  We prove a stability result of isometric immersions of hypersurfaces in
Riemannian manifolds, with respect to $L^p$-perturbations of their fundamental
forms: For a manifold $M^d$ endowed with a reference metric and a reference
shape operator, we show that a sequence of immersions $f_n:M^d\to N^{d+1}$,
whose pullback metrics and shape operators are arbitrary close in $L^p$ to the
reference ones, converge to an isometric immersion having the reference shape
operator. This result is motivated by elasticity theory and generalizes a
previous result by the authors to a general target manifold $N$, removing a
constant curvature assumption. The method of proof differs from that in Alpern
et al.: it extends a Young measure approach that was used in codimension-0
stability results, together with an appropriate relaxation of the energy and a
regularity result for immersions satisfying given fundamental forms. In
addition, we prove a related quantitative (rather than asymptotic) stability
result in the case of Euclidean target, similar to Ciarlet et al. (Anal. Appl.
2019) but with no a-priori assumed bounds.
