Non-uniformly elliptic variational problems on BV

We establish $\mathrm{W}^{1,1}$-regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on $\mathrm{BV}$. Unlike classical examples such as the minimal surface integrand, we only require linear growth from below but not necessarily from above. This typically comes with a non-uniformly degenerate elliptic behaviour, for which our results extend the presently available bounds from the superlinear growth case in a sharp way.

💡 Research Summary

The paper addresses variational integrals with linear growth from below but without an upper linear bound, allowing super‑linear growth of the form
γ|z| ≤ F(z) ≤ Γ(1+|z|^q) (γ,Γ>0, 1 ≤ q < ∞).
The integrand F is assumed C² and satisfies a non‑uniform ellipticity condition, called (µ,q)‑ellipticity: there exist constants λ,Λ>0 such that

λ(1+|z|²)^{−µ/2}|ξ|² ≤ ⟨∇²F(z)ξ,ξ⟩ ≤ Λ(1+|z|²)^{(q−2)/2}|ξ|²

for all z,ξ∈ℝ^{N×n}. This condition captures the degenerate behaviour typical of linear‑growth functionals (µ>1) and extends it to the super‑linear regime (q>1).

Because the lower linear bound forces minimizing sequences to be bounded only in W^{1,1}, compactness is lost in the Sobolev space. The authors therefore pass to the space of functions of bounded variation, BV(Ω;ℝ^N), and introduce the Lebesgue–Serrin–Marcellini (LSM) relaxation

F⁎_{u₀}