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The general form of local bilinear functions. (English) Zbl 0785.46034
A bilinear functional $$B(u,v)$$ in $$L_ 2(0,1)$$ or $$H^ 1(0,1)$$ is called a local bilinear functional (l.b.f.) if $$B(u,v)=0$$ for every pair $$u,v\in L_ 2(0,1)$$ or $$H^ 1(0,1)$$ such that $$\text{mess}(\text{supp }u \cap\text{supp }v)=0$$. Using some technical lemmas the author gives the general form of l.b.f. in the space $$L_ 2(0,1)$$, respectively $$H^ 1(0,1)$$. Thus, $$B(u,v)$$ is a continuous l.b.f. in $$L_ 2(0,1)$$ (in $$H^ 1(0,1)$$) if and only if there exists a bounded function $$\eta$$ (respectively, $$\eta_ 1\in L_ \infty(0,1)$$, $$\eta_ 0\in L_ 2(0,1)$$ and a number $$\alpha$$) such that $B(u,v)=\int_ 0^ 1 \eta(t)u(t)v(t)dt \quad \left(= \int_ 0^ 1 [\eta_ 1(t)u'(t) v'(t)+ \eta_ 0(t)(u(t)v(t))']dt+ \alpha u(1) \right)$ for all $$u,v\in L_ 2(0,1)$$ ($$u,v\in H^ 1(0,1)$$).
If the continuous l.b.f. in $$H^ 1(0,1)$$ is antisymmetric it has the form $B(u,v)= \int_ 0^ 1 \eta_ 2(t) (u(t)v'(t)- u'(t)v(t))dt$ where $$\eta_ 2\in L_ 2(0,1)$$.
In this way, the author obtains exactly those bilinear functions that correspond to standard boundary-value problems for elliptic differential operators.
##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 35J40 Boundary value problems for higher-order elliptic equations 46C99 Inner product spaces and their generalizations, Hilbert spaces
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##### References:
  V. Jarník: Differential Calculus. Publishing House of the Czech. Acad. Sci., Prague, 1953.
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