By Jean-Pierre Aubin

In order to stay at an introductory point, this quantity refrains from delving into technical problems and complex effects no longer in present use. functions are defined once attainable, and theoretical points are aimed toward useful use. themes diversity from very tender features to nonsmooth ones, from convex variational difficulties to nonconvex ones, and from economics to mechanics. historical past notes, reviews, bibliography, and indexes complement the text.

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**Example text**

By compactness, there will be some eo>0 and finite subcoverings , ^„) and { 'f i , . . , iT„) with the property that for all |f|<6o, there is a map cuf: such that ij/, — Since { 'f i , . . , 'f„) is a covering, it follows that is surjective for |f|<6o. We claim there is an Sj > 0 such that ij/, is one to one for |i| < «i. If it were not so, there would be a sequence f„->0 and points y„^y'„ such that fl/t„(yn)=^t„{yn)- By compactness, these sequences would have cluster points j and / , with tl/o{y)= >l/oiy'), so that y=y'.

Its inverse must be CO? on hence the result. ■ We now extend (f>, to the unit ball (1) to := ||x ||< ^,(x ||x ||-‘) by setting fo rllx N l The map defined in this way is clearly a diffeomorphism for |i| < s. We use it to compute the volume of + ^ Namely, we use the well-known formula for changing variables in a multiple integral. Vol (Vl + i^B"+‘)= [ |Det ^',{x)\dx jBm+1 There is no need to compute precisely the right-hand side. Note only that Det 0i(x) does not vanish, so that it has the sign of Det (l>oix\ which is positive.

The condition U{x)=0 breaks down into two pointwise conditions x^^(co)=0 for almost every co dx — (co)= +1 for almost every co d(o These conditions are clearly incompatible: There is no x minimizing {/. ■ Note that the functional U itself is lower semicontinuous on //¿, by example 1. Changing ( l —x^)^ to (1 — makes it continuous, and even but the corresponding minimization problem will still have no solution (argu ment unchanged). In other words, there is nothing pathological about the func tion U that causes this situation to occur.