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Approximation of Lipschitz Functions Preserving Boundary Values
Given an open subset of a Banach space and a Lipschitz function
we study whether it is possible to
approximate uniformly on by -smooth Lipschitz functions
which coincide with on the boundary of and
have the same Lipschitz constant as As a consequence, we show that every
-Lipschitz function defined on the
closure of an open subset of a finite dimensional
normed space of dimension , and such that the Lipschitz constant of
the restriction of to the boundary of is less than , can be
uniformly approximated by differentiable -Lipschitz functions which
coincide with on and satisfy the equation almost everywhere on This result does not hold in general without
assumption on the restriction of to the boundary of .Comment: Some cosmetic changes were made in this versio