Beyond Constant Truncations: The Bounded Sobolev Approximation Gap

We study the approximation of finite-energy Sobolev functions by bounded Sobolev functions for integral functionals associated with possibly signed Carathéodory integrands. The approximating sequence is required to converge strongly in the Sobolev space, while the corresponding energy densities converge in $L^1$. Ordinary truncations may fail because they replace the gradient by zero on the tails, where the integrand need not be integrable. Our main contribution is a general method based on spatially dependent barriers whose gradients can be adapted to the structure of the Lagrangian.