arXiv subjects

Yahong Guo

Publications and source records attributed to Yahong Guo.

16 recordsLinked to original sources

arXiv2026

Classification of global solutions to the singular equation $-Δu=f(X)\cdot u^{-γ}$ in a Lipschitz epigraphical cone

We construct and classify all global solutions to the singular equation $$-Δu=f(X)\cdot u^{-γ}$$ supported in a general Lipschitz epigraphical cone, where $f(X)$ is a locally Dini continuous function with $0<λ\leq f(X)\leqΛ$. The existence and non-existence of a global solution is solely determined by the exponent $γ$ of the equation and the ``frequency" of the cone. Moreover, in order to classify all global solutions, we introduce several new methods. First, we use the local data to estimate the global growth rate, which in turn establishes the boundedness of the ``asymptotic slope" of the global solution. Second, by establishing a nonlinear variant of Kemper's boundary Harnack principle, we classify all global solutions through an ``oscillation reduction" argument on the ``asymptotic slope".

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

Classification of solutions to a weighted singular fractional problem in the half space

We focus on the classification of positive solutions to $(-Δ)^s u=\frac{x_n^α}{u^γ}$ in the half space with $γ>0$, subject to the Dirichlet condition. We show that when $-2s<α<(γ-1)s$, all positive solutions exhibit one-dimensional symmetry and are monotone increasing in $x_n$. Moreover, we provide a complete classification of all such one-dimensional solutions via their ``asymptotic $s$-order slope". When $α$ lies outside this range, we demonstrate the nonexistence of global positive solutions.

math.AP↗

arXiv2026

Boundary regularity theory of the singular Lane-Emden-Fowler equation in a Lipschitz domain

We study the singular Lane-Emden-Fowler equation \begin{equation} -Δu=f(X)\cdot u^{-γ} \end{equation} in a bounded Lipschitz domain $Ω$, with the Dirichlet boundary condition and a positive, bounded function $f(X)$. A distinguishing feature is that the vanishing boundary condition introduces a singularity in the equation. We focus on the well-posedness of the equation and the growth rate of solutions near the boundary. The key is to classify the limiting cone of a boundary point into three categories based on its "frequency", and obtain distinct growth rate estimates for each case. Additionally, we discuss the boundary Harnack principle for the singular Lane-Emden-Fowler equation, which is essential in deriving the boundary growth rate estimate. To our knowledge, the boundary Harnack principle we derive is the first Kemper-type estimate for singular semi-linear equations. It notably differs from the classical one for linear equations, in particular, the boundedness of the ratio $u/v$ does not imply its continuity. To address the lack of a suitable upper barrier, we introduce new techniques, including constructing upper barriers iteratively. We also construct a subharmonic auxiliary function $V(X)$ related to the solution $u$ in the limiting cone. The growth rate of $u(X)$ is then obtained inductively from the growth rate of the auxiliary function $V(X)$. Our results and methods offer novel insights into the behavior of singular elliptic equations in non-smooth domains.

math.AP↗

arXiv2026

A convergence result for the master operator

In this paper, we establish a convergence result for the fully fractional heat operator $\ma{s}$, also known as the master operator, stated as follows: \[\mbox{If\ }u_i\to u\ \mbox{in}\ C^{2,1}_{x,t,loc}(\R^n\times\R),\ \mbox{then}\ \ma{s} u_i\to \ma{s}u-b\ \mbox{a.e. in}\ \R^n\times\R,\] for some nonnegative constant $b$. This result addresses a fundamental question in the blow-up and rescaling analysis, which are essential for establishing a priori estimates for solutions of master equations. Additionally, we present examples demonstrating that in certain cases, the constant $b$ can indeed be positive. This highlights a key distinction between nonlocal and local operators: for a local heat operator, such as $\partial_t - \lap$, it is well-known that $b \equiv 0$.

math.AP↗

arXiv2026

Monotonicity for the fractional semi-linear problem in a half space

In this paper, we study semilinear fractional equations $$(-Δ)^s u(x) = f(u(x))$$ in a half-space and prove that all positive solutions are strictly increasing in the $x_n$-direction. Previous results typically require the solution $u$ to be globally bounded in $\mathbb{R}^n$. We substantially weaken this condition by assuming only that $u$ be bounded in each slab. Moreover, our analysis relies solely on the local Lipschitz continuity of the nonlinearity $f$, which is weaker than the conditions imposed in earlier works. As a crucial ingredient, we obtained a boundary H\"{older} regularity estimate that requires only the boundedness of $u$ near the boundary. This represents a significant improvement over existing results, which often assumed global boundedness of $u$ throughout $\mathbb{R}^n$. The proof introduces a new idea that may be of independent interest. To derive the monotonicity, we employ the method of moving planes. We first obtain a narrow region principle in unbounded domains, which ensures that the moving plane procedure can be initiated from $x_n = 0$. We then establish two averaging effects for the solutions to fractional equations. These key ingredients guarantee that the planes can be moved continuously all the way to $x_n = \infty$. Previously, narrow region principle can only be applied to a single narrow region. It is for the first time that we establish a multiple narrow region principle that can be applied simultaneously to finitely many narrow regions. Compared with the traditional approaches, methods based on the {\em averaging effect} require substantially weaker regularity assumptions and can even accommodate unbounded solutions. We believe that these new ideas and techniques develop here will serve as powerful tools in studying qualitative properties of solutions to fractional equations.

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

Methods in studying qualitative properties of fractional equations

In this paper, we systematically review a series of effective methods for studying the qualitative properties of solutions to fractional equations. Beginning with the pioneering extension method and the method of moving planes in integral forms, we introduce a variety of direct methods, including the direct method of moving planes, the method of moving spheres, blow-up and rescaling techniques, the sliding method, regularity lifting, and approaches for interior and boundary regularity estimates. To elucidate the core ideas behind these methods, we employ simple examples that demonstrate how they can be applied to investigate qualitative properties of solutions. We also provide a comparative discussion of their respective strengths and limitations. It is our hope that this paper will serve as a useful handbook for researchers engaged in the study of fractional equations.

math.AP↗

arXiv2026

Master equations with indefinite nonlinearities

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator \begin{equation} (\partial_t -Δ)^{s} u(x,t) = x_1u^p(x,t)\ \ \mbox{in}\ \R^n\times\R , \end{equation} where $s\in(0,1)$, and $-\infty < p < \infty$. Under mild conditions, we prove that there is no positive bounded solutions. To this end, we first show that the solutions are strictly increasing along $x_1$ direction by employing the direct method of moving planes. Then by constructing an unbounded sub-solution, we derive the nonexistence of bounded solutions. To circumvent the difficulties caused by the fully fractional master operator, we introduced some new ideas and novel approaches that, as we believe, will become useful tool in studying a variety of other fractional elliptic and parabolic problems.

math.AP↗

arXiv2026

Regularity of solutions for fully fractional parabolic equations

In this paper, we study the fully fractional heat equation involving the master operator: $$ (\partial_t -Δ)^{s} u(x,t) = f(x,t)\ \ \mbox{in}\ \mathbb{R}^n\times\mathbb{R} , $$ where $s\in(0,1)$ and $f(x,t) \geq 0$. First we derive Hölder and Schauder estimates for nonnegative solutions of this equation. Due to the {\em nonlocality} of the master operator, existing results (cf. \cite{ST}) rely on global bounds of the solutions $u$ to control their higher local norms. However, such results are inadequate for blow-up and rescaling analysis aimed at obtaining a priori estimates for solutions to {\em nonlocal } equations on unbounded domains, as the global norms of the rescaled functions may diverge. This limitation raises to a natural and challenging question: {\em Can local bounds of solutions replace global bounds to control their higher local norms?} Here, we provide an affirmative answer to this question for nonnegative solutions. To achieve this, we introduced several new ideas and novel techniques. One of the key innovations is to use a {\em directional perturbation average} to derive an important estimate for the fully fractional heat kernel, as stated in Lemma \ref{key0}. We believe this estimate, along with other new techniques introduced here, will serve as powerful tools in regularity estimates for a wide range of nonlocal equations. Building on this breakthrough, we employ the blow-up and rescaling arguments to establish a priori estimates for solutions to a broader class of nonlocal equations in unbounded domains, such as $$(\partial_t -Δ)^{s} u(x,t) = b(x,t) |\nabla_x u (x,t)|^q + f(x, u(x,t))\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R}.$$ Under appropriate conditions, we prove that all nonnegative solutions, along with their spatial gradients, are uniformly bounded.

math.AP↗

arXiv2026

Boundary regularity and a priori estimates for fractional equations on unbounded domains

In this paper, we study the boundary Hölder regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \begin{equation*} \begin{cases} (-Δ)^s u(x) = g(x),&\text{in } Ω, u(x)=0, &\text{in } Ω^c. \end{cases} \end{equation*} Existing results rely on the global $L^{\infty}$ norm of solutions to control their boundary $C^s$ norm, which is insufficient for blow-up and rescaling analysis to obtain a priori estimates in unbounded domains. To overcome this limitation, we first derive a local version of boundary Hölder regularity for nonnegative solutions in which we replace the global $L^{\infty}$ norm by only a local $L^{\infty}$ norm. Then as an important application, we establish a priori estimates for nonnegative solutions to a family of nonlinear equations on unbounded domains with boundaries.

math.AP↗

arXiv2026

Liouville type theorems for dual nonlocal evolution equations involving Marchaud derivatives

In this paper, we establish a Liouville type theorem for the homogeneous dual fractional parabolic equation \begin{equation} \partial^α_t u(x,t)+(-Δ)^s u(x,t) = 0\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R} . \end{equation} where $0<α,s<1$. Under an asymptotic assumption $$\liminf_{|x|\rightarrow\infty}\frac{u(x,t)}{|x|^γ}\geq 0 \; ( \mbox{or} \; \leq 0) \,\,\mbox{for some} \;0\leqγ\leq 1, $$ in the case $\frac{1}{2}<s < 1$, we prove that all solutions in the sense of distributions of above equation must be constant by employing a method of Fourier analysis. Our result includes the previous Liouville theorems on harmonic functions \cite{ABR} and on $s$-harmonic functions \cite{CDL} as special cases and it is still novel even restricted to one-sided Marchaud fractional equations, and our methods can be applied to a variety of dual nonlocal parabolic problems. In the process of deriving our main result, through very delicate calculations, we obtain an optimal estimate on the decay rate of $\left[D_{\rm right}^α+(-Δ)^s\right] φ(x,t)$ for functions in Schwartz space. This sharp estimate plays a crucial role in defining the solution in the sense of distributions and will become a useful tool in the analysis of this family of equations.

math.AP↗

arXiv2023

A Liouville Theorem and Radial Symmetry for dual fractional parabolic equations

In this paper, we first study the dual fractional parabolic equation \begin{equation*} \partial^α_t u(x,t)+(-Δ)^s u(x,t) = f(u(x,t))\ \ \mbox{in}\ \ B_1(0)\times\R , \end{equation*} subject to the vanishing exterior condition. We show that for each $t\in\R$, the positive bounded solution $u(\cdot,t)$ must be radially symmetric and strictly decreasing about the origin in the unit ball in $\R^n$. To overcome the challenges caused by the dual non-locality of the operator $\partial^α_t+(-Δ)^s$, some novel techniques were introduced. Then we establish the Liouville theorem for the homogeneous equation in the whole space \begin{equation*}\label{B} \partial^α_t u(x,t)+(-Δ)^s u(x,t) = 0\ \ \mbox{in}\ \ \R^n\times\R . \end{equation*} We first prove a maximum principle in unbounded domains for anti-symmetric functions to deduce that $u(x,t)$ must be constant with respect to $x.$ Then it suffices for us to establish the Liouville theorem for the Marchaud fractional equation \begin{equation*} \partial^α_t u(t) = 0\ \ \mbox{in}\ \ \R . \end{equation*} To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator $\partial_t^α$, we bring in some new ideas and simpler approaches. Instead of disturbing the anti-symmetric function, we employ a perturbation technique directly on the solution $u(t)$ itself. This method provides a more concise and intuitive route to establish the Liouville theorem for one-sided operators $\partial_t^α$, including even more general Marchaud time derivatives.

math.AP↗

arXiv2023

Liouville theorem for fully fractional master equations and its applications

In this paper, we study the fully fractional master equation \begin{equation}\label{pdeq1} (\partial_t-Δ)^s u(x,t) =f(x,t,u(x,t)),\,\,(x, t)\in \mathbb{R}^n\times \mathbb{R}. \end{equation} First we prove a Liouville type theorem for the homogeneous equation \begin{equation}\label{pdeq0} (\partial_t-Δ)^s u(x,t) = 0,\,\,(x, t)\in \mathbb{R}^n\times \mathbb{R}, \end{equation} where $0<s<1$. When $u$ belongs to the slowly increasing function space $$\mathcal{L}^{2s,s}(\mathbb{R}^n\times\mathbb{R})=\left\{u(x,t) \in L^1_{\rm loc} (\mathbb{R}^n\times\mathbb{R}) \mid \int_{-\infty}^{+\infty} \int_{\mathbb{R}^n} \frac{|u(x,t)|}{1+|x|^{n+2+2s}+|t|^{\frac{n}{2}+1+s}}\operatorname{d}\!x\operatorname{d}\!t<\infty\right\} $$ and satisfies an additional asymptotic assumption $$\liminf_{|x|\rightarrow\infty}\frac{u(x,t)}{|x|^γ}\geq 0 \; ( \mbox{or} \; \leq 0) \,\,\mbox{for some} \;0\leqγ\leq 1, $$ in the case $\frac{1}{2}<s < 1$, we prove that all solutions of (\ref{pdeq0}) must be constant. This result includes the previous Liouville theorems on harmonic functions \cite{ABR} and on $s$-harmonic functions \cite{CDL} as special cases. Then we establish the equivalence between nonhomogeneous pseudo-differential equations (\ref{pdeq1}) and the corresponding integral equations. We believe that these integral equations will become very useful tools in further analysing qualitative properties of solutions, such as regularity, monotonicity, and symmetry. In the process of deriving the Liouville type theorem, through very delicate calculations, we obtain an optimal estimate on the decay rate of $(\partial_t-Δ)_{\rm right}^s φ(x,t)$. This sharp estimate will become a key ingredient and an important tool in investigating master equations.

math.AP↗

arXiv2023

Radial symmetry and Liouville theorem for master equations

This paper has two primary objectives. The first one is to demonstrate that the solutions of master equation \begin{equation*} (\partial_t-Δ)^s u(x,t) =f(u(x, t)), \,\,(x, t)\in B_1(0)\times \mathbb{R}, \end{equation*} subject to the vanishing exterior condition, are radially symmetric and strictly decreasing with respect to the origin in $B_1(0)$ for any $t\in \mathbb{R}$. Another one is to establish the Liouville theorem for homogeneous master equation \begin{equation*} (\partial_t-Δ)^s u(x,t)=0 ,\,\, \mbox{in}\,\, \mathbb{R}^n\times\mathbb{R}, \end{equation*} which states that all bounded solutions must be constant. We propose a new methodology for a direct method of moving planes applicable to the fully fractional heat operator $(\partial_t-Δ)^s$, and the proof of our main results based on this direct method involves the perturbation technique, limit argument as well as Fourier transform. This study opens up a way to investigate the geometric behavior of master equations, and provides valuable insights for establishing qualitative properties of solutions and even for deriving important Liouville theorems for other types of fractional order parabolic equations.

math.AP↗

arXiv2023

Sharp Eyes: A Salient Object Detector Working The Same Way as Human Visual Characteristics

Current methods aggregate multi-level features or introduce edge and skeleton to get more refined saliency maps. However, little attention is paid to how to obtain the complete salient object in cluttered background, where the targets are usually similar in color and texture to the background. To handle this complex scene, we propose a sharp eyes network (SENet) that first seperates the object from scene, and then finely segments it, which is in line with human visual characteristics, i.e., to look first and then focus. Different from previous methods which directly integrate edge or skeleton to supplement the defects of objects, the proposed method aims to utilize the expanded objects to guide the network obtain complete prediction. Specifically, SENet mainly consists of target separation (TS) brach and object segmentation (OS) branch trained by minimizing a new hierarchical difference aware (HDA) loss. In the TS branch, we construct a fractal structure to produce saliency features with expanded boundary via the supervision of expanded ground truth, which can enlarge the detail difference between foreground and background. In the OS branch, we first aggregate multi-level features to adaptively select complementary components, and then feed the saliency features with expanded boundary into aggregated features to guide the network obtain complete prediction. Moreover, we propose the HDA loss to further improve the structural integrity and local details of the salient objects, which assigns weight to each pixel according to its distance from the boundary hierarchically. Hard pixels with similar appearance in border region will be given more attention hierarchically to emphasize their importance in completeness prediction. Comprehensive experimental results on five datasets demonstrate that the proposed approach outperforms the state-of-the-art methods both quantitatively and qualitatively.

cs.CV↗