By Umberto Massari

This ebook provides a unified presentation of other mathematical instruments used to resolve classical difficulties like Plateau's challenge, Bernstein's challenge, Dirichlet's challenge for the minimum floor Equation and the Capillary challenge. the elemental thought is a fairly effortless geometrical definition of codimension one surfaces. The isoperimetric estate of the Euclidean balls, including the fashionable concept of partial differential equations are used to resolve the nineteenth Hilbert challenge. additionally integrated is a latest mathematical therapy of capillary difficulties.

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W e i Lac2 = 1. r 2 1 1 1 6 . 6 , =~ c~L 6 . 6 . v. i , . + h,i,j ~ Thus, w e g e t I n t h i s i d e n t i t y we s u b s t i t u t e 6iA with and what e l s e w i l l be necessary, r e c a l l i n g t h a t 6 . 6 . = 6 . 6 . + X ( v . 6 . v -vi6jvh)6h 1 1 3 1 x. fiivj A 6 . v - = j i ,J E l l h 6ivj6h6h6ivj 6ivj6hvi6hvk6kvj h , k , i tj = h,i,j n,i,j , i so 6iv j. 6 i hv j + DIFFERENTIAL PROPERTIES OF SURFACES 38 a t t h e f i x e d p o i n t . From Schwarz i n e q u a l i t y , we have 2 2 " i=l Thus, where e x i s t s , which i s t h e c a s e almost everywhere, we g e t 6c n i ,h=l L e t us observe now t h a t n 1 2 (6,6,Vi) " z = i,h=l 2 (ai"Vi' " + II " 1 (GhSiVi) 2 , i = l h=l i=l hCi moreover, a t t h e f i x e d p o i n t , f o r E6 2 o=-c v .

Nitsche's Review Article on Minimal Surfaces, E 7 4 1 ) . SLOPE OF MINIMAL GRAPHS 27 What w e f o l l o w h e r e i s e s s e n t i a l l y T r u d i n g e r ' s argument. VdH n ' ' T h i s i n e q u a l i t y , which h o l d s f o r a l l compact s u b s e t s KCAX R for which t h e d i v e r g e n c e theorem makes s e n s e , i m p l i e s t h a t g r a p h s w i t h bounded mean c u r v a t u r e h a v e l o c a l l y bounded area. i n our further considerations. 2 SLOPE ESTIMATE FOR GRAPHS.

I n f a c t , also h e r e w e have, i f relations a(X) <+m, t h e f o l l o w i n g sequence of 47 SETS O F FINITE PERIMETER t j > h a(XnM. d. m We have so proved t h a t m sets, then smallest aIB must c o n t a i n a l l B o r e l s e t s , t h a t i s t h e e l e m e n t s o f t h e f u n c t i o n with val u es i n + ") The r e s t r i c t i o n u - a l g e b r a o f s e t s , t o which open s e t s b e l o n g . a t o the family of 0 - a l g e b r a of s e t s c o n t a i n i n g a l l open is a % KO, o f Borel s e t s i s a c o m p l e t e l y a d d i t i v e a(%,) C E O , + + m 1 and l o c a l l y f i n i t e , t h a t i s * % A function l i k e t h a t , defined over f i n i t e over %o , LO, with values i n +a] and c o m p l e t e l y a d d i t i v e i s what one u s u a l l y means f o r a non n e g a t i v e Radon measure.