arXiv subjects
Publications and source records attributed to Shengyang Zang.
For $2\le k<n$, we prove pointwise gradient bounds for proper weak $k$-convex functions in terms of a local average and a Wolff potential of the full Hessian measure $F_k[u]$. The estimate yields local Lipschitz regularity for $F_k[u]=f$ with $f\in L^{n/k,1/k}_{\mathrm{loc}}$, and for $F_k[u]=|Du|^p$ when $0<p<nk/(n-k)$. Both the Lorentz index $1/k$ and the upper bound on $p$ are sharp.
We classify isolated singularities of $p$-Laplace and $k$-Hessian Lane--Emden systems on their Serrin critical curves without radial symmetry. Our approach reconstructs a radial profile from the source measures through a truncated Wolff potential, providing a nonlinear counterpart of spherical averaging. For upper inequalities at equal powers, we determine the sharp joint logarithmic lower bound on the zero-mass singular branch. A radial pair attains this bound while both normalized components oscillate between zero and infinity. For exact systems at equal powers, we obtain a common second-order profile with a shared phase and a sharp two-term logarithmic expansion of the component difference. At unequal powers, we establish power--logarithmic asymptotics with explicit leading constants. These asymptotics hold uniformly in direction.
We prove sharp bounds for the upper and lower intermediate dimensions of orthogonal and radial projections of compactly supported measures. Both dimension functions are preserved almost everywhere when the quasi-Assouad dimension does not exceed the target dimension. We obtain exceptional-set estimates using the entire source dimension function, with sharpness for explicit families. We also construct a fixed measure whose projected dimension functions vary on open direction sets, even when all source dimensions lie below the target dimension.
We prove that four cosines with distinct positive integer frequencies have the same sign on at least $1/33$ of a period, with equality precisely for positive integer dilates of $\{1,3,11,33\}$. If both odd and even frequencies occur, the optimal proportion is $1/24$; equality holds precisely for odd positive integer dilates of $\{1,3,4,12\}$ and $\{2,4,5,15\}$. For arbitrary $n$, we establish the universal lower bound $4^{1-n}$ and construct frequency sets for which this proportion is at most $16n^\theta4^{-n}$, where $θ=\log_2(3/2)$. These bounds determine the optimal exponential decay rate $1/4$.
We classify isolated singularities of nonpositive, $k$-admissible classical solutions to $k$-Hessian equations without radial symmetry assumptions. For $1\le k 0$ and $0\le\ell<n$. The proofs combine Hessian-measure compactness, Wolff-potential estimates, a global Bôcher theorem, and a bounded-variation analysis of spherical extrema.
In this paper, we study the asymptotic behavior of isolated singularities for quasilinear elliptic equations of $p$-Laplacian type. We focus on the $p$-Laplace equation with the critical Serrin exponent $n(p-1)/(n-p)$ and the $n$-Laplace equation with critical exponential growth. We prove that any positive solution either admits a $C^{1,α}$ extension to the origin or exhibits a precise singular asymptotic behavior. These results provide a complement to our previous study on $p$-Laplacian type inequalities.
We study intermediate coverings of spectral families, Cantor sets, and projections of measures. Two perturbation families of a fixed ReLU tangent kernel have identical source thresholds. Their Brownian images have equal box thresholds and distinct intermediate thresholds, almost surely along prescribed geometric scales. For Cantor sets with delayed contractions, we obtain entropy formulas for the intermediate dimensions of the sets and their fractional Brownian images. These formulas yield endpoint asymptotics and a sharp separation theorem for the first phase transition. An oscillating variant has distinct lower and upper intermediate dimensions while preserving the Assouad spectrum. Finally, almost every orthogonal projection preserves both intermediate dimensions of a nonzero finite compactly supported measure whose quasi-Assouad dimension does not exceed the target dimension.
We provide a complete classification of the asymptotic behavior of isolated singularities for solutions satisfying \[ 0\le-Δ_{p}u(x)\le τu^{\frac{n(p-1)}{n-p}} (x),\,\,u(x)\ge0,\,\,1<p<n,\,\,n\ge2, \]where $u(x)\in C^2(B(0,1)\backslash\{0\})$ and $B(0,1)\subset \mathbb R^n$.