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The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth

  • Simone Ciani , Eurica Henriques EMAIL logo and Igor I. Skrypnik
Published/Copyright: November 17, 2024
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 18 Issue 3

Abstract

In this work we prove that the non-negative functions u ∈ L loc s ⁢ ( Ω ) , for some s > 0 , belonging to the De Giorgi classes

⨍ B r ⁢ ( 1 - σ ) ⁢ ( x 0 ) | ∇ ( u - k ) - | p d x ⩽ c σ q Λ ( x 0 , r , k ) ( k r ) p ( | B r ( x 0 ) ∩ { u ⩽ k } | | B r ⁢ ( x 0 ) | ) 1 - δ ,

under proper assumptions on Λ, satisfy a weak Harnack inequality with a constant depending on the L s -norm of u. Under suitable assumptions on Λ, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying the above condition; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents.

Keywords: Unbounded minimizers; non-logarithmic conditions; weak Harnack’s inequality
MSC 2020: 35B40; 35B45; 35B65

Communicated by Ugo Gianazza

Funding source: Fundação para a Ciência e a Tecnologia

Award Identifier / Grant number: UIDB/00013/2020 and UIDP/00013/2020

Funding source: Simons Foundation

Award Identifier / Grant number: 1160640

Funding statement: Eurica Henriques was financed by Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia – within the Projects UIDB/00013/2020 and UIDP/00013/2020. Igor I. Skrypnik is partial supported by a grant from the Simons Foundation (Award 1160640, Presidential Discretionary-Ukraine Support Grants, Skrypnik I. I.)

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Received: 2024-03-20
Accepted: 2024-09-24
Published Online: 2024-11-17
Published in Print: 2025-07-01

© 2024 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-2024-0032
Keywords for this article
Unbounded minimizers; non-logarithmic conditions; weak Harnack’s inequality

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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