On the Elliptic Sturmian Theory for General Domains

Erich Müller-Pfeiffer
1988 Zeitschrift für Analysis und ihre Anwendungen  
4 Per bekannte Vergleichssatz von Sturm und Picone fur gewohnliche, seibstadjungierte Differentialgleichungen zweiter Ordnung wird auf sclbstadjungierte elliptische Differentialglcichungen verailgemeinert. Dabei sind dàs Grundgebiet G und die Koeffizienten der Differential • gleichung nicht notwendig-beschränkt, und es werden keine Regularithtsforderungen an den Rand c9G gesteilt. l43BecTIiaH TeopeMa cpaueun LHTypMa 11 fluione Ann o6bnul0Be6Hb1x caMoconpneHIlhLx Jui44epeHuItaJihHax ypaBHeHlin
more » ... JihHax ypaBHeHlin noporo nopnja o6o6I1aeTca iia caMoconpmIe1iHbIe anj11enTH qecII1e )e144epeuILHaJ1bHhIe ypaBMeHuM. ilpu aroM ocHoBHan o6JlacTb G It 1I1eHTb1 Ai444epeHuHaJIbHoro ypanieunn He 06n3aTeibII0 orpaHu4eHal, u 'ycioniin peryJ!np-HOCTH AJIFI rp4HI1Lai bU HO Tpe6y}0TcH. The well-known comparison theorem by Sturm and Picone for ordinary, self-adjoint, second order 'differential equations is extended to self-adjoint elliptic differential equations. The basic domain 0 and the coefficients of the equation are not necessarily bounded, and no regularity hypotheses on the boundary eo are required. / S • Consider the differential equations 4QU = ---( P(x) u')'+ Q(c) u = 0 (x b)), 4q = -(p(x) u')' q(x) u = 0 • where P, p E C'[a, b] and Q, q E C[a, b] are real-valued and P(x), p(x) > 0, x E [a, b]. A wellknown version of the Sturm-Picone theorem is the following one (compare [8: Cor. 1], [13: Theorem 1.5]). -• Theorem 1: If there exists a real solution u 0 of 4QU = 0 such that u(a) = 0 = u(b) and •f[p(u)2 +,qu2]d 0, ---Sthen every real solution v of 4qV = 0 is a constant multiple of u or has at least one zero in (a, b). --In the following this theorem will be extended to self-adjoint, second order, elliptic differential equations. The present investigation complements the paper [11] , where the extension of the following version of the Sturm-Picone theorem is handled. Theorem 1': Suppose p(x) P(x) and q(x) Q(x), z E [a,b]. I/ there exists a real solution U = 0 of s4QU = 0 with u(a) = 0 = u(b), then every real solution V of çv = 0 has at least one zero in (a, b) if -S -• -
doi:10.4171/zaa/284 fatcat:delwd663dbasxhr4kelqn4toge