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Borderline regularity in singular free boundary problems.

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Τίτλος: Borderline regularity in singular free boundary problems.
Συγγραφείς: Araújo, Damião J.1 (AUTHOR), Sobral, Aelson2 (AUTHOR), Teixeira, Eduardo V.3 (AUTHOR) eduardo.teixeira@okstate.edu, Urbano, José Miguel4,5 (AUTHOR)
Πηγή: Communications in Partial Differential Equations. 2026, Vol. 51 Issue 6, p619-636. 18p.
Θεματικοί όροι: *Calculus of variations, *Lipschitz continuity, *Smoothness of functions, *Boundary value problems
Περίληψη: In this article, we investigate the borderline regularity of local minimizers of energy functionals of the form ∫ 1 2 | Du | 2 + σ (u) , under minimal assumptions on the potential term σ . When σ is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which represents the optimal regularity in this general setting. In the one-phase case, however, we establish gradient bounds for minimizers along their free boundaries, revealing a structural gain in regularity. Most notably, we prove that if σ is continuous, then minimizers are of class C 1 along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential. [ABSTRACT FROM AUTHOR]
Βάση Δεδομένων: Academic Search Index
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  Data: Borderline regularity in singular free boundary problems.
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  Data: *<searchLink fieldCode="DE" term="%22Calculus+of+variations%22">Calculus of variations</searchLink><br />*<searchLink fieldCode="DE" term="%22Lipschitz+continuity%22">Lipschitz continuity</searchLink><br />*<searchLink fieldCode="DE" term="%22Smoothness+of+functions%22">Smoothness of functions</searchLink><br />*<searchLink fieldCode="DE" term="%22Boundary+value+problems%22">Boundary value problems</searchLink>
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  Data: In this article, we investigate the borderline regularity of local minimizers of energy functionals of the form ∫ 1 2 | Du | 2 + σ (u) , under minimal assumptions on the potential term σ . When σ is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which represents the optimal regularity in this general setting. In the one-phase case, however, we establish gradient bounds for minimizers along their free boundaries, revealing a structural gain in regularity. Most notably, we prove that if σ is continuous, then minimizers are of class C 1 along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential. [ABSTRACT FROM AUTHOR]
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    Identifiers:
      – Type: doi
        Value: 10.1080/03605302.2026.2704981
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      – Code: eng
        Text: English
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        PageCount: 18
        StartPage: 619
    Subjects:
      – SubjectFull: Calculus of variations
        Type: general
      – SubjectFull: Lipschitz continuity
        Type: general
      – SubjectFull: Smoothness of functions
        Type: general
      – SubjectFull: Boundary value problems
        Type: general
    Titles:
      – TitleFull: Borderline regularity in singular free boundary problems.
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            NameFull: Araújo, Damião J.
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            NameFull: Sobral, Aelson
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            NameFull: Teixeira, Eduardo V.
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            NameFull: Urbano, José Miguel
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            – D: 01
              M: 06
              Text: 2026
              Type: published
              Y: 2026
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              Value: 51
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              Value: 6
          Titles:
            – TitleFull: Communications in Partial Differential Equations
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