Related papers: Hessian Estimates for the Sigma-2 Equation with Va…

We obtain $C^{2,\beta}$ estimates up to the boundary for solutions to degenerate Monge-Amp\`ere equations of the type $$ \det D^2 u = f~~\text{in}~\Omega, \quad \quad ~f\sim \text{dist}^{\alpha}(\cdot,…

This paper mainly focus on optimal time decay estimation for large-solution about compressible magnetohydrodynamic equations in 3D whole space, provided that $(\sigma_{0}-1,u_{0},M_{0})\in L^1\cap H^2$. In [2](Chen et al.,2019), they proved…

We discuss recent advances in the theory of quasilinear equations of the type $ -\Delta_{p} u = \sigma u^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for…

We establish a new $W^{1,2\frac{n-1}{n-2}}$ estimate for the extremal solution of $-\Delta u=\lambda f(u)$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^n$, which is convex, for arbitrary positive and increasing nonlinearities $f\in…

We prove estimates for solutions of the $\bar \partial u=\omega $ equation in a strictly pseudo convex domain $ \Omega $ in ${\mathbb{C}}^{n}.$ For instance if the $ (p,q)$ current $\omega $ has its coefficients in $L^{r}(\Omega )$ with…

In this paper, uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by a epsilon-periodic lattice of highly conducting thin rods. The estimates are derived…

We give a brief account of two recent applications of the harmonic superspace method: (i) an off-shell description of torsionful $(4,4)$ supersymmetric $2D$ sigma models in the framework of $SU(2)\times SU(2)$ harmonic superspace and (ii)…

This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces $\mathbb{R}^m$ where $\mu$ is a positive constant. If the right-hand side $f$ is a rapidly converging series of separable functions, the solution $u$ can be…

Let $f \in L^1(\mathbb{R}^2)$ and let $\widehat f$ be its Fourier integral. We study summability of the partial integral $S_{\rho,\mathsf{H}}(x)=\int_{\{\|y\|_\mathsf{H} \le \rho\}} e^{i x\cdot y}\widehat f(y) dy$, where $\|y\|_\mathsf{H}$…

In this paper, we consider the exterior Dirichlet problem for Hessian quotient equations with the right hand side $g$, where $g$ is a positive function and $g=1+O(|x|^{-\beta})$ near infinity, for some $\beta>2$. Under a prescribed…

Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to…

We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Amp\`ere equations $$\det D^2u =f(x) \text{ in } \Omega , \quad D_{\beta}u = \phi(x) \text{ on } \partial \Omega.$$ In…

We extend classical results of Bridgeman-Taylor and McMullen on the Hessian of the Hausdorff dimension on quasi-Fuchsian space to the class of (1,1,2)-hyperconvex representations, a class introduced in arXiv:1902.01303 which includes small…

We prove existence results concerning equations of the type $-\Delta_pu=P(u)+\mu$ for $p>1$ and $F_k[-u]=P(u)+\mu$ with $1\leq k<\frac{N}{2}$ in a bounded domain $\Omega$ or the whole $\mathbb{R}^N$, where $\mu$ is a positive Radon measure…

In this paper, we establish a doubling argument to obtain Hessian estimates for the special Lagrangian equation under general phase constraints. In particular, our approach does not rely on the Michael-Simon mean value inequality. As an…

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution $u$ of $-|\nabla…

Investigating for interior regularity of viscosity solutions to the fully nonlinear elliptic equation $$F(x,u,\triangledown u,\triangledown ^2 u)=0,$$ we establish the interior $C^{1+1}$ continuity under the assumptions that $F$ is…

Recently, the Li-Yau-type gradient estimates for positive solutions to parabolic equations \begin{equation} \partial_t u=\Delta u+\mathcal{R}_1u+\mathcal{R}_2u^{\alpha}+\mathcal{R}_3u(\log u)^{\beta},\notag \end{equation} under the general…

We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and…

We provide a local classification of self-dual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, non-hyperk\"ahlerian structure compatible with the…