By Love A. E. H.

Such a lot entire remedy of classical elasticity in one quantity. assurance of pressure, pressure, bending, torsion, gravitational results

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**Sample text**

The reader, however, should not be misled into thinking that our new definition of inequality reduces to inequality in the almost everywhere sense. In fact, as we shall see in Section 6, inequality in the sense of Hl(Q) serves to determine the capacity of a set. A set of measure zero, for example a closed interval in 1R2, may have positive capacity. The next assertion illustrates the role of inequality in H J (Q) in the weak maximum principle. 3. Let u e Hl(Q) and suppose that Then for any k > M, max(u -k,0) € H^(Q)andmax(« -k,0) > 0in Q inHl(&).

A (RN be bounded and connected with smooth boundary <3Q and let atj e L°°(Q) satisfy Set and define the mapping by Recall that if a 0 (x) e C^H), then is an elliptic equation in the classical sense. e. in Q. 1. 2. 1. 1) implies that is, a(u, v) is coercive. 1 is the solution to the minimization problem 6 THE OBSTACLE PROBLEM I FIRST PROPERTIES 41 Now we give a useful characterization of our solution. 3. 1 is an L —/supersolution since u + £ e IK whenever £ > 0, £ e Ho(fi). e. 2). 4. 1 and suppose that g is a supersolution of L — f satisfying g > \l/ in Q and g > 0 on 5Q (in /^(Q)).

Observe (cf. e. in Q. 3) we find therefore £ = 0 in H£(Q), so u - M < 0 in Hl(ty. D. One extension of this maximum principle, which we prove in Appendix B, is that if with/ = /0 + £7 (/,)„,/, 6 LS(Q) for 0 < i < N, then provided s > N. 6. A function ueHl(£l) is a supersolution to L, or an L super solution, if Analogously, u is an L subsolution provided that For supersolutions, there is a minimum principle. 5, is omitted. 7. Let u e J/^Q) be an L supersolution. Then An interesting property of supersolutions is that if u, v are supersolutions to L, then min(w, i;) is also a supersolution.