Regularity for the evolution of \(p\)-harmonic maps
The author studies the regularity for \(p\)-harmonic map heat flows. A \(p\)-harmonic map is defined to be a critical point, which is a solution of the Euler-Lagrange equation of the \(p\)-energy \(-\mathrm{div}(\text{ }\!\!|\!\!\text{ Du}{{\text{ }\!\!|\!\!\text{ }}^{\text{p-2}}}\text{Du)= }\!\!|\!\!\text{ Du}{{\text{ }\!\!|\!\!\text{ }}^{\text{p-2}}}A(u)(\text{Du},\text{Du})\), where \(\Omega \subset {{\mathbb{R}}^{m}},\;m\geq 2\) be a bounded domain with a smooth boundary, \(Nu\) is an \(n\)-dimensional smooth compact Riemannian manifold without a boundary assumed to be isometrically embedded in \({{\mathbb{R}}^{l}}\), \(l>n\), \(u:\Omega \to \text{N}\), \(\text{Du}=\mathrm{grad}(u)\), \(A(\cdot )(\cdot ,\cdot )=(\sum\limits_{j,k=1}^{l}{A_{jk}^{i}}(u)\text{D}{{\text{u}}^{j}}\cdot \text{D}{{\text{u}}^{k}}),\;i=1,\dots,l\) is the second fundamental form of \(N \subset {{\mathbb{R}}^{l}}\). It is devised a monotonicity type formula of scaled energy and established a criterion for a uniform regularity estimate for the regular \(p\)-harmonic map heat flows. In additional as application of the obtained results are given the small data global in the time existence of regular \(p\)-harmonic map heat flow.
scientific article; zbMATH DE number 6806982