The stiffness of cells and of their nuclei is a biomarker of several pathological conditions. Current measurement methods rely on invasive physical probes that yield one or two stiffness values for the whole cell. However, the internal distribution of cells is heterogeneous. We propose a framework to estimate maps of intracellular and intranuclear stiffness inside deforming cells from fluorescent image sequences. Our scheme requires the resolution of two inverse problems. First, we use a novel optical-flow method that penalizes the nuclear norm of the Hessian to favor deformations that are continuous and piecewise linear, which we show to be compatible with elastic models. We then invert these deformations for the relative intracellular stiffness using a novel system of elliptic PDEs. Our method operates in quasi-static conditions and can still provide relative maps even in the absence of knowledge about the boundary conditions. We compare the accuracy of both methods to the state of the art on simulated data. The application of our method to real data of different cell strains allows us to distinguish different regions inside their nuclei.