Shape analysis has been of great interest in many areas such as medicine, materials science, industrial processes, computer vision, and computer graphics. Analysis of shapes highly depends on their representation. In this thesis, we consider implicit shape representations, fields or mappings from the shape domain to the real line, obtained by utilizing an elliptic PDE or its modifications. In the first part of the thesis, we present a novel shape characterization field called discrepancy as a local measure of deviation from a reference disk. Discrepancy is computed indirectly by comparing the numerical solution of an elliptic PDE on the arbitrary shape domain and the analytical solution of the same PDE on the reference disk. We demonstrate the potential of discrepancy via illustrative applications, namely, global characterization of the body roundness and the periphery thickness uniformity, context-dependent categorization, and shape decomposition. In the second part, we follow a framework involving high-dimensional feature space construction by solving the elliptic PDE multiple times varying either the diffusion parameter or the right hand side function and low-dimensional reduction to assign a distinctness value to each shape point. We utilize the obtained fields for non-structural representation of 2D shapes and saliency measurement of 3D mesh surfaces. In the third and the final part, we utilize the elliptic PDE modifications for bringing a pair of 3D shapes into comparable topology.