Related papers: Remarks on weak convergence of complex Monge-Amp\`…

Given a psh function $\varphi\in\mathcal{E}(\Omega)$ and a smooth, bounded $\theta\geq 0$, it is known that one can solve the Monge-Amp\`{e}re equation $\mathrm{MA}(\varphi_\theta)=\theta^n\mathrm{MA}(\varphi)$, with some form of Dirichlet…

Let $\Omega$ be an unbounded, pseudoconvex domain in $\Bbb C^n$ and let $\varphi$ be a $\mathcal C^2$-weight function plurisubharmonic on $\Omega$. We show both necessary and sufficient conditions for existence and compactness of a weighted…

In this paper, we prove a higher integrability result for very weak solutions of higher-order elliptic systems involving a double phase operator as the principal part. As a model case, we consider \begin{equation} \int_{\Omega} \left( |D^m…

The present work constitutes a first step towards establishing a systematic framework for treating variational problems that depend on a given input function through a mixture of its derivatives of different orders in different directions.…

In this paper, we extend the concept of finite entropy measures in K\"ahler geometry. We define the finite $p$-entropy related to $\omega$-plurisubharmonic functions and demonstrate their inclusion in an appropriate energy class. Our study…

The purpose of this paper is to study convergence of Monge-Ampere measures associated to sequences of plurisubharmonic functions defined on a hyperconvex subset of ${\mathbb C^n}$.

For $p \in (1,N)$ and $\Omega \subseteq \mathbb{R}^N$ open, the Beppo-Levi space $\mathcal{D}^{1,p}_0(\Omega)$ is the completion of $C_c^{\infty}(\Omega)$ with respect to the norm $\left( \int_{\Omega}|\nabla u|^p \right)^ \frac{1}{p}.$…

We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[ \mathcal{A} \mu = \sigma, \] where $\mu \in \mathcal{M}(\Omega;V)$ and $\sigma\in \mathcal{M}(\Omega;W)$ are vector measures and the polar…

We study the problem of minimizing the functional $$ I(\varphi)=\int\limits_{\Omega} W(x,D\varphi)\,dx $$ on a new class of mappings. We relax summability conditions for admissible deformations to $\varphi\in W^1_n(\Omega)$ and growth…

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. Let $\beta$ be a smooth real closed $(1,1)$ form such that there exists a function $\rho \in \mbox{PSH}(X,\beta)\cap L^{\infty}(X)$. We study the range of the…

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Amp\`ere mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. We then investigate stability properties of this condition with respect to…

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel…

Let $(E,\mathcal E,\mu)$ be a measure space and $G\colon E\times E\to [0,\infty]$ be measurable. Moreover, let $\mathcal F\!_{ui}$ denote the set of all $q\in\mathcal E^+$ (measurable numerical functions $q\ge 0$ on $E$) such that…

We study H\"older continuity of solutions to the Monge-Amp\`{e}re equations on compact K\"ahler manifolds. In [DNS] the authors have shown that the measure $\omega_u^n$ is moderate if $u$ is H\"older continuous. We prove a theorem which is…

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…

We prove that the image under any dominant meromorphic map of the Monge-Amp{\`e}re measure of a H{\"o}lder continuous quasi-psh function still possesses a H{\"o}lder potential. We also discuss the case of lower regularity.

We study the Dirichlet problem $-\div(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $\Omega$, with $u=f$ on $\partial \Omega$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $\Omega$. The main issue is to give a proper sense to what…

We consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth…

We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})+f(t,z,u)$ in domains of $\mathbb C^n$. This equation has a close connection with the K\"ahler-Ricci flow. In this paper, we consider the case of the boundary conditions are…

In this paper, we study a Dirichlet type problem for the non-pluripolar complex Monge - Amp\`ere equation with prescribed singularity on a bounded domain of $\mathbb{C}^n$. We provide a local version for an existence and uniqueness theorem…