Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators

Birindelli, Isabeau; Demengel, Françoise

doi:10.3934/cpaa.2007.6.335

Cited by 117 publications

(181 citation statements)

References 17 publications

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“…The eigenvalue problem for fully nonlinear PDE (1) has been paid much attention since the work of P.-L. Lions [28]. For the recent developments, we refer to [2,3,[6][7][8][9][10][11][12][13][14]18,19,21,22,26,[30][31][32]. See also [1,4,5,17,24] for the recent contributions and overviews on the eigenvalue problem for linear elliptic operators.…”

Section: Introductionmentioning

confidence: 99%

“…an eigenvalue μ and an eigenfunction u] of (7) and (8) of nth order if u has exactly n zeroes in [0, R) as a function on [0, R]. We note that a radial eigenpair (μ, u) is a principal eigenpair if and only if it is of zeroth order as an eigenpair of (7)- (8). The main contributions in this paper are described briefly as follows.…”

Section: Introductionmentioning

confidence: 99%

“…where f ∈ L q (Ω) is a given nonnegative function, with the boundary condition like (5) or (8) is closely related to the principal eigenvalues. We establish general theorems in this direction that relate the solvability of boundary value problems for the inhomogeneous PDE…”

Section: Introductionmentioning

confidence: 99%

“…[5,28]) that the principal eigenvalues are thresholds to the validity of the maximum principle for PDE (1). See also [1,3,[7][8][9][10]12,13,24,26,31,32]. Theorems 16 and 34 below state roughly that the principal eigenvalues have this role of threshold for our general boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

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Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

Ikoma

Ishii

2015

Bull. Math. Sci.

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This is a continuation of Ikoma and Ishii (Ann Inst H Poincaré Anal Non Linéaire 29:783-812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous of degree one, on finite intervals or balls. In the multi-dimensional case, we consider only radial eigenpairs. Our eigenvalue problem has a general first-order boundary condition which includes, as a special case, the Dirichlet, Neumann and Robin boundary conditions. Given a nonnegative integer n, we prove the existence and uniqueness, modulo multiplication of the eigenfunction by a positive constant, of an eigenpair whose eigenfunction, as a radial function in the multi-dimensional case, has exactly n zeroes. When an eigenfunction has n zeroes, we call the corresponding eigenvalue of nth order. Furthermore, we establish results concerning comparison of two eigenvalues, characterizations of nth order eigenvalues via differential inequalities, the maximum principle for the boundary value problem in connection with the principal eigenvalue, and existence of a solution having n zeroes, Communicated by Neil Trudinger.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

Ikoma

Ishii

2015

Bull. Math. Sci.

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“…Significant contributions on eigenvalues of continuous operators in nondivergence form in bounded domains include the fundamental work [70] for linear operators; [28] for convex fully nonlinear operators; [71] for nonlocal operators; [72], [73], [74] and the recent [75] for degenerate elliptic operators. Theorem 4.2 is a slight improvement to the general existence theory about nonconvex operators possessing first eigenvalues in [64] (see also [69]), since we are not supposing that our nonlinearity is uniformly continuous in x.…”

Section: C-viscosity Notions Of Solutionsmentioning

confidence: 99%

Methods of the Regularity Theory in the Study of Partial Differential Equations With Natural Growth in the Gradient

NORNBERG¹

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Fundamental solutions of homogeneous fully nonlinear elliptic equations

Armstrong

Smart

Sirakov

2011

Comm Pure Appl Math

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We prove the existence of two fundamental solutionsˆand ẑ of the PDEfor any positively homogeneous, uniformly elliptic operator F . Corresponding to F are two unique scaling exponents˛ ; z > 1 that describe the homogeneity ofˆand ẑ . We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation F .D 2 u/ D 0, which is bounded on one side. A Liouville-type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of F .D 2 u/ D 0 in R n n f0g that are bounded on one side in both a neighborhood of the origin as well as at infinity. Finally, we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two-player differential game.

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Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators

Cited by 117 publications

References 17 publications

Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

Methods of the Regularity Theory in the Study of Partial Differential Equations With Natural Growth in the Gradient

Fundamental solutions of homogeneous fully nonlinear elliptic equations

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