 By K.L. Teo and Z.S. Wu (Eds.)

ISBN-10: 0126854807

ISBN-13: 9780126854800

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The proof is complete. 311. 11. The next theorem is well known. 2. Let Q be an open subset in R” with compact closure, U be a nonempty compact convex subset in R”, and %! be the class of all those measurable functions from Q into U . Then, is a weak* compact subset in L A Q , R”). 7. Bibliographical Remarks To close this chapter we wish to indicate the main references. 11. 11. 51. 11. 11. 11. 11. 1. Introduction In this chapter our aim is to study first and second boundary-value problems for a linear second-order parabolic partial differential equation, in both general and divergence forms.

1) Since lui(x)l is bounded by a constant for all i and x , a subsequence of the sequence { x k } can be chosen, which is again denoted by { x k > , so that ui(xk)-+ u: for i = 1, . . , m. Since u(xk)E F ( x k ) and F is continuous on Q and F(x') is closed, u' = (u',, . . , uk) E F(x'). 1) and the continuity of the functions ui(x), i = 1, 2, . . , s - 1 on a,, it follows that u: = ui(x') u; Iu,(x') for i = 1,. . , s -- 1, - El. 2) I.?. Bibliographical Remarks 31 Taking limit in the identity g(xk,u1(xk),.

Bibliographical Remarks 31 Taking limit in the identity g(xk,u1(xk),. . , um(xk))= y(xk). u1(x’)7 * . 7 us- 1(x’)7u s ? . > ’ = y(x’) This contradicts the definition of u,(x). Thus the set { x E R , :u,(x) Ia } must be closed. Since p(R\ur= SZlIk)= 0, us is measurable on R. The proof is complete. 311. 11. The next theorem is well known. 2. Let Q be an open subset in R” with compact closure, U be a nonempty compact convex subset in R”, and %! be the class of all those measurable functions from Q into U .