A conditional regularity result for p-harmonic flows

Abstract

Abstract. We prove an ε-regularity result for a wide class of parabolic systemswith the right hand side B growing critically, like |∇u| p . It is assumed a priori that the solution u(t, ·) 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 the elliptic context by T. Rivière and the last named author.

$$L^\infty $$ L ∞ -Norm Estimates of Weak Solutions via Their Morse Indices for the m-Laplacian Problems

“…Inspired by the previously mentioned works, we are concerned to establish some L p and L ∞ estimates for weak solution of (1) via their Morse indices. However, in this work, we use a different approach to those explored in [15,20] because here we are working with the m-Laplacian operator which is a nonlinear degenerate operator and a standard boot-strap iteration do not yield any extra regularity of the solution which only gives C 1,α loc (Ω) regularity, for some α ∈ (0, 1) (we refer the interested readers to [4][5][6]12,16,18]). Therefore, Eq.…”