Mathematische Annalen

Math Ann 263, 303-312 (1983) ДШИУрП

( g ) Springer-Verlag 1983

A Sufficient Condition for a Critical Point

of a Functional to be a Minimum

and Its Application to Plateau's Problem

A . J. Tromba

Department of Mathematics, University of California, CA 95064, USA

0 . Introduction

It is a classical consequence of Taylor's theorem that if [/CiR" is an open subset, qeU Si critical point for a C^ mapping f:U->R and the second derivative of / at ^ or "Hessian" of/, D^f{q) : IR" x IR"-^i? is a positive quadratic form, then ^ is a strict minimum for /

If и С H is open in a Hilbert space Я, ^ a critical point for f:U-^R, and ^Y(^)(^5^)^<^ll^p5 c>0, then q is also a strict minimum for /

If we merely have DY{q){v,v)>0 for all v, the implication that ^ is a strict mmimum is, m general, false. To illustrate an example of such an / let Я = /2 the Hilbert space of square summable sequences. For x = (aCpX2,...) define f{x) =

" ^ 1 . 1

— Y, ^cosiXj. 0el2 is clearly a critical point and moreover DY(0)(i^, t^)= X^^f

1=1 i I i

which is obviously positive. /(0)= —^l/i^ However if >^„ = (0, ...,27г/п,0,...),

where 2n/n occurs in the n^^ place then /(y„) = /(0). Thus we have a sequence y„-^0 with f{y^) = /(0) and 0 is not a strict minimum for /.

It is also possible to find an / with D^/(0)(d, î;)>0 for 1;фО with 0 not even a

minimum consider for example f{x)= X ( "" ^ + ^ cosix^

Despite these examples there are many many problems in non-linear al analysis where all we know is that the Hessian is positive but not positive definite. In these cases it would be useful to know when the critical point is a minimum. The situation which we have in mind is Plateau's problem where this occurs. In the section below we shall describe a theory which shows how to deal with such situations and which also lays the foundation for an infinite dimensional catastrophe theory for a class of problems heretofore intractible.

An approach to such questions was given by the author in [8] and by Magnus in [4]. However the results in [8] were phrased in much too abstract a setting for them to be reachly applicable and in [4] applications seem not to be the immediate objective.