arXiv subjects

Truyen Nguyen

Publications and source records attributed to Truyen Nguyen.

21 records · Page 2Linked to original sources

arXiv2012

Global $W^{2,p}$ estimates for solutions to the linearized Monge--Ampère equations

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Ampère equations.

math.AP↗

arXiv2012

Interior second derivative estimates for solutions to the linearized Monge--Ampère equation

Let $Ω\subset \R^n$ be a bounded convex domain and $ϕ\in C(\barΩ)$ be a convex function such that $ϕ$ is sufficiently smooth on $\partialΩ$ and the Monge--Ampère measure $\det D^2ϕ$ is bounded away from zero and infinity in $Ω$. The corresponding linearized Monge--Ampère equation is \[ \trace(ΦD^2 u) =f, \] where $Φ:= \det D^2 ϕ~ (D^2ϕ)^{-1}$ is the matrix of cofactors of $D^2ϕ$. We prove a conjecture in \cite{GT} about the relationship between $L^p$ estimates for $D^2 u$ and the closeness between $\det D^2ϕ$ and one. As a consequence, we obtain interior $W^{2,p}$ estimates for solutions to such equation whenever the measure $\det D^2ϕ$ is given by a continuous density and the function $f$ belongs to $L^q(Ω)$ for some $q> \max{\{p,n\}}$.

math.AP↗

arXiv2012

Geometric properties of boundary sections of solutions to the Monge--Ampère equation and applications

In this paper, we establish several geometric properties of boundary sections of convex solutions to the Monge-Ampère equations: the engulfing and separating properties and volume estimates. As applications, we prove a covering lemma of Besicovitch type, a covering theorem and a strong type $p-p$ estimate for the maximal function corresponding to boundary sections. Moreover, we show that the Monge-Ampère setting forms a space of homogeneous type.

math.AP↗