We prove a weighted version of the Bourgain–Brezis–Mironescu (BBM) formula, both in the pointwise and Γ-convergence sense, together with a compactness criterion for energy-bounded sequences. The non-negative weights need only be L∞ convergent to a bounded and uniformly continuous limit. We apply the BBM formula to show a Poincaré-type inequality and the stability of the first eigenvalues relative to the energies. Finally, we discuss a non-local analogue of the weighted BBM formula.
On a weighted version of the BBM formula
Stefani, Giorgio
2025
Abstract
We prove a weighted version of the Bourgain–Brezis–Mironescu (BBM) formula, both in the pointwise and Γ-convergence sense, together with a compactness criterion for energy-bounded sequences. The non-negative weights need only be L∞ convergent to a bounded and uniformly continuous limit. We apply the BBM formula to show a Poincaré-type inequality and the stability of the first eigenvalues relative to the energies. Finally, we discuss a non-local analogue of the weighted BBM formula.
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