Abstract
We investigate some of the effects of the lack of compactness in the critical Folland–Stein–Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi’s Γ-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe [M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 1984, 4, 511–517].
Funding statement: The authors are supported by INdAM projects “Fenomeni non locali in problemi locali”, Grant No. CUP_E55F22000270001. The first two authors are supported by “Problemi non locali: teoria cinetica e non uniforme ellitticità”, Grant No. CUP_E53C22001930001 and “Problemi ellittici e sub-ellittici: singolarità e crescita critica”, Grant No. CUP_E53C23001670001. The second author is also supported by the Project “Local vs Nonlocal: mixed-type operators and nonuniform ellipticity”, Grant No. CUP_D91B21005370003. The results in this paper have been announced in the preliminary research report .
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© 2024 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case
Artikel in diesem Heft
- Frontmatter
- A characterization of ℓ1 double bubbles with general interface interaction
- The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth
- On the behavior in time of the solutions to total variation flow
- Zaremba problem with degenerate weights
- Point-wise characterizations of limits of planar Sobolev homeomorphisms and their quasi-monotonicity
- Struwe’s global compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group
- On the variational nature of the Anzellotti pairing
- Strongly nonlinear Robin problems for harmonic and polyharmonic functions in the half-space
- Upper semicontinuity of index plus nullity for minimal and CMC hypersurfaces
- A variational approach to the Navier–Stokes equations with shear-dependent viscosity
- Hölder regularity for the fractional p-Laplacian, revisited
- Poincaré inequality and energy of separating sets
- On a class of obstacle problems with (p, q)-growth and explicit u-dependence
- Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups
- Parabolic Lipschitz truncation for multi-phase problems: The degenerate case