Concentrations and Fine Properties of PDE-constrained measures

Project Information

ConFine

Grant agreement ID: 101078057

DOI

10.3030/101078057

EC signature date 22 October 2023

Start date 1 March 2024

End date 28 February 2029

Funded under

• European Research Council (ERC)

Total cost

€ 1 439 816,00

EU contribution

€ 1 439 816,00

Coordinated by

UNIVERSITA DI PISA
Italy

Periodic Reporting for period 1 - ConFine (Concentrations and Fine Properties of PDE-constrained measures)

Reporting period: 2024-03-01 to 2026-08-31

Summary of the context and overall objectives of the project

The interaction between microscopic behavior and macroscopic observables lies at the heart of modern nonlinear Partial Differential Equations (PDEs). In physical and materials sciences, macroscopic deformations often arise from microscopic phenomena like oscillations, mass concentrations, and fine geometric patterns. For example, solid materials under extreme stress may develop microscopic plane-like dislocations, fractures, or phase transitions. Mathematically, these complex behaviors are modeled using weak forms of convergence and measure-valued solutions constrained by constant-coefficient linear PDEs (A-free measures). While ubiquitous in describing elasticity, electromagnetism, and fluid dynamics, the profound geometric and analytical nature of these underlying singularities remains a great mathematical frontier. The ConFine project was motivated by the urgent need to bridge the Calculus of Variations, Geometric Measure Theory (GMT), and Harmonic Analysis to decode the rigid structural rules governing these PDE-constrained measures.

The overarching goal of ConFine is to investigate the qualitative and quantitative nature of these concentrations and fine geometries to solve longstanding, mathematically rigid problems. The project is structured around two ambitious, intersecting themes:

- Theme I (Concentrations & Variational Properties): Investigates the exact nature of PDE-constrained mass concentrations. The primary objective is to advance Compensated Integrability and address the longstanding Vanishing Mass Conjecture. By imposing pointwise constraints on wave cones, the project aims to determine whether singular mass concentrations can be strictly prevented or rigidly structured.
- Theme II (Fine Properties of PDEs): Seeks to describe the exact geometric structure and dimensional size of singularities in PDE solutions. Key objectives include resolving the σ-finite conjecture for maps with bounded deformation (BD) and exploring the generalized Luzin property for linear operators. Following early progress, this theme was strategically expanded to tackle the 1-dimensional Flat Chain Conjecture from a PDE perspective.

The project aims to deliver significant advances in pure mathematics, pushing Geometric Measure Theory beyond the state of the art. Scientifically, resolving conjectures like the Vanishing Mass and Flat Chain conjectures provides researchers with a powerful analytical toolkit, changing how mathematicians and physicists understand limits of integration, optimal transport, and current geometries. Beyond pure mathematics, providing sharp dimensional estimates and rigidity proofs for A-free measures has profound downstream implications. By rigorously explaining how singularities form, ConFine equips materials scientists and physicists with the precise mathematical language needed to better model material failures, crystalline slips, and complex geometric currents.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

In its first half, ConFine achieved the following outcomes:

1. Compensated Integrability & Variational Properties (Theme I): We proved a quantitative L^1 compensated compactness result: A-free sequences with uniformly bounded mass are strictly equi-integrable on cones with quantified aperture, preventing singular mass concentrations [1]. With S. Conti [7], we expanded our variational framework to investigate the rigidity of functions with additive oscillations. Postdocs F. Nobili and I. Y. Violo explored compactness failures in geometric PDEs driven by Riemannian Sobolev inequalities [9].

2. Fine Properties and Dimensional Estimates (Theme II.A): We established optimal dimensional estimates for BV^A-functions [6], proving that a measure Au cannot concentrate on sets of dimension below n−1 to describe the exact Hausdorff size of discontinuity sets for complex-elliptic PDEs. Using this, we characterized the jump part of functions with bounded A-variation, including one-sided approximate limits, and established one-dimensional plane wave rigidity for Au as a valuable tool for future studies [3].

3. The Luzin Property (Theme II.C): We resolved Problem II.C by establishing the Lebesgue-Lusin property for first- and second-order linear operators [2]. This prompt success allowed us to formally request an amendment to pivot the project's remaining resources.

4. The Flat Chain Conjecture & Optimal Transport (Theme II.C'): We addressed the 1D flat chain conjecture across metric spaces by providing necessary and sufficient conditions for its validity [4]. This led us to develop the Monge-Kantorovich tangent bundle [8], introducing a novel tool for studying currents via the duality between Lipschitz functions and the Arens-Eells space. Postdoc F. Nobili co-authored a study [11] on robust disintegration techniques arising from L1 optimal transport.

5. Foundational Methodologies: We developed an analytical toolkit for non-local operators, introducing a general difference quotient representation and a generalized Gauss-Green Formula [5]. Postdoc I. Y. Violo co-authored a robust framework for obtaining quantitative second-order estimates for low-regularity elliptic equations [12].

6. Knowledge Transfer & Internationalisation: We actively shared our methodologies globally, highlighted by the PI's invited talk at the National Congress of the Calculus of Variations (Riccione, 2025). The PI presented collaborative work at major events in Padua, Madrid, Bonn, Cortona, and Oaxaca. Our postdocs also promoted these findings at esteemed institutions: F. Nobili (ESI Wien, Warwick, Nice, Oxford), I. Y. Violo (Padova, Genova, Penn State), and A. Aussedat (SISSA, Padua).

Bibliography

[1] Arroyo-Rabasa A, Aussedat A. Compensated compactness of A-free measures on cones with quantified aperture.
[2] Arroyo-Rabasa A. A Lebesgue-Lusin property for linear operators of first and second order.
[3] Arroyo-Rabasa A, Skorobogatova A. A look into some fine properties of functions with bounded A-variation.
[4] Arroyo-Rabasa A, Bouchitté G. Structural properties of one-dimensional metric currents: SBV-representations, connectedness and the flat chain conjecture.
[5] Arroyo-Rabasa A. Functional and variational aspects of nonlocal operators associated with linear PDEs.
[6] Arroyo-Rabasa A. Complex-ellipticity, dimensional estimates, and plane wave rigidity in BV^A.
[7] Arroyo-Rabasa A, Conti S. Relaxation and fine properties of PDE-constrained microstructures.
[8] Aussedat A, Bouchitté G, Arroyo-Rabasa A. The Monge-Kantorovich Tangent Bundle to a Measure.
[9] Nobili F, Violo IY. Generalized existence of extremizers for the sharp p-Sobolev inequality on Riemannian manifolds with nonnegative curvature.
[10] Nobili F, Renzi F, Vitillaro F. Mosco-convergence of Cheeger energies on varying spaces satisfying curvature-dimension conditions.
[11] Caputo E, Nobili F, Rossi T. Comparison estimates on nonsmooth spaces with integrable Ricci lower bounds via localization.
[12] Nobili F, Violo IY. Fine Pólya–Szegő rearrangement inequalities in metric spaces and applications.
[13] Schulz SM, Violo IY. Regularity for quasilinear elliptic equations in metric measure spaces.
[14] Fässler K, Violo IY. On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part II.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

ConFine advanced Geometric Measure Theory and PDEs beyond the state of the art:

- 1D Flat Chains & Tangent Bundles: Resolving the 1D flat chain conjecture [4] was a key step. Showing the Alberti-Marchese differentiability bundle matches Bouchitté's optimal transport bundle [8] offers a new pathway for PDE constraints.

- Vanishing Mass Conjecture: Our quantitative cone rigidity [1] addresses Rindler's problem, providing an alternative De Philippis-Rindler Theorem proof. Our Monge-Kantorovich bundle findings [8] should help geometrically resolve this conjecture.

- Regularity & Stability: We established unified variational frameworks for geometric PDE stability [10], refined nonsmooth symmetrization [12], and developed quantitative second-order regularity strategies for quasilinear elliptic operators [13].

- Optimal Transport in Singular Geometries: New stability principles for functional inequalities and flexible disintegration procedures [9, 11] provide robust tools to study singular spaces under weak regularity.

- Quantitative Rectifiability: We extended Euclidean techniques to general metric spaces, establishing Carleson-type criteria to measure set flatness [14] and detect regular structures in irregular spaces.

Scientifically, these advances provide a versatile toolkit (e.g. generalized Gauss-Green formula [5], bundle unification). Downstream, our estimates and rigidity proofs for A-free measures equip physicists to model material failures.

Our environment also actively supports early-career researchers; notably, postdoc Francesco Nobili secured a Tenure Track position at the University of Naples Federico II.