Problems of stability and well-posedness in the calculus of variations and related PDEs

We present a collection of three closely related topics regarding the stability and well-posedness of minimization problems in the calculus of variations, namely the generic Tykhonov well-posedness with respect to linear perturbations, the generalized Hadamard well-posedness for a class of Dirichlet quasiconvex variational problems, and the d-rectifiability of the set of minimizers for a singularly perturbed quasiconvex variational integral.