Sharp Born--Infeld Sobolev inequality and extremal functions on the lattice graph ℤ^N

Abstract

In this paper, we study the sharp Sobolev-type inequality associated with the Born--Infeld energy on the lattice graph ℤ^N, N≥3: 1/2∑_x∈ℤ^N∑_{y∼ x} (1−√(1−|∇_xyu|²)) ≥ C_N,α (∑_x∈ℤ^N|u(x)|^α)^N/(N+α), for u∈ D^1,2(ℤ^N)∩ℓ^α(ℤ^N) satisfying |∇_xyu|≤1 for every x∼ y, where C_N,α denotes the optimal constant. We determine the exact positivity threshold and prove that C_N,α>0 if and only if α≥2^*:=2N/(N−2). At the Sobolev critical exponent α=2^*, we identify the optimal constant as C_N,2^*=1/2𝒮₂, where 𝒮₂ is the optimal discrete Sobolev constant, and show that it is not attained. In the supercritical regime, there exists ε₀=ε₀(N)>0 such that C_N,α is attained for every 2^*<α<2^*+ε₀, with a nonnegative Schwarz symmetric extremal function. The main compactness difficulties stem from the lack of a suitable scaling on ℤ^N and from the nonhomogeneity of the Born--Infeld energy. We overcome them by combining discrete Schwarz rearrangement with a Q₁ discrete-to-continuum comparison and by establishing a strict separation as the ℓ^α-norm tends to infinity.

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BibTeX RIS

BibTeX

@misc{ji2026sharp,
  title = {{Sharp Born--Infeld Sobolev inequality and extremal functions on the lattice graph $\mathbb Z^N$}},
  author = {Chao Ji and Kai Sheng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18142},
  eprint = {2609.18142},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18142},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Sharp Born--Infeld Sobolev inequality and extremal functions on the lattice graph ℤ^N
AU  - Chao Ji
AU  - Kai Sheng
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18142
UR  - https://arxiv.org/abs/2609.18142
AB  - In this paper, we study the sharp Sobolev-type inequality associated with the Born--Infeld energy on the lattice graph ℤ^N, N≥3: 1/2∑_x∈ℤ^N∑_y∼ x (1−√(1−|∇_xyu|^2)) ≥ C_N,α (∑_x∈ℤ^N|u(x)|^α)^N/(N+α), for u∈ D^1,2(ℤ^N)∩ℓ^α(ℤ^N) satisfying |∇_xyu|≤1 for every x∼ y, where C_N,α denotes the optimal constant. We determine the exact positivity threshold and prove that C_N,α>0 if and only if α≥2^*:=2N/(N−2). At the Sobolev critical exponent α=2^*, we identify the optimal constant as C_N,2^*=1/2𝒮_2, where 𝒮_2 is the optimal discrete Sobolev constant, and show that it is not attained. In the supercritical regime, there exists ε_0=ε_0(N)>0 such that C_N,α is attained for every 2^*<α<2^*+ε_0, with a nonnegative Schwarz symmetric extremal function. The main compactness difficulties stem from the lack of a suitable scaling on ℤ^N and from the nonhomogeneity of the Born--Infeld energy. We overcome them by combining discrete Schwarz rearrangement with a Q_1 discrete-to-continuum comparison and by establishing a strict separation as the ℓ^α-norm tends to infinity.
ER  -

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