On a weighted version of the BBM formula

We prove a weighted version of the Bourgain-Brezis-Mironescu (BBM) formula, both in the pointwise and $\mathrm{\Gamma }$-convergence sense, together with a compactness criterion for energy-bounded sequences. The non-negative weights need only be $L^{\mathrm{\infty }}$ 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.