and fine properties of functions in $\mathrm{BV}^{\mathcal A}$, the space of functions with bounded $\mathcal A$-variation. Here, $\mathcal A$ is a homogeneous linear differential operator with constant coefficients (of arbitrary order). Our main result is the characterization of all $\mathcal A$ satisfying the following one-dimensional structure theorem: every $u \in \mathrm{BV}^{\mathcal A}$ can be sliced into one-dimensional $\mathrm{BV}$-sections. Moreover, decomposing $\mathcal A u$ into an absolutely continuous part $\mathcal A^a u$, a Cantor part $\mathcal A^c u$ and a jump part $\mathcal A^j u$, each of these measures can be recovered from the corresponding classical $D^a,D^c$ and $D^j$ $BV$-derivatives of its one-dimensional sections. By means of this result, we are able to analyze the set of Lebesgue points as well as the set of jump points where these functions have approximate one-sided limits. Thus, proving a structure and fine properties theorem in $\mathrm{BV}^{\mathcal A}$. Our results extend most of the classical fine properties of $\mathrm{BV}$ (and all of those known for $\mathrm{BD}$). In particular, we establish a slicing theory and fine properties for $\mathscr {BV}^k, \mathrm{BD}^k$ and a whole class of $\mathrm{BV}^{\mathcal A}$-spaces that is not covered by the existing theory.

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arXiv2020

Debye screening for the stationary Vlasov-Poisson equation in interaction with a point charge

We prove that the Debye screening length emerges in a spatially homogeneous plasma described by the nonlinear Vlasov-Poisson equation interacting with a point charge. While screened stationary states as well as the time-dependent problem are well-studied for the linearized equation, there are few rigorous results on screening in the nonlinear setting. As such, the results presented here cover the stationary case under the assumptions predicted by the linearized theory.

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arXiv2019

A look into some of the fine properties of functions with bounded $\mathscr A$-variation

We establish certain fine properties for functions of bounded $\mathscr A$-variation known in the classical $BV$ setting. Here, $\mathscr A$ is a $k$th order constant-coefficient homogeneous linear differential operator with a finite-dimensional kernel (also known as a complex-elliptic operator). We prove that if $\mathscr Au$ can be represented by a finite Radon measure, then the potential $u$ has one-sided $L^p$-approximate limits on Lipschitz hypersurfaces, and, more generally, on countably rectifiable sets of codimension one. We use this to give pointwise characterizations of the (functional) interior and exterior traces. We also establish a quantitative scale-dependent continuity result, which allows us to prove that the Lebesgue discontinuity set has zero $(n-1)$-dimensional Riesz capacity. Lastly, we introduce a decomposition that reduces the complexity of analyzing $k$th-order operators to that of first-order methods and allows us to establish the $k$th order $L^p$-differentiability of $BV^{\mathscr A}$ maps.

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arXiv2019

Characterization of generalized Young measures generated by $\mathcal A$-free measures

We give two characterizations, one for the class of generalized Young measures generated by $\mathcal A$-free measures, and one for the class generated by $\mathcal B$-gradient measures $\mathcal Bu$. Here, $\mathcal A$ and $\mathcal B$ are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The characterization places the class of generalized $\mathcal A$-free Young measures in duality with the class of $\mathcal A$-quasiconvex integrands by means of a well-known Hahn--Banach separation property. A similar statement holds for generalized $\mathcal B$-gradient Young measures. Concerning applications, we discuss several examples that showcase the rigidity or the failure of $\mathrm{L}^1$-compensated compactness when concentration of mass is allowed. These include the failure of $\mathrm{L}^1$-estimates for elliptic systems and the failure of $\mathrm{L}^1$-rigidity for the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set $\Omega$, the inclusions \[ \mathrm{L}^1(\Omega) \cap \ker \mathcal A \hookrightarrow \mathcal M(\Omega) \cap \ker \mathcal A, \] \[ \{\mathcal B u\in \mathrm{C}^\infty(\Omega)\} \hookrightarrow \{\mathcal B u\in \mathcal M(\Omega)\}, \] are dense with respect to area-functional convergence of measures

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arXiv2019

Generalized multi-scale Young measures

This paper is devoted to the construction of generalized multi-scale Young measures, which are the extension of Pedregal's multi-scale Young measures [Trans. Amer. Math. Soc. 358 (2006), pp. 591-602] to the setting of generalized Young measures introduced by DiPerna and Majda [Comm. Math. Phys. 108 (1987), pp. 667-689]. As a tool for variational problems, these are well-suited objects for the study (at different length-scales) of oscillation and concentration effects of convergent sequences of measures. Important properties of multi-scale Young measures such as compactness, representation of non-linear compositions, localization principles, and differential constraints are extensively developed in the second part of this paper. As an application, we use this framework to address the $\Gamma$-limit characterization of the homogenized limit of convex integrals defined on spaces of measures satisfying a general linear PDE constraint.

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arXiv2018

An elementary approach to the dimension of measures satisfying a first-order linear PDE constraint

We give a simple criterion on the set of probability tangent measures $\mathrm{Tan}(\mu,x)$ of a positive Radon measure $\mu$, which yields lower bounds on the Hausdorff dimension of $\mu$. As an application, we give an elementary and purely algebraic proof of the sharp Hausdorff dimension lower bounds for first-order linear PDE-constrained measures; bounds for closed (measure) differential forms and normal currents are further discussed. A weak structure theorem in the spirit of [Ann. Math. 184(3) (2016), pp. 1017-1039] is also discussed for such measures.

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