How to recognize constant functions. Connections with Sobolev spaces

By Haim Brezis. 2002

236 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5fea1c6177be5e7b45ba75cce4d59183863746ea

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics (2008, background): “A related question has been investigated by Bourgain, Brezis, Mironescu in several papers (see [ 6 ] and references therein) regarding the lifting problem for harmonic maps.”
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces (2013, background): “4) is known to be finite if and only if u is constant (see [4]).”
An estimate in the spirit of Poincaré's inequality (2004, background): “In this case, we have the following pointwise limit (see [2], see also [6] for a simpler proof) lim n→∞ ∫”
The best constant in a fractional Hardy inequality (2008, background): “This corresponds to the fact that the only function u ∈ Cc(Ω) for which the left hand side of (1) is finite for α = 2 is the zero function, see [12, 16].”
On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, unknown): “d (4.16) we therefore obtain D2(J) 2 ZZ RN×RN ρ σ(z) (ϕσ(x+z) −ϕσ(x))2 |z|2 dzdx− Z RN a(x)ϕ2(x)dx ≤ (λp(Mσ,2,Ω +a)+2δ) Z RN ϕ2 σ(x)dx. (4.17) On the other hand, inspired by the proof of Theorem 2 in [14], since ϕσ∈ C∞ c (RN), by Taylor’s expansion, for all x,z∈ RN, we have |ϕσ(x+z) −ϕσ(x) −z·∇ϕσ(x)”
A new approach to Sobolev spaces and connections to $\mathbf\Gamma$-convergence (2004, methodology): “Our approach not only unifies the proofs of some well-known results, including in the BV -case, but it also deals with more general families (ρε) ⊂ L(1)(R ) (see [1, 2, 3, 6, 10, 11, 14]).”