Related papers: Viscosity solutions to quaternionic Monge-Amp\`{e}…
In this work we consider viscosity solutions to second order parabolic PDEs $u_{t}+F(t,x,u,du,d^{2}u)=0$ defined on compact Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the…
Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem $\Delta_\infty u=f(x,u)$ where $u$ takes a prescribed continuous data on the boundary of bounded domains. We employ a combination of…
We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})-Au+f(z,t)$ 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 where the boundary condition is…
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…
In this paper we investigate the regularity and solvability of solutions to Dirichlet problem for fully non-linear elliptic equations with gradient terms on Hermitian manifolds, which include among others the Monge-Amp\`ere equation for…
Numerical issues arising in computations of viscous flows in corners formed by a liquid-fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of…
We shall consider the regularity problem of solutions for complex Monge-Ampere equations. First we prove interior $C^2$ estimates of solutions in a bounded domain for complex Monge-Ampere equation with assumption of certain $L^p$ bound for…
We study a nonlinear porous medium type equation involving the infinity Laplacian operator. We first consider the problem posed on a bounded domain and prove existence of maximal nonnegative viscosity solutions. Uniqueness is obtained for…
In this paper, we study the Dirichlet problem of Hessian quotient equations in exterior domains. By estimating the eigenvalues of the solution, the necessary and sufficient conditions on existence of radial solutions are obtained. Applying…
In \cite{GL21a} we have developed a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations, when the reference form is closed. This simplifying assumption was used to ensure the constancy of the…
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}…
We study the Dirichlet problem for the following prescribed mean curvature PDE $$ \begin{cases} -\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v) \text{ in }\Omega\\ v=\varphi \text{ on }\partial\Omega. \end{cases} $$…
In this paper we study the following eigenvalue boundary value problem for Monge-Amp\`{e}re equations: {equation} \{{array}{l} \det(D^2u)=\lambda^N f(-u)\,\, \text{in}\,\, \Omega, u=0,\,\text{on}\,\, \partial \Omega. {array}. {equation} We…
In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Amp\`ere equation $\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(\Omega) = \Omega^*$ is $C^{2,\alpha}$…
We consider the asymptotic behavior of solutions to the Monge--Amp\`ere equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao--Li--Zhang [Calc. Var PDE. 52(2015). pp. 39-63].…
We prove the long time existence and uniqueness of solutions to the parabolic Monge-Amp\`ere equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$…
We study the problem of approximating plurisubharmonic functions on a bounded domain $\Omega$ by continuous plurisubharmonic functions defined on neighborhoods of $\bar\Omega$. It turns out that this problem can be linked to the problem of…
We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term $$ -{\rm div}(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=f(x,u,Du),\quad 1<p\le q<\infty, a(x)\ge0. $$ We find some…
In this paper we study existence and nonexistence of solutions for a Dirichlet boundary value problem whose model is $$ \begin{cases} -\sum_{m=1}^{\infty} a_m \Delta u^m= f&\text{in}\ \Omega \newline u=0 & \text{on}\ \partial\Omega\,,…
This paper is devoted to a complete classification on the existence and nonexistence results of viscosity solutions to the general Dirichlet problem for a class of eigenvalue type equations. With the distance function included in the…