A conditional regularity result for p-harmonic flows

We prove an $\varepsilon$-regularity result for a wide class of parabolic systems $$u_t-\text{div}\big(|\nabla u|^{p-2}\nabla u) = B(u, \nabla u)$$ with the right hand side $B$ growing like $|\nabla u|^p$. It is assumed that the solution $u(t,\cdot)$ is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in an elliptic context by T. Rivi\`ere and the last named author.Comment: To appear in NoDEA. Referee suggestions implemente

Boundary regularity for minimizing biharmonic maps

We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Scheven’s interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong–Lamm–Wang