The L p Neumann problem for the heat equation in non-cylindrical domains
Steve Hofmann, John L. Lewis · Journées Équations aux dérivées partielles · 1998
I shall discuss joint work with John L. Lewis on the solvability of boundary value problems for the heat equation in non-cylindrical (i.e., time-varying) domains, whose boundaries are in some sense minimally smooth in both space and time. The emphasis will be on the Neumann problem with data in L p . A somewhat surprising feature of our results is that, in contrast to the cylindrical case, the optimal results hold when p = 2 , with the situation getting progressively worse as p approaches 1 . In particular, in our setting, the Neumann problem fails to be solvable when the data is taken to belong to the Hardy space H 1 .