On the Dirichlet-to-Neumann Map for the p-Laplacian on a Metric Measure Space
arxiv(2024)
摘要
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-Poincaré 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.
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