Download PDF by Yuan Y.: A Bernstein problem for special Lagrangian equations

By Yuan Y.

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Extra resources for A Bernstein problem for special Lagrangian equations

Example text

O°). Lis called the minimal unclosed differential operator, defined by the operation (1). Its closure L will be called the minimal differential operator defined by the operation (1). As is known (see /7/), the defect number of this operator satisfies the inequality a < Def L < 2n. Any self -adjoint extension L of the operator L will be called a self -adjoint operator produced by the operation 1. By virtue of Theorem 4, all the self-adjoint operators, gen- erated by a given operation (1) have the same continuous spectrum C(Z)=C(L).

G), which together with (32) leads to the relation (Ag g)2 < M (Ktf. f) (Ag. g) or (hg. g) < M (Ktf f). (33) On the other hand, as in the proof of the previous theorem, we obtain relations (28), (29) and (30). Thus, from (28) we obtain the following equality for any f E HA Tf-A 'Kf, since the sequence of the vectors A-'tpR converges in HA, provided the sequence T. E HA is A-convergent. Let Fbe a bounded set of elements hEH. Then according to the conditions of the theorem the set f=B-'h (hEF) is K,-compact, therefore, from (33), it follows that the set of elements g=TB-'h (hEF) is compact in HA and a fortiori, also in H.

In combination with (31), this inequality leads, for any f, inequality g E ZA, to the (Kf. g)2

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A Bernstein problem for special Lagrangian equations by Yuan Y.


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