ﻻ يوجد ملخص باللغة العربية
We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-Delta)^m u = K(x) u^{frac{n+2m}{n-2m}} ~~~~~~ in ~ Omega backslash Lambda $$ near their singular set $Lambda$, where $Omega subset mathbb{R}^n$ is an open set, $K(x)$ is a positive continuous function on $Omega$, $Lambda$ is a closed subset of $mathbb{R}^n$, $2 leq m < n/2$ and $m$ is an integer. Under certain flatness conditions at critical points of $K$ on $Lambda$, we prove that $u(x) leq C [{dist}(x, Lambda)]^{-(n-2m)/2}$ when the upper Minkowski dimension of $Lambda$ is less than $(n-2m)/2$.
This paper is focused on the local interior $W^{1,infty}$-regularity for weak solutions of degenerate elliptic equations of the form $text{div}[mathbf{a}(x,u, abla u)] +b(x, u, abla u) =0$, which include those of $p$-Laplacian type. We derive an ex
We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ partial_t^alpha u - Lu= f quad mathrm{in} quad (0,T) times mathbb{R}^d,$$ where $partial_t^alpha u$ is the Caputo fractional derivative of order $alph
We give a unified approach to weighted mixed-norm estimates and solvability for both the usual and time fractional parabolic equations in nondivergence form when coefficients are merely measurable in the time variable. In the spatial variables, the l
In this paper, we derive a priori interior Hessian estimates for Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives.
We have an idea on the influence of a nonlinear term (tending to 0) on the prescribed scalar curvature equation to have an uniform estimate.