On p-Harmonic Map Heat Flows for 1<=p<∞ and Their Finite Element Approximations

Motivated by emerging applications from imaging processing, the heat flow of a generalized p-harmonic map into spheres is studied for the whole spectrum, 1 ≤ p <∞, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a BV -solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms A ·∇v and A ∧∇v; which pair a divergence-L1, or a divergence-measure, tensor field A, and a BV -vector field v. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the p-harmonic map heat flow, and the convergence of the proposed numerical method is also established.