Related papers: A priori estimates for parabolic Monge-Amp\`ere ty…
We introduce the so-called $d$-concavity, $d \geq 0,$ and prove that the nonsymmetric Monge-Amp\`{e}re type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for…
We consider the Monge-Amp\`ere equation $\det(D^2u)=f$ where $f$ is a positive function in $\mathbb R^n$ and $f=1+O(|x|^{-\beta})$ for some $\beta>2$ at infinity. If the equation is globally defined on $\mathbb R^n$ we classify the…
We extend the study of inverse boundary value problems to the setting of fully nonlinear PDEs by considering an inverse source problem for the Monge-Amp\`ere equation \[ \det D^2 u = F. \] We prove that, on a convex Euclidean domain in the…
In this paper, we study the Neumann problem of Monge-Amp\`ere equations in Semi-space. For two dimensional case, we prove that its viscosity convex solutions must be a quadratic polynomial. When the space dimension $n\geq 3$, we show that…
We demonstrate that $C^{2,\alpha}$ estimates for the Monge-Amp\`{e}re equation depend in a highly nonlinear way both on the $C^{\alpha}$ norm of the right-hand side and $1/\alpha$. First, we show that if a solution is strictly convex, then…
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 correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge-Amp\`{e}re equation \begin{eqnarray}…
In this paper, we consider the following parabolic-parabolic-elliptic system } \begin{align*} \left\{\aligned & u_t=\Delta u-\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w)+au-\mu u^{\alpha}, && x\in\Omega, t>0,\\ & v_t=\Delta…
In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and…
First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data.…
In this paper, we study the existence of uniform a priori estimates for positive solutions to Navier problems of higher order Lane-Emden equations \begin{equation*} (-\Delta)^{m}u(x)=u^{p}(x), \qquad \,\, x\in\Omega \end{equation*} for all…
In this work, we study the so-called Allen-Cahn-Navier-Stokes equations, a diffuse-interface model for two-phase incompressible flows with different densities. We first prove the local-in-time existence and uniqueness of classical solutions…
We prove the H\"{o}lder continuity of the unique solution to quaternionic Monge-Amp\`{e}re equation with densities in $L^{p},$ $p>2,$ on a bounded strictly pseudoconvex domains.
We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.
We prove global existence and blow-up of solutions of initial-boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for…
Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Amp\`ere operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of…
In this paper we consider the generalised solutions to the Monge-Amp{\`{e}}re type equations with general source terms. We firstly prove the so-called comparison principle and then give some important propositions for the border of…
We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Amp\`ere type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this…
As a sequel to the paper [9], we study the existence and properties of Lipschitz solutions to the initial-boundary value problem of some forward-backward parabolic equations with diffusion fluxes violating Fourier's inequality.
In this paper, we are concerned with the monotonic and symmetric properties of convex solutions to fully nonlinear elliptic systems. We mainly discuss Monge-Amp\`ere type systems for instance, considering \begin{equation*}…