By T. J. I'a; Bromwich
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Extra info for Quadratic forms and their classification by means of invariant-factors
Example text
Let fo, fi, . 2). 1 in H, for f = fo denote the measure determined b y u. Then there exists a constant C > 0 such that p ( E ) 5 C(cap E)’” for E c R compact. + I n particular, ifcap E = 0, then p ( E ) = 0. Proof: We assume, of course, that 06, # Let 0 I( E C&l) satisfy ( 2 1 on E. Then 0, that is, that 9 I0 on 8R. l,Az,n, = C(cap E)'''. Sn 47 f o r d x . D. 12. 1 and ler I denote its set of coincidence. If cap I = 0, then Lu = f in Q. 1 1 may be weakened since the best constant C satisfies where C , depends on the bilinear form a and Q, and therefore C depends on only through I I U , I I ~ ~ ( ~ , .
A set of measure zero, for example a closed interval in R2, may have positive capacity. The next assertion illustrates the role of inequality in H'(R) in the weak maximum principle. 3. Let u E H'(R) and suppose that -= s u p u = M = i n f ( m ~ R : u ~ m o n a R i n H ' ( R ) ) +a. dR Then for any k 2 M , max(u - k, 0) E H;(R) and max(u - k, 0) 2 0 in R in H ' ( 0 ) . Proof: In order to prove that max(u - k, 0) E HA(S1) it suffices to prove the existence of a sequence u, E HA*"(R) such that u, + max(u - k, 0) weakly in H'(Q).
Sobolev Spaces and Boundary Value Problems The use of Sobolev spaces is essential to our method. Here we do not intend to develop in detail the properties of these spaces but only to recall their definitions. Some aspects of them relevant for our study are described in Appendix A of this chapter. More information is available in many books and papers. In particular, we refer frequently to the book of Morrey [l], where such spaces are investigated very deeply. 1 may be invoked to obtain their solutions.