Sharp Conditions for the BBM Formula and Asymptotics of Heat Content-Type Energies

Abstract Given $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) , we provide sufficient and necessary conditions on the non-negative measurable kernels $$(\rho _t)_{t\in (0,1)}$$ ( ρ t ) t ∈ ( 0 , 1 ) ensuring convergence of the associated Bourgain–Brezis–Mironescu (BBM) energies $$(\mathscr {F}_{t,p})_{t\in (0,1)}$$ ( F t , p ) t ∈ ( 0 , 1 ) to a variant of the p -Dirichlet energy on $$\mathbb {R}^N$$ R N as $$t\rightarrow 0^+$$ t → 0 + both in the pointwise and in the $$\Gamma $$ Γ -sense. We also devise sufficient conditions on $$(\rho _t)_{t\in (0,1)}$$ ( ρ t ) t ∈ ( 0 , 1 ) yielding local compactness in $$L^p(\mathbb {R}^N)$$ L p ( R N ) of sequences with bounded BBM energy. Moreover, we give sufficient conditions on $$(\rho _t)_{t\in (0,1)}$$ ( ρ t ) t ∈ ( 0 , 1 ) implying pointwise and $$\Gamma $$ Γ -convergence and equicoercivity of