Positive thermie majorization of temperatures on infinite strips

Initial and relative limiting behaviour of temperatures on a strip

“…so that u has a positive thermic majorant given by the expression on the right. This expression therefore majonzes the least positive thermic majorant of M, SO that by [18,Theorem 2], which shows that | h \ < /4 F K , as required.…”

“…As a consequence of Theorem 6, we can extend a result of Gehring [6,Theorem 10] to the case of an arbitrary n, and thus sharpen [18,Theorem 3] and extend [17,Theorem 5] to R" for all n. for p-almost every x G R". Another application of Theorem 6 now shows that u > 0, and the result is proved.…”

“…Finally, if u is a temperature and v is a non-negative temperature such that u < v on R" X ]0, c[, then t; is called a positive thermic majorant of u on R" X ]0, c[. For details and references, see [18].…”

“…on R" X ]0, c[, and let C = {Xj} J>x be a sequence of points in R". / / there is a real constant A, and a non-negative constant K, such that(18) liminf«(.x, t) < A exp(K||jc|| 2 )for m-almost all x G R", and(19) liminfM(x,/) < oo for all x G R" \ C, then u can be written in the formu(x, t) =AV K (x, t) -h(x, t)+ 2f* + {{xj})w(x -Xj, t) 7 = 1 on R" X ]0,min{c, (4K)"'}[ if K > 0, on R" X ]0, c%if K = 0,where h is a non-negative temperature and V K is as defined in Section 1. PROOF.…”