By Della Dora J.

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**Example text**

73), we have dz + C~h -2 _ ~2+v~2 dx < C,yh -6 f a 2+'Y dx ,~ C~h ~-2. 15. 70). 81) that L V (Rh -- I) g- V (~h - I)'~ dx = - L V (Rh -- I) g. V r dx L = - 1L ~ + ~ Iv (R~ - I)gl ~ d~ + ~ (Zxo ~+~) [(Rh - I) 9] ~ d~. 72) we get L a2+~ IV (Rh - I) g 2 dx <_ - L v ( R h - I ) g . V ( r r h - I ) t b d x + C ~ ~ a ~ [ ( R h - Z ) g ] 12 -< -2 o~+~ Iv (R. - I) gl ~ d~ + ] 2 dx ~ - ~ - ~ IV (~h - I)r ~ d~ +C~ f ~ [(~, - Z) g]~ d~. Ja This implies L o-~+'Iv (R~ - i) 91~ d~ - L ~-~-~ Iv ( ~ - ,)~1~ d~+ C~L ~ [(R~- ~)g]~ dx.

E [194]. For the parametric case, Tsuchiya [389], [390], [391] showed the convergence of the discrete minimal surfaces constructed by the finite element method. 4. Descloux's iemma was shown in [111]. 7 were given by Crouzeix and Thom~e [102]. Related works are also referred to there. For Riesz-Thorin's interpolation theorem, see Adams [1], Folland [127], and so forth. Other inequalities driven from the inverse assumption are given in Ciarlet [83]. 5. The L ~ stability was proven by Nitsche [297] and Scott [349].

V ( r r h - I ) t b d x + C ~ ~ a ~ [ ( R h - Z ) g ] 12 -< -2 o~+~ Iv (R. - I) gl ~ d~ + ] 2 dx ~ - ~ - ~ IV (~h - I)r ~ d~ +C~ f ~ [(~, - Z) g]~ d~. Ja This implies L o-~+'Iv (R~ - i) 91~ d~ - L ~-~-~ Iv ( ~ - ,)~1~ d~+ C~L ~ [(R~- ~)g]~ dx. 75). 88) 34 1. 88). 89) is absorbed into the left-hand side by the duality argument. 13. This gives f ~-~-, ID~l~ d. ~]2) d,. 90)imply / = <_ m - L v ( ~ - i ) v . v ( ~ - i) 0 d, (~ ,~+" IV (R,,. fo,~-~-, ID~vl ~ d. _< c-,,~-= f (o-~+" IV (R,,- I):jl ~ + ,-,-~[(/~,,- I)~]~) d..