Related papers: The Monge-Ampere system: convex integration in arb…
We prove a convex integration result for the Monge-Ampere system in dimension $d=2$ and arbitrary codimension $k\geq 1$. We achieve flexibility up to the Holder regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$, improving hence the previous…
We prove a convex integration result for the Monge-Amp\`ere system, in case of dimension $d=2$ and arbitrary codimension $k\geq 1$. Our prior result stated flexibility up to the H\"older regularity $\mathcal{C}^{1,\frac{1}{1+ 4/k}}$,…
We revisit the convex integration constructions for the Monge-Amp\`ere system and prove its flexibility in dimension $d=2$ and codimension $k=3$, up to $\mathcal{C}^{1,1-1/\sqrt{5}}$. To our knowledge, it is the first result in which the…
We prove that $\mathcal{C}^{1,\alpha}$ solutions to the Monge-Amp\`ere system in dimension $d$ and codimension $k= d_*-d+1$, where $d_*$ denotes the Janet dimension, are dense in the space of continuous functions, for every H\"older…
This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Amp\`ere equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e.…
We prove that every $\mathcal{C}^1(\bar\omega)$-regular subsolution of the Monge-Amp\`ere system posed on a $2$-dimensional domain $\omega$ and with target codimension $2$, can be uniformly approximated by its exact solutions with…
For any $\theta<\frac{1}{3}$, we show that very weak solutions to the two-dimensional Monge-Amp\`ere equation with regularity $C^{1,\theta}$ are dense in the space of continuous functions. This result is shown by a convex integration scheme…
In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Amp\'ere equation with H\"older-continuous first derivatives of exponent $\beta<1/5$. Our approach is based on combining the…
By developing an integral approach, we present a new method for the interior regularity of strictly convex solution of the Monge-Amp\`{e}re equation $\det D^2 u = 1$.
The Monge-Amp\`ere type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Amp\`ere type equations, which can be degenerate or singular near…
We study the Dirichlet problem for Monge-Amp\`ere equation in bounded convex polytopes. We give sharp conditions for the existence of global $C^2$ and $C^{2,\alpha}$ convex solutions provided that a global $C^2$, convex subsolution exists.
The convexity of solutions to boundary value problems for fully nonlinear elliptic partial differential equations (such as real or complex $k$-Hessian equations) is a challenging topic. In this paper, we establish the power convexity of…
In this article, we implement the algorithm based on the convex integration result proved in [Lewicka-Pakzad Analysis and PDE (2017)] and obtain visualizations of the first iterations of the Nash-Kuiper scheme, approx- imating the anomalous…
This paper develops a rigorous analytic framework for the hyperbolic Monge-Amp\`ere equation on strip-like domains, which model wrinkled patterns in thin elastic sheets. Our work addresses the rigid side of the classical…
We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function…
We propose an extension to our monotone and convergent method for the Monge-Amp\`{e}re equation in dimension $d \geq2$, that incorporates the idea of filtered schemes. The method combines our original monotone operator with a more accurate…
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-Amp\`ere equation with weak initial data. As…
We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…
We prove that if a smoothly bounded strongly pseudoconvex domain $D \subset \mathbb C^n$, $n \geq 2$, admits at least one Monge-Amp\`ere exhaustion smooth up to the boundary (i.e. a plurisubharmonic exhaustion $\tau: \overline D \to [0,1]$,…
This paper concerns with the convergence analysis of a fourth order singular perturbation of the Dirichlet Monge-Amp\`ere problem in the $n$-dimensional radial symmetric case. A detailed study of the fourth order problem is presented. In…