Download PDF by Harold R. Parks: Explicit Determination of Area Minimizing Hypersurfaces, II

By Harold R. Parks

ISBN-10: 0821823396

ISBN-13: 9780821823392

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Example text

5. For j = 1,2,... set H. equal to the union of the connected components of « n {x : a(j) + 66 (j) <_ w. (x) £ &(j) - 6 6 (j) } which intersect r, and set G. = F . n H. 3 3 3 Because the functions w. are piecewise linear, the sets explicitly computable and, thus, the sets (1) G. H. are are explicitly computable. 5(1). (2i,ii) We claim that, for each a(j+l) £ s <_ 3(j + D j = 1,2, .. , :Lf_ s satisfies r then (spt T g u spt S ) C H. holds. To see this, we fix a, 6, 6, e, w replaced by respectively, to obtain j and s as above.

Thus we can find x' e ft, q' e A* with a(j) < q* < r, |x-x'| x' 6 Q n u^Cq") , 2"1T < dist(x < p, f / D. Since £ hold, n [E(q',r)] we c o n c l u d e , dist(x,ft 13 Z2 " j/n < a, a < Y10(2""1T)n by 4 . 3 ( 2 ) , n UQ1(r)) Finally, we have Y Y that <_ 7 L 1 c r I / n + p = Y-i ? 3(1), that there exists a unique y e ft n u" 1 ^) 31 with |y-x| = dist(x,ft n u" (r)). 5. 0 REMARKS R. Schoen and L. Simon have given a proof of the regularity theorem for rectifiable currents minimizing parametric elliptic functions in which the constants involved in the a priori estimates can be explicitly computed (see [SS]).

A - Let us assume for definiteness that q < r holds. _L] < a. It follows that Un(x,p) ~ (fi n u"1(q)) ^ 0, where p = (2a(n) 1 a ) 1 / n < T / 2 . Thus we can find x' e ft, q' e A* with a(j) < q* < r, |x-x'| x' 6 Q n u^Cq") , 2"1T < dist(x < p, f / D. Since £ hold, n [E(q',r)] we c o n c l u d e , dist(x,ft 13 Z2 " j/n < a, a < Y10(2""1T)n by 4 . 3 ( 2 ) , n UQ1(r)) Finally, we have Y Y that <_ 7 L 1 c r I / n + p = Y-i ? 3(1), that there exists a unique y e ft n u" 1 ^) 31 with |y-x| = dist(x,ft n u" (r)).

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Explicit Determination of Area Minimizing Hypersurfaces, II by Harold R. Parks


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