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Papa Badiane

Publications and source records attributed to Papa Badiane.

4 recordsLinked to original sources
arXiv2026

The Eigenvalue Problem for the complex Monge-Ampère operator

We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Ampère operator on a bounded strongly pseudoconvex domain in $\C^n$. We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in $Ω$ and boundary values $0$. Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P.L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge-Ampère equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck \cite{CKNS85} and B. Guan \cite{GuanB98}. Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge-Ampère operator.

math.CV↗
arXiv2025

Generalized study of the operator $α\partial^k \bar{\partial}^{k} + β\bar{\partial}^k +γ\partial^k + c$ in weighted Hilbert space $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$

By Hörmander's $L^2$-method, we study the operator $α\partial^k \bar{\partial}^{k} + β\bar{\partial}^k +γ\partial^k + c$ for any order $k$ with $α, β, γ\in \mathbb{R}$ such that $(α, β, γ) \neq(0,0,0)$ in the weighted Hilbert space $L^2(\mathbb{C}, \mathrm{e}^{-|z|^2})$. We prove the existence of its right inverse which is also a bounded operator. Subsequently we will study two cases that arise from this operator, namely: (1) Case where $α= γ=0$: The operator $β\bar{\partial}^{k} + c$ with $\vert β\vert \geq 1$. (2) Case where $β= γ=0$: The operator $α\partial^{k} \bar{\partial}^{k} + c$ with $\vert α\vert \geq 1$.

math.CV↗
arXiv2023

A variational approach to the eigenvalue problem for complex Hessian operators

Let $1 \leq m \leq n$ be two integers and $Ω\Subset \C^n$ a bounded $m$-hyperconvex domain in $\C^n$. Using a variational approach, we prove the existence of the first eigenvalue and an associated eigenfunction which is $m$-subharmonic with finite energy for general twisted complex Hessian operators of order $m$. Under some extra assumption on the twist measure we prove Hölder continuity of the corresponding eigenfunction. Moreover we give applications to the solvability of more general degenerate complex Hessian equations with the right hand side depending on the unknown function.

math.CV↗