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Jiongduo Xie

Publications and source records attributed to Jiongduo Xie.

10 recordsLinked to original sources
arXiv2026

A Quantitative Gidas--Ni--Nirenberg Theorem for the Fractional Laplacian

We establish a quantitative Gidas--Ni--Nirenberg theorem for the pure fractional Dirichlet problem \[ (-Δ)^s u=k(x)g(u)\quad\text{in }B_1, \qquad u=0\quad\text{in }\mathbb R^n\setminus B_1, \qquad 0<s<1. \] For positive bounded solutions under a two-sided $L^\infty$ normalization, we prove \[ \mathcal D(u)\le C\,\mathcal D(k)^γ,\qquad 0<γ<1, \] where $\mathcal D$ measures spherical oscillation and outward radial increase. To the best of our knowledge, this provides the first quantitative Gidas--Ni--Nirenberg estimate for the pure fractional Dirichlet problem. The key new ingredient is a shift-uniform, weak Harnack inequality for shifted antisymmetric supersolutions, which overcomes the exterior-tail obstruction in the quantitative moving-plane argument. This mechanism extends to nonseparable nonlinearities.

math.AP↗
arXiv2026

Mass Bounds for Confined Area-Minimizing Minimal Surfaces

We establish codimension-independent mass bounds for geometrically confined area-minimizing rectifiable currents. In Euclidean space, we combine the confined-volume doubling theorem of Colding--Minicozzi with a current-theoretic squashing argument. This gives an affirmative answer to Lin's interior mass-bound problem for every algebraic projection multiplicity $Q$: the interior mass is bounded by $C(n)Q$, without an a priori mass bound at a larger scale. This result yields a degree-one Bernstein-type rigidity result under sublinear confinement. For the hyperbolic application, we make the curvature modification of the fixed-scale argument needed in a thin tubular neighborhood of a totally geodesic copy of $\mathbb{H}^n$. Combining this auxiliary estimate with a localized squashing estimate removes the doubly exponential local mass-growth condition from the boundary regularity results in [13].

math.DG↗
arXiv2026

Asymptotics of high-codimensional area-minimizing currents in hyperbolic space

We investigate the asymptotic behavior of high-codimensional area-minimizing locally rectifiable currents in hyperbolic space, addressing a problem posed by F.H. Lin and establishing ``boundary regularity at infinity" results for such currents near their asymptotic boundaries under the standard Euclidean metric. Intrinsic obstructions to high-order regularity arise for odd-dimensional minimal surfaces, revealing a constraint dependent on the geometry of the asymptotic boundary. Our work advances the asymptotic theory of high-codimensional minimal surfaces in hyperbolic space.

math.DG↗
arXiv2026

Asymptotic Expansions of the Fundamental Solution for a Mixed Local-Nonlocal Operator

In this paper, we study the fundamental solution of the mixed local-nonlocal operator $ -Δ+(-Δ)^s,$ $ 0<s<1. $ We derive precise asymptotic expansions of the fundamental solution and its gradient, up to the first nontrivial correction term, both near the origin and at infinity. The expansions describe the transition between the local and nonlocal diffusion scales and include the critical regimes in which logarithmic terms occur. As applications of these asymptotics, we establish a distributional Bôcher-type theorem for nonnegative supersolutions with an isolated singularity. We further obtain quantitative positive and antisymmetric maximum principles on punctured balls.

math.AP↗
arXiv2026

Liouville Theorems for the Lane-Emden Equation Involving a Mixed Local-Nonlocal Operator

In this paper, we investigate the existence of positive supersolutions for the following mixed local-nonlocal Lane-Emden type equation: $$ -Δu+(-Δ)^s u=u^q\quad\text{in }\mathbb R^n, $$ where $n\geq 3$, $s\in(0,1)$, and $q>1$. More precisely, we prove that the equation admits positive distributional supersolutions if and only if $q>\frac{n}{n-2s}$. In the process, we establish several novel properties of mixed local-nonlocal operators, including sharp asymptotic estimates for the fundamental solution, a maximum principle in the distributional sense, and an equivalent integral inequality for supersolutions.

math.AP↗
arXiv2026

Pointwise regularity of solutions for fully fractional parabolic equations

This paper investigates the higher pointwise regularity of nonnegative classical solutions for fully fractional parabolic equations $(\partial_t -Δ)^{s} u = f,$ where $s\in(0,1)$. We establish $C^{k+α+2s}$ or $C^{k+α+2s,\ln} (k\geq 0,α\in[0,1))$ pointwise regularity according to $α+2s\notin \mathbb{Z}$ or $α+2s\in \mathbb{Z}$, which imply the classical local regularity directly. We provide a simplified and unified proof by introducing novel equivalent definitions for pointwise function spaces. Moreover, the equivalent integral representation and directional average for fractional heat kernel play an important role in our discussion.

math.AP↗
arXiv2026

Solutions of the singular Yamabe problem near singular boundaries

In this paper, we investigate the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive optimal estimates, where the background metrics are not assumed to be conformally flat. Specifically, we demonstrate that for a wide class of Lipschitz domains with asymptotic conical structure, the local positive solutions are well approximated by the positive solutions in the tangent cones at singular boundary points. This extends the results of [10, 12, 26].

math.AP↗