Let u be a weak solution to a p-harmonic system with vanishing Neumann data on a portion of the boundary of a domain which is convex. We show that subsolution type arguments for some uniformly elliptic PDE’s can be used to deduce that the modulus of the gradient is bounded depending on the Lipschitz character of the domain. In this context, I would like to mention that classical results on the boundedness of the gradient require the domain to be C^{1, Dini}. However, in our case, since the domain is convex, one can make use of the fundamental inequality of Grisvard which can be thought of as an analogue of the use of the barriers for Dirichlet problems in convex domains. Our arguments replaces an argument based on level sets in recent important works of Mazya, Cianchi-Mazya and Geng-Shen involving similar problems. I also intend to indicate some open problems in the regularity theory of degenerate elliptic and parabolic systems. This is a joint work with Prof. John L. Lewis.

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Last updated: 19 Jul 2024