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On a class of obstacle problems with (p, q)-growth and explicit u-dependence

  • Andrea Gentile ORCID logo , Teresa Isernia ORCID logo EMAIL logo and Antonia Passarelli di Napoli ORCID logo
Published/Copyright: April 2, 2025
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 18 Issue 3

Abstract

In this paper, we establish the higher differentiability of the gradient of solutions to variational obstacle problems of the type

min ⁡ { ∫ Ω F ⁢ ( x , u , D ⁢ u ) ⁢ 𝑑 x : u ∈ 𝒦 ψ ⁢ ( Ω ) } .

Here Ω ⊂ ℝ n is a bounded open set, ψ ∈ W 0 1 , q ⁢ ( Ω ) is a fixed function called obstacle, and 𝒦 ψ ⁢ ( Ω ) is the class of admissible functions. The main feature of the energy densities under consideration here is that they satisfy non-standard growth conditions with respect to the gradient variable and that they explicitly depend on the pair ( x , u ) . Assuming that ψ ∈ L loc ∞ ⁢ ( Ω ) ∩ W loc 2 , 2 ⁢ q - p ⁢ ( Ω ) , we are able to prove a second order regularity result for the solution.

Keywords: Elliptic equations; regularity of solutions; higher differentiability; obstacle problem
MSC 2020: 35J87; 49J40; 47J20

Communicated by Juha Kinnunen

Funding statement: The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Antonia Passarelli di Napoli have been partially supported through the INdAM-GNAMPA 2024 Project “Interazione ottimale tra la regolarità dei coefficienti e l’anisotropia del problema in funzionali integrali a crescite non standard” (CUP: E53C23001670001). Antonia Passarelli di Napoli has also been supported by the Centro Nazionale per la Mobilità Sostenibile (CN00000023) – Spoke 10 Logistica Merci (CUP: E63C22000930007).

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Received: 2024-10-25
Accepted: 2025-03-02
Published Online: 2025-04-02
Published in Print: 2025-07-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. A characterization of ℓ1 double bubbles with general interface interaction
  3. The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
  4. On the behavior in time of the solutions to total variation flow
  5. Zaremba problem with degenerate weights
  6. Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
  7. Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
  8. On the variational nature of the Anzellotti pairing
  9. Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
  10. Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
  11. A variational approach to the Navier–Stokes equations with shear-dependent viscosity
  12. Hölder regularity for the fractional p-Laplacian, revisited
  13. Poincaré inequality and energy of separating sets
  14. On a class of obstacle problems with (p, q)-growth and explicit u-dependence
  15. Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
  16. Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
https://doi.org/10.1515/acv-2024-0111
Keywords for this article
Elliptic equations; regularity of solutions; higher differentiability; obstacle problem

Articles in the same Issue

  1. Frontmatter
  2. A characterization of ℓ1 double bubbles with general interface interaction
  3. The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
  4. On the behavior in time of the solutions to total variation flow
  5. Zaremba problem with degenerate weights
  6. Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
  7. Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
  8. On the variational nature of the Anzellotti pairing
  9. Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
  10. Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
  11. A variational approach to the Navier–Stokes equations with shear-dependent viscosity
  12. Hölder regularity for the fractional p-Laplacian, revisited
  13. Poincaré inequality and energy of separating sets
  14. On a class of obstacle problems with (p, q)-growth and explicit u-dependence
  15. Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
  16. Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
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