Multiple integrals of the calculus of variations with vector-valued unknown are considered. Strong Tikhonov well-posedness within \(W^{1,1}_0\) of the corresponding minimum problems is proved assuming strict convexity of the integrand at the gradient of the minimizer. Then well-posedness under perturbations is obtained with respect to the bounded Hausdorff convergence of the integral functionals. Applications are presented to integrals depending only on the gradient, and to radially symmetric functionals.