arXiv subjects
Publications and source records attributed to Soufian Abja.
In this note we continue our investigation of geodesics in the space of convex and plurisubharmonic functions. We show optimal regularity for geodesics joining two smooth strictly convex functions. We also investigate the regularity theory in the $C^{1,α}$ realm. Finally we discuss the regularity of geodesics joining two toric strictly plurisubharmonic functions.
We study the viscosity solutions to the first eigenvalue equation. We consider $Ω$ a bounded B-regular domain in $\mathbb{C}^n$ and we prove that the Dirichlet problem $Λ_{1}(D_{\mathbb{C}}^2 u)=f$ in $Ω$ and $u=φ$ on $\partialΩ$ admits a unique viscosity solution. We also deal with viscosity theory for operators which are comparable to the first eigenvalue operator.
In this paper, we establish a local regularity result for $W^{2,p}_{\mathrm{loc}}$ solutions to complex degenerate nonlinear elliptic equations $F(D^2_{\mathbb{C}} u)=f$ when they are comparable to the Monge-Ampère equation. Notably, we apply our result to the so-called $k$-Monge-Ampère equation.
We prove sharp uniform estimates for strong supersolutions of a large class of fully nonlinear degenerate elliptic complex equations. Our findings rely on ideas of Kuo and Trudinger who dealt with degenerate linear equations in the real setting. We also exploit the pluripotential theory for the complex Monge-Ampère operator as well as suitably tailored theory of $L^p$-viscosity subsolutions.
In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C^{1,1} regularity. We construct examples which prove that the global C^{1,1} regularity fails both in the real and complex case in contrast to the Kähler manifold setting. Finally we show a necessary and sufficient conditions for existence of a smooth geodesic between two smooth strictly convex functions.
Let $Ω$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $Ω$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local Kähler-Einstein metrics.