Differentiability of Solutions to the Neumann Problem with Low-regularity Data via Dynamical Systems
We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to satisfy the square-Dini condition and the boundary is assumed to be differentiable with derivatives also having this modulus of continuity. Additional conditions for the solution to be Lipschitz continuous or differentiable at a point on the boundary depend upon the stability of a dynamical system that is derived from the coefficients of the elliptic equation.
KeywordsDifferentiability Lipschitz continuity weak solution elliptic equation divergence form co-normal boundary condition modulus of continuity square-Dini condition dynamical system asymptotically constant uniformly stable
Mathematics Subject Classification (2010)Primary 35A08
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