Global H\"older estimates for linearized Monge-Amp\`ere equations in divergence form with applications to dual semigeostrophic equations and periodic homogenization
We establish interior and global H\"older estimates for linearized Monge-Amp\`ere equations in divergence form in all dimensions $n\geq 3$, when the Hessian determinant of the convex Monge-Amp\`ere potential is bounded above and below by positive constants and the vector field on the right-hand side is bounded. The estimates use the $L^p$ norm of the solution for any $p>1$. A key ingredient is an $L^1$ estimate, with appropriate decay rates, for the gradient of the Green's function of the linearized Monge-Amp\`ere operator in small sections, uniform with respect to the pole. As applications, we obtain uniform H\"older estimates for the time derivatives of the primal and dual potentials in the three-dimensional periodic dual semigeostrophic system when the initial density is bounded away from zero and infinity, and establish a linear convergence rate for periodic homogenization of the Monge-Amp\`ere equation.
We establish an interior $C^2$ estimate for locally bounded plurisubharmonic Bedford-Taylor solutions of the complex Monge--Amp\`ere equation $\det(u_{i\bar j})=f$ under weak quadratic growth bounds on sufficiently large balls, where $f$ is positive and Dini-continuous. We assume a distributional lower bound for the La...
We investigate interior regularity for stable solutions of the semilinear elliptic equation $-\Delta u=f(u)$ in an open set $\Omega\subseteq \mathbb R^n$, where $f$ is locally Lipschitz and may change sign. We establish sharp interior H\"older estimates for stable classical solutions in every dimension $n\le 9$, thereb...
We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate aux...
We study the free-boundary Euler--Monge--Amp\`ere system as a conservative relaxation of ideal incompressible fluid motion. The force is determined on the actual fluid domain by a Monge--Amp\`ere equation coupled to a nonlinear boundary condition. For each fixed relaxation parameter, we establish local well-posedness f...
We prove that a spatial semidiscrete finite element method preserves the stochastic maximal regularity of the heat equation on bounded convex polygonal and polyhedral domains in dimensions two and three. A main difficulty is to establish suitable Green-function estimates for stochastic convolutions, whose paths general...
We establish universal interior H\"older and fractional energy estimates for stable solutions of $(-\Delta)^su=f(u)$, $0<s<1$, with $f\in C^1_{\mathrm{loc}}(\mathbb{R})$ nonnegative and convex. The bounds depend only on a weighted $L^1$ norm of the solution and hold under optimal conditions on the dimension and the fra...
Yi-Yang Xiao· 0 citations
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