Furushima M.'s A Birational Construction of Projective Compactifications of PDF

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By Furushima M.

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Plane-like minimizers in periodic media, preprint, mp_arc 00-368. , C 0; a regularity of o-minima, Boll. Un. Mat. Ital. A (7) 10 (1996), no. 1, 113–125. , Introduction to regularity theory for nonlinear elliptic systems, Birkha¨user-Verlag, Basel 1993. , On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46. , Trudinger, N. , Elliptic partial di¤erential equations of second order, Second edition, Springer-Verlag, Berlin-New York 1983. , On the theory of superconductivity, Zh.

Math. 48 (1995), no. 1, 1–12. Ca¤arelli, L. , Plane-like minimizers in periodic media, preprint, mp_arc 00-368. , C 0; a regularity of o-minima, Boll. Un. Mat. Ital. A (7) 10 (1996), no. 1, 113–125. , Introduction to regularity theory for nonlinear elliptic systems, Birkha¨user-Verlag, Basel 1993. , On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46. , Trudinger, N. , Elliptic partial di¤erential equations of second order, Second edition, Springer-Verlag, Berlin-New York 1983.

1. This proof is a variation of the one in [CC], where the result was proved in the case ai; j ¼ di; j . Here, we simplify a bit the proof and correct some minor flawns of [CC]. Let g be a radial smooth function that is identically equal to À1 in BrÀ1 and takes value 1 on qBr . Define u ? ¼ minfu; gg. Since u is a minimizer,   Ð j‘u ? j 2 þ r nÀ1 JBr ðuÞ e JBr ðu ? Þ e const Br ÀBrÀ1 e const  Ð ðj‘uj 2 þ j‘gj 2 Þ þ r nÀ1  Br ÀBrÀ1 e const Ð j‘uj 2 þ r nÀ1 : Br ÀBrÀ1 Let us now cover Br À BrÀ1 with balls B1 ; .

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A Birational Construction of Projective Compactifications of C 3 with Second Betti Number Equal to One by Furushima M.


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