2023/08/30 by Hupp, Erik, Miśkiewicz, Michał · 1 citation
#35B44 #35K92 #53E99 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.16096
We construct and analyze solutions to a regularized homogeneous p-harmonic map flow equation for general p ≥ 2. The homogeneous version of the problem is new and features a monotonicity formula extending the one found by Struwe for p = 2; such a formula is not available for the nonhomogeneous equation. The construction itself is via a Ginzburg-Landau-type approximation à la Chen-Struwe, employing tools such as a Bochner-type formula and an ε-regularity theorem. We similarly obtain strong subsequential convergence of the approximations away from a concentration set with parabolic codimension at least p. However, the quasilinear and non-divergence nature of the equation presents new obstacles that do not appear in the classical case p = 2, namely uniform-time existence for the approximating problem, and thus our basic existence result is stated conditionally.