Oscillation criteria for selfadjoint elliptic equations

Kreith, Kurt; Travis, Curtis C.

doi:10.2140/pjm.1972.41.743

Cited by 24 publications

(19 citation statements)

References 9 publications

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“…The conclusion follows by combining the oscillation results due to Kreith and Travis [9] with Corollary 5.3. From Corollary 5.5 it follows that every solution v ∈ C 2 (R 2 ;R) of (5.12) is oscillatory in R 2 .…”

Section: Oscillation Theorems For (16)supporting

confidence: 52%

Picone-type inequalities for nonlinear elliptic equations with first-order terms and their applications

Jaroš

Kusano

Yoshida

2006

Journal of Inequalities and Applications

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Section: Oscillation Theorems For (16)supporting

confidence: 52%

Picone-type inequalities for nonlinear elliptic equations with first-order terms and their applications

Jaroš

Kusano

Yoshida

2006

Journal of Inequalities and Applications

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“…and dS denotes the surface element on p(x) = t. Then L -k oscillates at 3G. Theorem 1 can easily be established by adapting the averaging procedures introduced in [7], [8], once we observe that near 3G we have by construction: dx = (|grad p\)~x dSdp.…”

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confidence: 87%

Spectral estimates and oscillations of singular differential operators

Allegretto¹

1979

Proc. Amer. Math. Soc.

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Abstract. Several estimates are established for the lower spectrum of singular second order elliptic operators. These estimates are obtained by considering the oscillation properties of the operators involved. This paper establishes estimates on the lower spectrum of operators generated by singular elliptic equations. These estimates are obtained by considering the oscillation properties of the operators involved. There is a considerable amount of literature on this subject in the special case where the only singularity comes from the unboundedness of the domain or, equivalently, from the behaviour of the coefficients at a single point. We refer the reader to the books by Glazman , where further references may be found. If the singularity comes from the behaviour of the coefficients on components of the boundary of the domain, the case of interest to us, then there are fewer results. We recall the criteria established some time ago by Kreith [6] for bounded domains and which we shall complement. The spectral properties with which we are concerned are completely determined by the behaviour of the operator near the boundary. An example below will show that due to the error introduced it is in general undesirable to proceed by further localizing our considerations to the behaviour near individual points of the boundary. This fact shows the need to modify the "point" procedures previously employed, and is one of the motivations for our considerations.Let Em denote Euclidean /«-space and let G denote a domain in Em with boundary 9G. When G is unbounded we shall always consider the topology of the one point compactification of G in Em, so that oo E 3G. We let D¡ denote differentiation with respect to x, for i = I, . . . , m and introduce in C™(G) the locally uniformly elliptic operator: m lu= -S DfajDju) + cu.(1) ',7=1 We shall always assume that the coefficients are real and that: a¡¡ = aß, atJ E CX(G), c E C(G). If / is not bounded below then neither will be its selfadjoint extensions and, for the problems we consider, the situation will beReceived by the editors February 13, 1978.A MS (MOS) subject classifications (1970). Primary 35B05.

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“…Recently, Noussair and Swanson [6,7,8] have developed some oscillation criteria for the Schrödinger equation. We also refer to Kreith and Travis [4] and others [1,3,5] for related results. Especially, Noussair and Swanson [7] get a necessary and sufficient condition for nonoscillation of a sublinear Emden-Fowler equation in the case of n > 3.…”

mentioning

confidence: 94%

“…We write the y, in (2) simply by y. We may take a constant k such that (4) 0 < y < k < 1 and 1 -k < y.…”

mentioning

confidence: 99%