The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold 𝑀

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain Ω ⊂ Rn, n ≥ 2, and a, b ∈ L∞(Ω). If the equation Lu = au+ − bu− + λu (where λ ∈ R and u±(x) = max{±u(x), 0}) has a non-trivial solution u, then λ is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are ‘simple’.We also consider the semilinear problem where f : Ω × R → R is a Carathéodory function such that, for a.e. x ∈ Ω, and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).

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The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

Applications of Mathematics ◽

10.1007/s10492-005-0020-4 ◽

2005 ◽

Vol 50 (3) ◽

pp. 323-329 ◽

Cited By ~ 7

Author(s):

Ivana Pultarova

Keyword(s):

Finite Element ◽

Differential Operator ◽

Partial Differential Operator ◽

Second Order ◽

Partial Differential

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Surjectivity of linear operators from a Banach space into itself

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700039538 ◽

2007 ◽

Vol 76 (1) ◽

pp. 143-154 ◽

Cited By ~ 3

Author(s):

Dimosthenis Drivaliaris ◽

Nikos Yannakakis

Keyword(s):

Banach Space ◽

Hilbert Space ◽

Differential Operator ◽

Partial Differential Operator ◽

Second Order ◽

Linear Operators ◽

Partial Differential ◽

Space Case

We show that linear operators from a Banach space into itself which satisfy some relaxed strong accretivity conditions are invertible. Moreover, we characterise a particular class of such operators in the Hilbert space case. By doing so we manage to answer a problem posed by B. Ricceri, concerning a linear second order partial differential operator.

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Partial Differential Operator on Vector Space of Polynomials

d'CARTESIAN ◽

10.35799/dc.4.2.2015.10537 ◽

2015 ◽

Vol 4 (2) ◽

pp. 218

Author(s):

Chriestie Montolalu

Keyword(s):

Differential Equation ◽

Differential Operator ◽

Vector Space ◽

Matrix Representation ◽

Partial Differential Operator ◽

Second Order ◽

Partial Differential ◽

Partial Differentiation

A differential operator which acts on partial differentiation is defined as Partial Differential Operator (PDO). PDO works based on the order of the differential equation which then can solve the eigenvalues of the operator. On vector space of polynomials, PDO can be written in matrix representation. This can be helpful in finding the general form of eigenvalues of vector space polynomials. On this paper, a second order PDO: will be operated on two and three variable vector space polynomials.

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The growth of the positive solutions of Lu = 0 near the boundary of an inner NTA domain

Nagoya Mathematical Journal ◽

10.1017/s0027763000002907 ◽

1988 ◽

Vol 110 ◽

pp. 129-135

Author(s):

Katsunori Shimomura

Keyword(s):

Differential Operator ◽

Euclidean Space ◽

Bounded Domain ◽

Positive Solutions ◽

Partial Differential Operator ◽

Second Order ◽

Partial Differential ◽

Hölder Continuous

Let D be a bounded domain in the Euclidean space Rn (n ≧ 2) and L a uniformly elliptic partial differential operator of second order with α-Hölder continuous coefficients (0 < α ≦ 1) on D.

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