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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:
[1] V. Jarník: Differential Calculus. Publishing House of the Czech. Acad. Sci., Prague, 1953.
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