By Gillespie L. J.
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The dual problem means evaluating (/») suP{-$'(0,P')\P' e X'}. 43. (i) - o o < sup P' < inf P < +oo. (ii) If the problem P is nontrivial, then supP* < i n f P < + o o . (iii) If the problem P* is nontrivial, then - o o < s u p P * < inf P . (iv) If the problems P and P* are nontrivial, then - o o < sup P' < inf P < +oo . □ Recalling that sup{-$*(0,p*)|p* € X'} = - inf{*'(0,p*)|p" € X'} , we can formulate the dual problem P" of P': (P**) inf{4>**(w,0)|we V} . , $** e T(V x X). By r ( V x X) we denote the space of all convex lower semicontinuous functions 32 Mathematical preliminaries defined on V x X.
The properties of the space Hj^ are studied by Lions and Magenes (1968). The regularity of the boundary, including weaker assumptions and singular cases is studied by Adams (1975), cf. also NeCas (1976), Nazarov and Plamenevskii (1991) and Movchan and Movchan (1995). 10 of the Hausdorff measure follows Giusti 14 Mathematical preliminaries (1984), cf. also Part II of the book by Morel and Solomini (1995). 12 is proved in standard textbooks on functional analysis, already mentioned. For more details on Poincare' and Poincar6-Wirtinger inequalities the reader is referred to Brezis (1983) and Morrey (1966).
For all v £ D(A) and all x" G D(A'). similarly D(A') is the domain of A*. 47) Here D(A) denotes the domain of the operator A; J(u) = G{Au) + F{u). 48) The perturbed functional (u, p) may be assumed in the form *(u,p) = G(Au + p) + P ( u ) . 49) It can easily be shown that the dual problem P* means evaluating sup{-G*(p*) - P*(-A*p*)|p* e X'} . 50) We observe that: (i) if F and G are. convex then $ is also convex; (ii) if F € r 0 (V) and G G r0(A") then $ £ r 0 (V x X). xX , . 53) These relations are obviously equivalent to -A"p* £dF{u), p* G dG{Au) .
An Equation for the Haber Equilibrium by Gillespie L. J.
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