Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.16686

Rigidity results for overdetermined problems of p-Laplace equation with spatially dependent data

Xi Dai
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $p>1$ and $Ω\subset\mathbb{R}^n$ be a bounded smooth domain. We study the following overdetermined problem for the p-Laplacian: \begin{equation*} \left\{ \begin{array}{ll} -Δ_p u=f(x)&x\inΩ,\\ u=0&x\in\partialΩ,\\ \frac{\partial u}{\partialν}=g(x)&x\in\partialΩ, \end{array} \right. \end{equation*} where $ν$ is the inward unit normal to $\partialΩ$ and both data may depend on position. A boundary-local version of the moving-plane argument is developed for the possibly degenerate p-Laplace operator. The advantage of this argument is that it avoids the analysis of the degenerate set of the p-Laplacian. Under some reasonable structural conditions, we prove that the solvability of the above mentioned problem forces the domain to be a ball, the source to be radial, and the Neumann datum to be a suitable constant. We also give an explicit formula for the solution. Further consequences include one directional domain symmetry, the conclusion that full rigidity follows from reflection information in any set of directions whose linear reflections generate a dense subgroup of the orthogonal group $O(n)$, and a constant mean curvature characterization.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xi Dai. 2026-10-12. Rigidity results for overdetermined problems of p-Laplace equation with spatially dependent data. https://arxiv.org/abs/2610.16686

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantization analysis of Moser-Trudinger equations in the Poincaré disk and applications

In this paper, we first establish the quantitative properties for positive solutions to the Moser-Trudinger equation in the two-dimensional Poincaré disk $\mathbb{B}^2$: \begin{equation*}\label{mt1} \left\{ \begin{aligned} &-Δ_{\mathbb{B}^2}u=λue^{u^2},\ x\in\mathbb{B}^2, &u\to0,\ \text{when}\ ρ(x)\to\infty, &\|\nabla_{\mathbb{B}^2} u\|_{L^2(\mathbb{B}^2)}^2\leq M_0, \end{aligned} \right. \end{equation*} where $0<λ<\frac{1}{4}=\inf\limits_{u\in W^{1,2}(\mathbb{B}^2)\backslash\{0\}}\frac{\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2}{\|u\|_{L^2(\mathbb{B}^2)}^2}$, $ρ(x)$ denotes the geodesic distance between $x$ and the origin and $M_0$ is a fixed large positive constant (see Theorem \ref{thm1}). Furthermore, by carefully analyzing the asymptotic expansion of the Dirichlet energy \(\|\nabla_{\mathbb{B}^2} u\|_{L^2(\mathbb{B}^2)}^2\) as \(λ\to 0\), and establishing that \[ \lim_{\substack{λ\to 1/4}} \int_{\mathbb{B}^2} |\nabla_{\mathbb{B}^2} u_λ|^2 \, dV_{\mathbb{B}^2} = +\infty \] for~the ~nonzero~least energy solution, we are able to prove the existence of a critical point for the Moser-Trudinger functional \(F(u) = \int_{\mathbb{B}^2} (e^{u^2} - 1) \, dV_{\mathbb{B}^2}\) under the constraint \(\int_{\mathbb{B}^2} |\nabla_{\mathbb{B}^2} u|^2 \, dV_{\mathbb{B}^2} = Λ\) for \(Λ\in (4π, +\infty)\), up to some Möbius transformation. This result is substantially different from the non-existence of critical points for the Moser-Trudinger functional in the Euclidean disk. Finally, by applying a local Pohozaev identity around the origin and exploiting the uniqueness of solutions to the Cauchy initial value problem for ODEs, we prove that the Moser-Trudinger equation has a unique positive solution up to some Möbius transformation when $λ$ is close to $0$.

math.AP↗
arXiv2026

Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus

We consider the Cauchy problem for semi-linear Schrödinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb T)$ for $s>\frac 52$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem.

math.AP↗
arXiv2026

Lipschitz regularity of fractional $p$-Laplacian

In this article, we investigate the Hölder regularity of the fractional $p$-Laplace equation of the form $(-Δ_p)^s u=f$ where $p>1, s\in (0, 1)$ and $f\in L^\infty_{\rm loc}(Ω)$. Specifically, we prove that $u\in C^{0, γ_\circ}_{\rm loc}(Ω)$ for $γ_\circ=\min\{1, \frac{sp}{p-1}\}$, provided that $\frac{sp}{p-1}\neq 1$. In particular, it shows that $u$ is locally Lipschitz for $\frac{sp}{p-1}>1$. Moreover, we show that for $\frac{sp}{p-1}=1$, the solution is locally Lipschitz, provided that $f$ is locally Hölder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems.

math.AP↗