On the Dirichlet-to-Neumann Map for the $p$-Laplacian on a Metric Measure Space

Abstract: In this note, we construct a Dirichlet-to-Neumann map, from a Besov space of functions, to the dual of this class. The Besov spaces are of functions on the boundary of a bounded, locally compact uniform domain equipped with a doubling measure supporting a $p$-Poincar\'e inequality so that this boundary is also equipped with a Radon measure that has a codimensional relationship with the measure on the domain. We construct this map via the following recipe. We show first that solutions to Dirichlet problem for the $p$-Laplacian on the domain with prescribed boundary data in the Besov space induce an operator that lives in the dual of the Besov space. Conversely, we show that there is a solution, in the homogeneous Newton-Sobolev space, to the Neumann problem for the $p$-Laplacian with the Neumann boundary data given by a continuous linear functional belonging to the dual of the Besov space. We also obtain bounds on its operator norm in terms of the norms of trace and extension operators that relate Newton-Sobolev functions on the domain to Besov functions on the boundary.