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Yuanyuan Lian

Publications and source records attributed to Yuanyuan Lian.

At least 19 recordsLinked to original sources
arXiv2026

Bifurcation of overdetermined capillary problems in a strip domain

In this paper, we consider the classical overdetermined capillary problem: \begin{equation*} \begin{cases} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) - bu =0 &~~\mbox{in}~~ Ω, \partial_ν u=κ&~~\mbox{on}~~\partialΩ, u=c &~~\mbox{on}~~\partialΩ, \end{cases} \end{equation*} where $b$, $c$ and $κ$ are positive constants, and $Ω\subset \mathbb{R}^2$. When $Ω$ is an infinite strip, i.e., a domain bounded by two parallel straight lines, there exists a unique one-dimensional solution (called the trivial solution) to this problem. By means of a bifurcation argument, we establish the existence of a critical period $T_*$ at which a branch of non-trivial solutions bifurcates from the trivial one. These solutions are genuinely two-dimensional and are defined in unbounded periodic domains $Ω$ that are diffeomorphic to an infinite strip, yet whose boundaries are no longer straight lines. This result offers a significant physical interpretation in the context of capillary phenomena.

math.AP↗
arXiv2026

Asymptotic behavior at infinity of Weingarten surfaces

We derive the asymptotic expansion at infinity for embedded ends of uniformly elliptic Weingarten surfaces with finite total curvature in $\mathbb{R}^3$, and we establish a maximum principle at infinity. Furthermore, we solve the Dirichlet problem for the uniformly elliptic Weingarten equation in dimension two on strictly convex bounded domains.

math.DG↗
arXiv2026

Pointwise Regularity for Fully Nonlinear Elliptic Equations in General Forms

In this paper, we develop systematically the pointwise regularity for viscosity solutions of fully nonlinear elliptic equations in general forms. In particular, the equations with quadratic growth (called natural growth) in the gradient are covered. We obtain a series of interior and boundary pointwise $C^{k,α}$ regularity ($k\geq 1$ and $0<α<1$). In addition, we also derive the pointwise $C^k$ regularity ($k\geq 1$) and $C^{k,\mathrm{lnL}}$ regularity ($k\geq 0$), which correspond to the end points $α=0$ and $α=1$ respectively. Some regularity results are new even for the linear equations. Moreover, the minimum requirements are imposed to obtain above regularity and our proofs are simple.

math.AP↗
arXiv2025

Bifurcating domains for an overdetermined eigenvalue problem in cylinders

We study an overdetermined eigenvalue problem for domains $Ω$ contained in the half-cylinder $Σ=ω\times (0, +\infty)$, based on a bounded regular domain $ω\subset \mathbb{R}^{N-1}$. It is easy to see that in any bounded cylinder $Ω_{t}=ω\times (0, t)$, $t > 0$, the eigenvalue problem admits a one-dimensional positive eigenfunction which satisfies the overdetermined boundary conditions. The aim of the paper is to construct other domains $Ω\subset Σ$ for which there exists a positive eigenfunction that is a solution of the overdetermined problem. This is achieved by showing that branches of such domains bifurcate from the ``trivial'' domains $Ω_{t_j}$ at the values $t_{j} = \fracπ{2\sqrt{σ_j}}$ where $σ_j$ ($j\geq 1$) is a simple Neumann eigenvalue of the Laplace operator on $ω\subset \mathbb{R}^{N-1}$. The solutions can be reflected with respect to $ω$ to generate nontrivial solutions in a cylinder.

math.AP↗
arXiv2025

Boundary Regularity for Fully Nonlinear Parabolic equations on $C^{1,\mathrm{Dini}}$ Domains

We establish the boundary pointwise Lipschitz regularity on exterior $C^{1,\mathrm{Dini}}$ domains and the Hopf lemma on interior $C^{1,\mathrm{Dini}}$ domains for fully nonlinear parabolic equations by a unified perturbation method. In fact, above two regularity hold for more general solution sets, i.e., the Pucci's class $S^*(λ, Λ, f)$. Furthermore, based on the boundary pointwise Lipschitz regularity, we obtain the global ${W}^{2,δ}$ regularity on exterior $C^{1,\mathrm{Dini}}$ domains for any $0<δ<1$, which is new even for the harmonic functions.

math.AP↗
arXiv2025

Modica type estimates and curvature results for overdetermined $p$-Laplace problems

In this paper we prove Modica type estimates for the following overdetermined $p$-Laplace problem \begin{equation*} \begin{cases} \mathrm{div} \left(|\nabla u|^{p-2}\nabla u\right)+f(u) =0& \mbox{in $Ω$, } u>0 &\mbox{in $Ω$, } u=0 &\mbox{on $\partialΩ$, } \partial_ν u=-κ&\mbox{on $\partialΩ$, } \end{cases} \end{equation*} where $1 2$ we also assume that if $F(u_0)=0$, $F(u)=O(|u-u_0|^p)$ as $u\rightarrow u_0$), then either the mean curvature of $\partial Ω$ is strictly negative or $Ω$ is a half-space.

math.AP↗
arXiv2025

Rigidity results for the capillary overdetermined problem

In this paper we obtain rigidity results for bounded positive solutions of the general capillary overdetermined problem \begin{equation} \left\{ \begin{array} {ll} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + f(u) = 0 & \mbox{in }\; Ω,\\[1mm] u= 0 & \mbox{on }\; \partial Ω,\\[1mm] \partial_ν u=κ&\mbox{on }\; \partial Ω, \end{array}\right. \end{equation} where $f$ is a given $C^1$ function in $\mathbb{R}$, $ν$ is the exterior unit normal, $κ$ is a constant and $Ω\subset \mathbb{R}^n$ is a $C^1$ domain. Our main theorem states that if $n=2, κ\neq 0$, $\partial Ω$ is unbounded and connected, $|\nabla u|$ is bounded and there exists a nonpositive primitive $F$ of $f$ such that $F(0)\geq \left(1+κ^2\right)^{-\frac12} -1$, then $Ω$ must be a half-plane and $u$ is a parallel solution. In other words, under our assumptions, if a capillary graph has the property that its mean curvature depends only on the height, then it is the graph of a one dimensional function. We also prove the boundedness of the gradient of solutions of the above problem when $f'(u) <0$. Moreover we study a Modica type estimate for the above overdetermined problem that allows us to prove that, unless $Ω$ is a half-space, the mean curvature of $\partial Ω$ is strictly negative under the assumption that $κ\neq 0$ and there exists a nonpositive primitive $F$ of $f$ such that $F(0)\geq \left(1+κ^2\right)^{-\frac12} -1$. Our results have an interesting physical application to the classical capillary overdetermined problem, i.e., the case where $f$ is linear.

math.AP↗
arXiv2025

Time derivative estimates for parabolic $p$-Laplace equations and applications to optimal regularity

We establish the boundedness of time derivatives of solutions to parabolic $p$-Laplace equations. Our approach relies on the Bernstein technique combined with a suitable approximation method. As a consequence, we obtain an optimal regularity result with a connection to the well-known $C^{p'}$-conjecture in the elliptic setting. Finally, we extend our method to treat global regularity results for both fully nonlinear and general quasilinear degenerate parabolic problems.

math.AP↗
arXiv2024

Pointwise regularity for locally uniformly elliptic equations and applications

In this paper, we study the regularity for viscosity solutions of locally uniformly elliptic equations and obtain a series of interior pointwise $C^{k,α}$ ($k\geq 1$, $0<α<1$) regularity with smallness assumptions on the solution and the right-hand term. As applications, we obtain various interior pointwise regularity for several classical elliptic equations, i.e., the prescribed mean curvature equation, the Monge-Ampère equation, the $k$-Hessian equations, the $k$-Hessian quotient equations and the Lagrangian mean curvature equation. Moreover, the smallness assumptions are necessary in most cases (Remark 2.6, Remark 3.5, Remark 4.7, Remark 5.4 and Remark 6.5).

math.AP↗
arXiv2024

Interior pointwise regularity for elliptic and parabolic equations in divergence form and applications to nodal sets

In this paper, we obtain the interior pointwise $C^{k,α}$ ($k\geq 0$, $0<α<1$) regularity for weak solutions of elliptic and parabolic equations in divergence form. The compactness method and perturbation technique are employed. The pointwise regularity is proved in a very simple way and the results are optimal. In addition, these pointwise regularity can be used to characterize the structure of the nodal sets of solutions.

math.AP↗
arXiv2024

Interior pointwise $C^α$ regularity for elliptic and parabolic equations with divergence-free drifts

We investigate the interior pointwise $C^α$ regularity for weak solutions of elliptic and parabolic equations with divergence-free drifts. For such equations, the integrability condition on the drift can be relaxed and the interior $C^α$ regularity for some $0<α<1$ has been obtained previously with the aid of Harnack inequality. In this paper, we prove the interior pointwise $C^α$ regularity for any $0<α<1$ provided that the drift is small. We obtain the regularity under three different types conditions on the drift. The proof is based on the energy inequality and the perturbation technique.

math.AP↗
arXiv2023

Boundary Lipschitz Regularity and the Hopf Lemma for Fully Nonlinear Elliptic Equations

In this paper, we study the boundary regularity for viscosity solutions of fully nonlinear elliptic equations. We use a unified, simple method to prove that if the domain $Ω$ satisfies the exterior $C^{1,\mathrm{Dini}}$ condition at $x_0\in \partial Ω$ (see Definition 1.2), the solution is Lipschitz continuous at $x_0$; if $Ω$ satisfies the interior $C^{1,\mathrm{Dini}}$ condition at $x_0$ (see Definition 1.3), the Hopf lemma holds at $x_0$. The key idea is that the curved boundaries are regarded as perturbations of a hyperplane. Moreover, we show that the $C^{1,\mathrm{Dini}}$ conditions are optimal.

math.AP↗
arXiv2022

Boundary Hölder Regularity for Elliptic Equations on Reifenberg Flat Domains

In this paper, we investigate the boundary Hölder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic equations) on Reifenberg flat domains. We prove that for any $0<α<1$, there exists $δ>0$ such that the solution is $C^α$ at $x_0\in \partial Ω$ provided that $Ω$ is $δ$-Reifenberg flat at $x_0$ (see Definition 1.1). In particular, for any $0 < α< 1$, if $\partial Ω$ is $C^1$ and $u=g$ on $\partial Ω$ with $g\in C^α(x_0)$, then $u\in C^α(x_0)$. A similar result for the Poisson equation has been proved by Lemenant and Sire, where the Alt-Caffarelli-Friedman's monotonicity formula is used.

math.AP↗