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August 5, 2026Communications in Partial Differential EquationsOpen Access

Borderline regularity in singular free boundary problems

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Authors

DADamião J. AraújoASAelson SobralETEduardo V. Teixeira

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Overview

This article investigates regularity in energy function minimizers with implications for differentiability thresholds.

Key Points

  • The aim is to explore the regularity properties of local minimizers in free boundary problems with minimal assumptions on the potential term.
  • Investigated the regularity of local minimizers of energy functionals of the form ∫12|Du|2+σ(u) with σ bounded and measurable.
  • Established Log-Lipschitz continuity for sign-changing minimizers under these conditions.
  • Proved gradient bounds and C1 class regularity for minimizers along free boundaries when σ is continuous.
  • Demonstrated that sign-changing minimizers are Log-Lipschitz continuous under minimal potential assumptions.
  • Established that minimizers exhibit gradient bounds along their free boundaries, indicating improved regularity structure.
  • Identified a sharp threshold for differentiability where minimizers are of class C1 if σ is continuous.

Cite This Study

Araújo et al. (2026) studied this question.

synapsesocial.com/papers/6a72e79c226790f370656cc4https://doi.org/10.1080/03605302.2026.2704981
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