A new $p$-harmonic map flow with Struwe monotonicity

Erik Hupp; Michał Miśkiewicz

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arXiv · 2308.16096

A new $p$-harmonic map flow with Struwe monotonicity

Erik Hupp·Michał Miśkiewicz

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Abstract

We construct and analyze solutions to a regularized homogeneous $p$-harmonic map flow equation for general $p \geq 2$. 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 \`a la Chen-Struwe, employing tools such as a Bochner-type formula and an $\varepsilon$-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.

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Erik Hupp, Michał Miśkiewicz. 2023-08-30. A new $p$-harmonic map flow with Struwe monotonicity. https://arxiv.org/abs/2308.16096

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