arXiv subjects

Jiali Lan

Publications and source records attributed to Jiali Lan.

5 recordsLinked to original sources

arXiv2026

A $C^2$-Perturbative Bernstein Theorem for Anisotropic Entire Minimal Graphs

We prove a Bernstein theorem for $\Phi$-anisotropic minimal hypersurfaces in dimensions $1\leq n\leq 7$ that the only entire smooth solutions $u$ of $\Phi$-anisotropic minimal hypersurfaces equation are affine functions provided the anisotropic area functional integrand $\Phi$ is sufficiently $C^{2}$--close to the Euclidean area integrand. This settles the $C^2$ entire--graph version of anisotropic Bernstein problem posed by Mooney and Yang \cite{MooneyYang2024}, and the proof uses a compactness--rigidity argument combined with Figalli's regularity theorem established in \cite{Figalli2017}.

math.AP↗

arXiv2026

Anisotropic minimal surface equation with Dirichlet boundary condition

This paper investigates the Dirichlet problem for the anisotropic minimal surface equation in a bounded domain. Under the natural assumption of non-negative boundary anisotropic mean curvature, we establish the unique solvability of the Dirichlet problem for continuous boundary data. To achieve this, an essential a priori gradient estimate is established, which also allows us to prove a weak version of Bernstein's theorem for entire solutions under a sharp, one-sided linear growth assumption. Moreover, using the direct method in the calculus of variations, we prove the existence and local Lipschitz regularity of generalized minimizers in $BV(\Omega)$ with $L^1(\partial\Omega)$ boundary data. We also find that this variational formulation naturally yields a Neumann-type boundary condition, geometrically explaining why the Neumann problem requires no boundary curvature constraints.

math.AP↗

arXiv2026

$C^{\infty}$ regularity of the Alt-Phillips Functional for negative powers

In this paper, we study the regularity of the free boundary for minimizers of the Alt-Phillips functional with negative powers \[\mathcal{E}_{\gamma}(u)=\int_{\Omega}\frac{1}{2}|\nabla u|^2+\frac{1}{\gamma}u^{-\gamma}\chi_{\{u>0\}}dx,\quad\gamma\in(0,2).\] We proved that the free boundaries are $C^{\infty}$ at regular points. A key technical tool is the linearized operator for the PDE satisfied by the partial derivatives of a solution to the Alt-Phillips Euler-Lagrange equation in the negative power case. For this operator we establish a comparison principle, which may have further applications to the Alt-Phillips problem with negative powers.

math.AP↗

arXiv2026

Sharp capillary Sobolev inequality and Moser-Trudinger inequality outside convex domain

The theory of sharp geometric inequality in $\mathbb{R}^n$ and inside convex cone has been well-developed, much less known for sharp capillary geometric inequality outside convex domain. Recently, Fusco-Julin-Morini-Pratelli \cite{FJMP} obtained sharp capillary isoperimetric inequality and make it possible to obtain the sharp capillary geometric inequality outside convex domain. In this paper, we establish the sharp capillary Sobolev inequality and Moser-Trudinger inequality outside convex domain, which can be seen as geometric inequality on the Finsler manifold to some extent. Our method is based on constructing capillary P\'{a}lya-Szeg\"{o} rearrangement inequality outside convex domain. Finally, we also consider the capillary Talenti-Comparison principle and Bossel-Daners inequality.

math.AP↗

arXiv2025

The weighted isoperimetric inequality and Sobolev inequality outside convex sets

In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $\lambda_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $\alpha$, such that $w^{1/\alpha}$ is concave on $\R^n$. Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted P\'{o}lya-Szeg\"{o} principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality in the half-space established by Ciraolo-Figalli-Roncoroni in \cite{CFR}.

math.AP↗