Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators

Abstract

For singular elliptic fully-nonlinear operators we prove that the eigenfunctions corresponding to the principal eigenvalues in bounded domains are simple. The proof uses in particular a regularity result obtained in the first part of the paper which is of independent interest.

Global $$W^{2,\delta }$$ W 2 , δ estimates for a type of singular fully nonlinear elliptic equations

“…The investigation of equations of this type has made much progress in recent years. I. Birindelli and F. Demengel proved comparison principle [BD04] and C 1,α estimate [BD10]. G. Dávila, P. Felmer and A. Quaas proved Alexandroff-Bakelman-Pucci (ABP for short) estimate [DFQ09] and Harnack inequality [DFQ10].…”

“…Following the same idea, J.-P. Daniel [Dan15] proved an estimate equivalent to local W 2,δ estimate for uniformly parabolic equation. For singular fully nonlinear elliptic equations, intuitively, once we have a universal control of Du L ∞ , for instance C 1,α estimate (see [BD10]), the W 2,δ estimate will then be a natural corollary of the traditional results of [Caf89], [CC95] and [Win09]. But our method does not depend on any a priori estimate of Du and it does not use maximum principles, so we can deal with a large class of equations as illustrated above.…”

Regularity for radial solutions of degenerate fully nonlinear equations

“…Finally let us recall that in a previous paper [8] we proved that for F such that F (∇u, D 2 u) := |∇u| αF (D 2 u) and for α ∈ (−1, 0], all solutions of…”

Eigenvalues for radially symmetric fully nonlinear singular or degenerate operators

“…See the work by Quaas and Sirakov [3], Armstrong [4], Ishii and Yoshimura [5] and Patrizi [6]. In the case of (1 + α)-homogeneous operators, Birindelli and Demengel have developed a theory in a series of papers [7][8][9][10][11]. Of particular interest to our work are the recent regularity results for the degenerate case by Imbert and Silvestre [12] and, for radial solutions, by Birindelli and Demengel [13] (see Lemma 5.2).…”