Paper
Integral Formulation and the Brézis-Ekeland-Nayroles-Type Principle for Prox-Regular Sweeping Processes
Authors
Juan Guillermo Garrido, Emilio Vilches
Abstract
We study sweeping processes in a Hilbert space driven by time-dependent uniformly prox-regular sets, allowing the moving constraint to exhibit discontinuities of bounded variation. We introduce a new integral formulation for bounded-variation trajectories, given by a global variational inequality tested against continuous admissible trajectories, and we compare it with the standard differential-measure formulation, in which the differential measure of the trajectory is constrained by the proximal normal cone. In the prox-regular (generally nonconvex) framework, the variational inequality necessarily includes a quadratic correction term reflecting the hypomonotonicity of proximal normal cones. Under mild regularity assumptions on the moving set, including lower semicontinuity in time, uniform prox-regularity of the values, and a selection-extension property guaranteeing a rich class of test trajectories (satisfied, for instance, in the convex case and for bounded prox-regular sets), we prove that the new integral formulation is equivalent to the differential-measure formulation. This yields a unified bounded-variation notion of solution for prox-regular sweeping processes. We further establish a Brézis-Ekeland-Nayroles-type variational characterization via a prox-regular variational residual: the residual is nonpositive along every admissible trajectory, and solutions are exactly those trajectories for which this residual attains its maximal value, namely zero. As a consequence, we prove a stability result: a uniform limit of admissible trajectories with vanishing residual is a solution of the limit sweeping process. The resulting variational framework provides a robust tool for stability and approximation analyses in the prox-regular, nonconvex setting.
Metadata
Related papers
Cosmic Shear in Effective Field Theory at Two-Loop Order: Revisiting $S_8$ in Dark Energy Survey Data
Shi-Fan Chen, Joseph DeRose, Mikhail M. Ivanov, Oliver H. E. Philcox • 2026-03-30
Stop Probing, Start Coding: Why Linear Probes and Sparse Autoencoders Fail at Compositional Generalisation
Vitória Barin Pacela, Shruti Joshi, Isabela Camacho, Simon Lacoste-Julien, Da... • 2026-03-30
SNID-SAGE: A Modern Framework for Interactive Supernova Classification and Spectral Analysis
Fiorenzo Stoppa, Stephen J. Smartt • 2026-03-30
Acoustic-to-articulatory Inversion of the Complete Vocal Tract from RT-MRI with Various Audio Embeddings and Dataset Sizes
Sofiane Azzouz, Pierre-André Vuissoz, Yves Laprie • 2026-03-30
Rotating black hole shadows in metric-affine bumblebee gravity
Jose R. Nascimento, Ana R. M. Oliveira, Albert Yu. Petrov, Paulo J. Porfírio,... • 2026-03-30
Raw Data (Debug)
{
"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.05376v1</id>\n <title>Integral Formulation and the Brézis-Ekeland-Nayroles-Type Principle for Prox-Regular Sweeping Processes</title>\n <updated>2026-03-05T17:01:28Z</updated>\n <link href='https://arxiv.org/abs/2603.05376v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.05376v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>We study sweeping processes in a Hilbert space driven by time-dependent uniformly prox-regular sets, allowing the moving constraint to exhibit discontinuities of bounded variation. We introduce a new integral formulation for bounded-variation trajectories, given by a global variational inequality tested against continuous admissible trajectories, and we compare it with the standard differential-measure formulation, in which the differential measure of the trajectory is constrained by the proximal normal cone. In the prox-regular (generally nonconvex) framework, the variational inequality necessarily includes a quadratic correction term reflecting the hypomonotonicity of proximal normal cones.\n Under mild regularity assumptions on the moving set, including lower semicontinuity in time, uniform prox-regularity of the values, and a selection-extension property guaranteeing a rich class of test trajectories (satisfied, for instance, in the convex case and for bounded prox-regular sets), we prove that the new integral formulation is equivalent to the differential-measure formulation. This yields a unified bounded-variation notion of solution for prox-regular sweeping processes.\n We further establish a Brézis-Ekeland-Nayroles-type variational characterization via a prox-regular variational residual: the residual is nonpositive along every admissible trajectory, and solutions are exactly those trajectories for which this residual attains its maximal value, namely zero. As a consequence, we prove a stability result: a uniform limit of admissible trajectories with vanishing residual is a solution of the limit sweeping process. The resulting variational framework provides a robust tool for stability and approximation analyses in the prox-regular, nonconvex setting.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.OC'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math.DS'/>\n <published>2026-03-05T17:01:28Z</published>\n <arxiv:primary_category term='math.OC'/>\n <author>\n <name>Juan Guillermo Garrido</name>\n </author>\n <author>\n <name>Emilio Vilches</name>\n </author>\n </entry>"
}