Related papers: Interior estimates for Monge-Amp\`ere type fourth …
A number of geometric problems, including affine hyperbolic spheres, Hilbert metrics and Minkowski type problems, are reduced to a singular Monge-Amp\`ere equation which can be written locally as a class of Monge-Amp\`ere equations with…
Monge--Amp\`ere equation plays an important part in Analysis. For example, it is instrumental in mass transport problems. On the other hand, the Bellman function technique appeared recently as a way to consider certain Harmonic Analysis…
In this paper, we establish boundary H\"older gradient estimates for solutions to the linearized Monge-Amp\`ere equations with $L^{p}$ ($n<p\leq\infty$) right hand side and $C^{1,\gamma}$ boundary values under natural assumptions on the…
Uniform $L^\infty$ and H\"older estimates were proved by the Kolodziej for complex Monge-Amp\`ere equations on compact K\"ahler manifolds with $L^p$ volume measure with $p>1$. On the other hand, establishing H\"older estimates on singular…
We consider three fundamental classes of compact almost homogeneous manifolds and show that the complements of singular complex orbits in such manifolds are endowed with plurisubharmonic exhaustions satisfying complex homogeneous…
We prove the local boundedness of the solutions to degenerate second order partial differential equations of Kolmogorov type with measurable coefficients in divergence form, under minimal integrability assumption on the lower order…
In this paper, we consider the Monge-Amp\`{e}re type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence…
We solve the Dirichlet problem for the quaternionic Monge-Amp\`ere equation with a continuous boundary data and the right hand side in $L^p$ for $p>2$. This is the optimal bound on $p$. We prove also that the local integrability exponent of…
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.…
In this paper, we explore the fundamental role of the Monge-Amp\`ere equation in deep learning, particularly in the context of Boltzmann machines and energy-based models. We first review the structure of Boltzmann learning and its relation…
It is proved a theorem providing necessary and sufficient conditions enabling one to map a nonlinear system of first order partial differential equations, polynomial in the derivatives, to an equivalent autonomous first order system…
An extension of the ambient metric construction of Fefferman-Graham to infinite order in even dimensions is described. The main ingredients are the introduction of "inhomogeneous ambient metrics" with asymptotic expansions involving the…
We present a proof of strict $g$-convexity in 2D for solutions of generated Jacobian equations with a $g$-Monge-Amp\`ere measure bounded away from 0. Subsequently this implies $C^1$ differentiability in the case of a $g$-Monge-Amp\`ere…
We prove the existence and regularity of convex solutions to the first initial-boundary value problem for the parabolic Monge-Amp\`ere equationn $$ \left\{\begin{eqnarray} &&-u_t+\det D^2u= \psi(x,t) \quad\quad\ \text{ in } Q_T,\newline…
We provide the sharp $C^0$ estimate for the quaternionic Monge-Ampere equation on any hyperhermitian manifold. This improves previously known results concerning this estimate in two directions. Namely, it turns out that the estimate depends…
This paper is about quantitative linearization results for the Monge-Amp\`ere equation with rough data. We develop a large-scale regularity theory and prove that if a measure $\mu$ is close to the Lebesgue measure in Wasserstein distance at…
We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Amp\`ere equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating…
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…
This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order…
In this paper, we shall study the boundary case for complex Monge-Amp\`ere type equations under certain geometric assumptions.