Uniqueness of extendable temperatures


  • N. A. Watson


Temperature, supertemperature, fine topology, caloric measure


Let E and D be open subsets of Rn+1 such that D is a compact subset of E, and let v be a supertemperature on E. A temperature u on D is called extendable by v if there is a supertemperature w on E such that w = u on D and w = v on E\D. We know that either there is a unique temperature extendable by v, or there are infinitely many. We also know that a neces- sary condition for uniqueness is that the generalised solution of the Dirichlet problem on D corresponding to the restriction of v to ∂eD, is equal to the greatest thermic minorant of v on D. In this paper, we first give a condition for non-uniqueness and an example to show that this necessary condition is not sufficient. We then give a uniqueness theorem involving the thermal and cothermal fine topologies, and deduce a corollary involving only parabolic and coparabolic tusks.