On the extension problem for semiconcave functions with fractional modulus

Consider a locally Lipschitz function u on the closure of a possibly unbounded open subset Omega of R-n with nonempty boundary. Suppose u is (locally) semiconcave on Omega with a fractional semiconcavity modulus. Is it possible to extend u in a neighborhood of any boundary point retaining the same semiconcavity modulus? We show that this is indeed the case and we give two applications of this extension property. First, we derive an approximation result for semiconcave functions on closed domains. Then, we use the above extension property to study the propagation of singularities of semiconcave functions at boundary points. (C) 2021 Elsevier Ltd. All rights reserved.