We establish global $C^{1,β}$ and $W^{2, p}$ regularity for singular Monge-Ampère equations of the form \[\det D^2 u \sim \text{dist}^{-α}(\cdot,\partialΩ),\quad α\in (0, 1),\] under suitable conditions on the boundary data and domains. Our results imply that the convex Aleksandrov solution to the singular Monge-Ampère equation \[\det D^2 u=|u|^{-α}\quad \text{in}\quadΩ,\quad u=0\quad \text{in}\quad \partialΩ, \quad α\in (0, 1),\] where $Ω$ is a $C^3$, bounded, and uniformly convex domain, is globally $C^{1,β}$ and belongs to $W^{2, p}$ for all $p<1/α$.
In this paper, we prove the regularity of the free boundary in the Monge-Ampère obstacle problem $\det D^2 v= f(y)χ_{\{v>0\}}. $ By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of the solution to the singular Monge-Ampère equation $\det D^2 u = 1/f (Du)+δ_0$ at the origin. We first establish an asymptotic estimate for the solution $u$ near the singular point, then use a partial Legendre transform to change the Monge-Ampère equation to a singular, fully nonlinear elliptic equation, and establish the regularity of solutions to the singular elliptic eq…
By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as $ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $ with zero boundary data, have unexpected degenerate nature.
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.
In this paper, we establish interior $C^{1,α}$ estimates for solutions of the linearized Monge-Amp$\grave{e}$re equation $$\mathcal{L}_ϕu:=\mathrm{tr}[ΦD^2 u]=f,$$ where the density of the Monge-Amp$\grave{e}$re measure $g:=\mathrm{det}D^2ϕ$ satisfies a $\mathrm{VMO}$-type condition and $Φ:=(\mathrm{det}D^2ϕ)(D^2ϕ)^{-1}$ is the cofactor matrix of $D^2ϕ$.
In this paper, we establish global $W^{2,p}$ estimates for solutions of the linearized Monge-Amp$\grave{e}$re equation $$\mathcal{L}_ϕu:=\mathrm{tr}[ΦD^2 u]=f,$$ where the density of the Monge-Amp$\grave{e}$re measure $g:=\mathrm{det}D^2ϕ$ satisfies a $\mathrm{VMO}$-type condition, and $Φ:=(\mathrm{det}D^2ϕ)(D^2ϕ)^{-1}$ is the cofactor matrix of $D^2ϕ$.
In this paper, we establish the global $C^{2,α}$ and $W^{2,p}$ regularity for the Monge-Ampère equation $\det\,D^2u = f$ subject to boundary condition $Du(Ω) = Ω^*$, where $Ω$ and $Ω^*$ are bounded convex domains in the Euclidean space $\mathbb{R}^n$ with $C^{1,1}$ boundaries, and $f$ is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.
We consider Monge-Ampère equations with right hand side $f$ that degenerate to $\infty$ near the boundary of a convex domain $Ω$, which are of the type $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;Ω,\quad\quad f\sim d^{-α}_{\partialΩ}\quad\mathrm{near}\;\partialΩ,$$ where $d_{\partialΩ}$ represents the distance to $\partial Ω$ and $-α$ is a negative power with $α\in(0,2)$. We study the boundary regularity of the solutions and establish a localization theorem for boundary sections.
We construct a counterexample to $W^{2,1}$ regularity for convex solutions to $$\det D^2u \leq 1, \quad u|_{\partial Ω} = 0$$ in two dimensions. We also prove a result on the propagation of singularities in two dimensions that are logarithmically slower than Lipschitz. This generalizes a classical result of Alexandrov and is optimal by example.
In this paper, we establish global $C^{1+α,\frac{1+α}{2}}$ estimates for solutions of the linearized parabolic Monge-Amp$\grave{e}$re equation $$\mathcal{L}_ϕu(x,t):=-u_t\,\mathrm{det}D^2ϕ(x)+\mathrm{tr}[Φ(x) D^2 u]=f(x,t)$$ under appropriate conditions on the domain, Monge-Amp$\grave{e}$re measures, boundary data and $f$, where $Φ:=\mathrm{det}(D^2ϕ)(D^2ϕ)^{-1}$ is the cofactor of the Hessian of $D^2ϕ$.
We survey the (old and new) regularity theory for the Monge-Ampère equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Ampère type equations arising in that context.
In this paper, by providing the uniform gradient estimates for a sequence of the approximating equations, we prove the existence, uniqueness and regularity of the conical parabolic complex Monge-Ampère equation with weak initial data. As an application, we prove a regularity estimates, that is, any $L^{\infty}$-solution of the conical complex Monge-Ampère equation admits the $C^{2,α,β}$-regularity.
In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Ampère equation $\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(Ω) = Ω^*$ is $C^{2,α}$ smooth up to the boundary. We establish the estimate under the sharp conditions that the inhomogeneous term $f\in C^α$ and the domains are convex and $C^{1,α}$ smooth. When $f\in C^0$ (resp. $1/C<f<C$ for some positive constant $C$), we also obtain the global $W^{2,p}$ (resp. $W^{2,1+ε}$) regularity.
We obtain the Hölder regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity. Our main tool is an interior Hölder estimate in two dimensions for an inhomogeneous linearized Monge-Ampère equation with right hand side being the divergence of a bounded vector field. As a further application of our Hölder estimate, we prove the Hölder regularity of the polar factorization for time-dependent maps in two dimensions with densities bounded away from zero and infinity. Our applications i…
In this paper, we study the interior C^{1,1} regularity of viscosity solutions for a degenerate Monge-Ampère type equation \det[D^{2}u-A(x, u, Du)]=B(x, u, Du) when B \geq 0 and B^{\frac{1}{n-1}}\in C^{1,1}(\barΩ\times\mathbb{R}\times \mathbb{R}^n). We prove that u\in C^{1,1}(Ω) under the A3 condition and A3w^+ condition respectively. In the former case, we construct a suitable auxiliary function to obtain uniform {\it a priori} estimates directly. In the latter case, the main argument is to establish the Pogorelov type estimates, which are interesting independently.
In this note we revisit previous Pogorelov type interior and global second derivative estimates of the author, F. Jiang and J. Liu for solutions of Monge-Amp`ere type partial differential equations. Taking account of recent strict convexity regularity results of Guillen, Kitagawa and Rankin, and following our earlier work in the optimal transportation case, we remove the monotonicity assumptions in the more general case of generated Jacobian equations and consequently in the subsequent application to classical solvability and global regularity for second boundary value problems.
In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above $C^{1,1}$ estimate is valid for a kind of complex Monge-Ampère type equations. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of …
This text contains the material discussed by the author in the Bourbaki seminar of June 2018, on the recent developments in the theory of the Monge-Ampère equation.
We present a proof of strict $g$-convexity in 2D for solutions of generated Jacobian equations with a $g$-Monge-Ampère measure bounded away from 0. Subsequently this implies $C^1$ differentiability in the case of a $g$-Monge-Ampère measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge-Ampère case. Thus, like theirs, our argument is local and yields a quantitative estimate on the $g$-convexity. As a result our differentiability result is new even in the optimal transport case: we weaken previously required domain convexity conditions. Moreover in the optimal…
The existence and regularity of the classical plurisubharmonic solution for complex Monge-Ampère equations subject to the semilinear oblique boundary condition which is C^1 perturbation of the Neumann boundary condition, are proved in the certain strictly pseudoconvex domain in C^n.
We obtain a quantitative expansion at infinity of solutions for a kind of Monge-Ampère type equations that origin from mean curvature equations of Lagrangian graph $(x,Du(x))$ and refine the previous study on zero mean curvature equations and the Monge-Ampère equations.
In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are boundary versions of Caffarelli and Gutiérrez's interior Harnack inequality for the linearized Monge-Ampère equations. As an application, we obtain sharp upper bound and global $L^p$-integrability for the Green's function of the linearized Monge-Ampère operator.
We provide counterexamples to regularity of optimal maps in the classical Monge problem under various assumptions on the initial data. Our construction is based on a variant of the counterexample in \cite{LSW} to Lipschitz regularity of the monotone optimal map between smooth densities supported on convex domains.
In this paper, we study the eigenvalue problem for the Monge-Ampère operator on general bounded convex domains. We prove the existence, uniqueness and variational characterization of the Monge-Ampère eigenvalue. The convex Monge-Ampère eigenfunctions are shown to be unique up to positive multiplicative constants. Our results are the singular counterpart of previous results by P-L. Lions and K. Tso in the smooth, uniformly convex setting. Moreover, we prove the stability of the Monge-Ampère eigenvalue with respect to the Hausdorff convergence of the domains. This stability property makes it pos…
We study a Monge-Ampère type equation that interpolates the classical {σ_2} -Yamabe equation in conformal geometry and the 2-Hessian equation in dimension 4.
In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.
In this paper, we prove a Moser-Trudinger type inequality for pluri-subharmonic functions vanishing on the boundary. Our proof uses a descent gradient flow for the complex Monge-Ampere functional.
