The Spectrum of Second Order Elliptic Operator and Useful Integral Inequality<数学・自然科学>

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We consider the second order elliptic equation A(x, ∂)u(x)+q(x)u(x)=f(x), and also consider the integral inequality that provides us with a lower bound estimate of the associated quadratic form. In so doing, we need to introduce newly a positive continuous function λ(x) on the region in question. This type of estimate is quite very useful if we apply the result to the elliptic equation to obtain estimates of eigenfunctions for the equation. The principal purpose of this article is to settle down the positive continuous function λ(x) in a concrete manner. It suffices to involve the spectra of operator and the compactness argument in functional spaces, in order to realize the above-mentioned program.

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