arXiv subjects

Guowei Dai

Publications and source records attributed to Guowei Dai.

At least 19 recordsLinked to original sources

arXiv2026

Uncertainty-Guided Dual-Domain Learning for Reliable Skin Lesion Segmentation

Accurate skin lesion segmentation is vital for dermoscopic Computer-Aided Diagnosis. However, visual ambiguity and morphological irregularity often defeat spatial modeling, necessitating multi-domain architectures. Existing paradigms frequently overlook the active use of prediction uncertainty, leading to deterministic frameworks that suffer from blind cross-domain fusion and overfit to label noise. To address these issues, we propose the Uncertainty-Guided Dual-Domain Network (UGDD-Net). UGDD-Net introduces a novel "Glance-and-Gaze" mechanism to transform uncertainty into an active guiding signal. Specifically, the Uncertainty-Guided Bi-directional Feature Fusion (UGBFF) module uses pixel-level uncertainty to modulate spatial-spectral interactions. The Uncertainty-Guided Graph Refinement (UGGR) module constructs a topology-aware graph to propagate reliable semantic consensus and refine uncertain nodes. Finally, the Uncertainty-Guided Margin-Adaptive Loss (UGML) enforces strict constraints on confident pixels while relaxing penalties on uncertain ones to improve statistical calibration. Extensive experiments on ISIC2017, ISIC2018, PH2, and HAM10000 datasets demonstrate that UGDD-Net achieves state-of-the-art performance, especially on "Hard Samples". Our uncertainty maps align with expert inter-observer variability, providing robust interpretability for human-machine collaborative diagnosis.

eess.IV↗

arXiv2025

Upper bound estimation for the ratio of the first two eigenvalues of Robin Laplacian

The celebrated conjecture by Payne, P\'{o}lya and Weinberger (1956) states that for the fixed membrane problem, the ratio of the first two eigenvalues, $\lambda_2/\lambda_1$, is maximized by a disk. A more general dimensional version of this conjecture was later resolved by Ashbaugh and Benguria in the 1990s. For the Robin Laplacian, Payne and Schaefer (2001) formulated an analogous conjecture, positing that the ratio $\mu_2/\mu_1$ is also maximized by a disk for a range of the boundary parameter $\sigma$. This was later restated by Henrot in 2003. In this work, under some suitable conditions, we affirm this conjecture for all dimensions $N\geq2$ and for all $\sigma>0$. Furthermore, we prove that the maximum value of $\mu_2/\mu_1$ is strictly decreasing in $\sigma$ over the entire interval $(0,+\infty)$. Our result provides a positive answer to a variant of Yau's Problem 77: by measuring the ratio of the first two eigenfrequencies, one can determine whether an elastically supported drum is circular.

math.AP↗

arXiv2025

On the Schiffer and Berenstein conjectures with high-frequency for convex domains in the plane

In this paper, by introducing two-point stationary-phase amplitude defect, we provide a partial positive answer to the Schiffer and Berenstein conjectures in $\mathbb{R}^2$. More precisely, assuming that a bounded uniformly convex domain $\Omega \subset \mathbb{R}^2$ has a connected boundary of class $C^{2,\epsilon}$ with $\epsilon \in (0,1)$, we show that if, for some nonzero constant $c_D$, the overdetermined elliptic problem \begin{equation} -\Delta u = \alpha u \ \text{ in } \ \Omega, \qquad u = 0 \ \text{ on } \ \partial\Omega, \qquad \frac{\partial u}{\partial \nu} = c_{D} \ \text{ on } \ \partial\Omega \nonumber \end{equation} admits a nontrivial solution corresponding to a large eigenvalue $\alpha$, then the domain $\Omega$ must be a disk. Similarly, we establish that if a domain $\Omega \subset \mathbb{R}^2$ has a connected Lipschitz boundary and the problem \begin{equation} -\Delta u = \alpha u \ \text{ in } \ \Omega, \qquad \frac{\partial u}{\partial \nu} = 0 \ \text{ on } \ \partial\Omega, \qquad u = c_{N} \ \text{ on } \ \partial\Omega \nonumber \end{equation} has a nontrivial solution corresponding to a large eigenvalue $\alpha$, then $\Omega$ is a disk as well.

math.AP↗

arXiv2025

Complete spectrum of the Robin eigenvalue problem on the ball

We investigate the following Robin eigenvalue problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=\mu u\,\, &\text{in}\,\, B,\\ \partial_\texttt{n} u+\alpha u=0 &\text{on}\,\, \partial B \end{array} \right. \end{equation*} on the unit ball of $\mathbb{R}^N$. We obtain the complete spectral structure of this problem. In particular, for $\alpha>0$, the first eigenvalue is $k_{\nu,1}^2$ and the second eigenvalue is $k_{\nu+1,1}^2$, where $k_{\nu+l,m}$ is the $m$th positive zero of $kJ_{\nu+l+1}(k)-(\alpha+l) J_{\nu+l}(k)$. Moreover, when $\alpha\in(-l,1-l)$ with any $l\in \mathbb{N}$, one has $l$ negative (strictly increasing) eigenvalues $-\widehat{k}_{\nu+i,1}^2$ with $i\in\{0,\ldots,l-1\}$ where $\widehat{k}_{\nu+l,1}$ denotes the unique zero of $\alpha I_{\nu+l}(k)+lI_{\nu+l}(k)+kI_{\nu+l+1}(k)$; while, for $\alpha=-l$, besides $l$ negative (increasing) eigenvalues, $0$ is also an eigenvalue.

math.AP↗

arXiv2025

Confirmed answer to the Schiffer conjecture and the Berenstein conjecture

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N+1}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha u\,\, \text{in}\,\,\Omega, \,\, u=0\,\,\text{on}\,\, \partial\Omega,\,\,\frac{\partial u}{\partial n} =c\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} has a nontrivial solution, then $\Omega$ is a ball, which is exactly the affirmative answer to the Berenstein conjecture. Similarly, we show that, if $\Omega$ has a Lipschitz connected boundary and the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha u\,\, \text{in}\,\,\Omega, \,\, \frac{\partial u}{\partial n}=0\,\,\text{on}\,\, \partial\Omega,\,\,u =c\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} has a nontrivial solution, then $\Omega$ is also a ball, which is exactly the affirmative answer to the Schiffer conjecture.

math.AP↗

arXiv2024

Regularity and uniqueness to multi-phase problem with variable exponent

In this paper, we consider a new class of multi phase operators with variable exponents, which reflects the inhomogeneous characteristics of hardness changes when multiple different materials are combined together. We at first deal with the corresponding functional spaces, namely the Musielak-Orlicz Sobolev spaces, hence we investigate their regularity properties and the extension of the classical Sobolev embedding results to the new context. Then, we focus on the regularity properties of our operators, and prove that these operators are bounded, continuous, strictly monotone, coercive and satisfy the (S_+)-property. Further, we discuss suitable problems driven by such operators. In particular, we deal with Dirichlet problems in which the nonlinearity is gradient dependent. Under very general assumptions, we establish the existence of a nontrivial solution for such problems. Also, we give additional conditions on the nonlinearity which guarantee the uniqueness of solution. Lastly, we produce some local regularity results (namely, Caccioppoli-type inequality, Sobolev-Poincar\'e-type inequalities and higher integrability) for minimizers of the integral functionals corresponding to the operators.

math.AP↗

arXiv2024

Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space

We construct nontrivial unbounded domains $\Omega$ in the hyperbolic space $\mathbb{H}^N$, $N \in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -\Delta_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} has a positive bounded solution in $C^{2,\alpha}\left(\Omega\right) \cap H^1\left(\Omega\right)$. We also give a condition under which this construction holds for larger dimensions $N$. This is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems, and, as far as we know, is the first nontrivial example of solution to an overdetermined elliptic problem in the hyperbolic space.

math.AP↗

arXiv2023

Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder

Fix $R\in(0,1)$ and $N\geq2$, and let $\lambda_k$ be the $k$th eigenvalue of the radial Dirichlet problem on the annulus $A_R:=\{x\in\mathbb R^N:R<|x|<1\}$. For every $k\geq1$ we construct a smooth local family of rotationally symmetric, axially periodic perturbations of the hollow cylinder $A_R\times\mathbb R$ for which \begin{equation} -\Delta u=\lambda u\quad\hbox{in }\Omega,\qquad u=0\quad\hbox{on }\partial\Omega_{\mathrm{out}}\cup\partial\Omega_R,\qquad \partial_\nu u=\hbox{constant}\quad\hbox{on }\partial\Omega_{\mathrm{out}} \nonumber \end{equation} admits a nonradial periodic solution. The branch bifurcates from the $k$th radial eigenfunction. If $k\geq2$, the solution is sign-changing: it has precisely $k-1$ smooth nested nodal hypersurfaces and hence exactly $k$ nodal domains. We also give an explicit connected-strip analogue when $N=1$. The proof is based on a mean-free Dirichlet-to-Neumann operator. A Sturm--Liouville Weyl function yields a unique simple crossing in the first spectral gap, so that the Crandall--Rabinowitz theorem produces a smooth local branch. This gives a partially overdetermined construction on an unbounded hollow domain with disconnected complement; it lies outside the positivity and connected-complement hypotheses of the Berestycki--Caffarelli--Nirenberg conjecture.

math.AP↗

arXiv2023

Sign-changing solutions to Schiffer's overdetermined problem on wavy cylinder

In this paper, we prove the existence of $k$ families of smooth unbounded domains $\Omega_s\subset\mathbb{R}^{N+1}$ with $N\geq1$, where \begin{equation} \Omega_s=\left\{(x,t)\in \mathbb{R}^N\times \mathbb{R}:\vert x\vert<1+s\cos \left(\frac{2\pi}{T(s)}t\right)+s w_s\left(\frac{2\pi}{T(s)}t\right)\right\},\nonumber \end{equation} such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, \partial_\nu u=0,\,\,u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} admits a bounded sign-changing solution with exactly $k+1$ nodal domains. These results can be regarded as counterexamples to the Schiffer conjecture on unbounded domain. These results also indicate that there exist non-spherical unbounded regions without Pompeiu property. Our construction shows that the condition "$\partial\Omega$ is homeomorphic to the unit sphere" is necessary for Williams conjecture to hold. In addition, these conclusions may have potential applications in remote sensing or CT.

math.AP↗

arXiv2023

Sufficient conditions for the existence of path-factors with given properties

A spanning subgraph $H$ of a graph $G$ is called a $P_{\geq k}$-factor of $G$ if every component of $H$ is isomorphic to a path of order at least $k$, where $k\geq2$ is an integer. A graph $G$ is called a $(P_{\geq k},l)$-factor critical graph if $G-V'$ contains a $P_{\geq k}$-factor for any $V'\subseteq V(G)$ with $|V'|=l$. A graph $G$ is called a $(P_{\geq k},m)$-factor deleted graph if $G-E'$ has a $P_{\geq k}$-factor for any $E'\subseteq E(G)$ with $|E'|=m$. Intuitively, if a graph is dense enough, it will have a $P_{\geq 3}$-factor. In this paper, we give some sufficient conditions for a graph to be a $(P_{\geq 3},l)$-factor critical graph or a $(P_{\geq 3},m)$-factor deleted graph. In this paper, we demonstrate that (i) $G$ is a $(P_{\geq 3},l)$-factor critical graph if its sun toughness $s(G)>\frac{l+1}{3}$ and $\kappa(G)\geq l+2$. (ii) $G$ is a $(P_{\geq 3},l)$-factor critical graph if its degree sum $\sigma_3(G)\geq n+2l$ and $\kappa(G)\geq l+1$. (iii) $G$ is a $(P_{\geq 3},m)$-factor deleted graph if its sun toughness $s(G)\geq \frac{m+1}{m+2}$ and $\kappa(G)\geq 2m+1$. (iv) $G$ is a $(P_{\geq 3},m)$-factor deleted graph if its degree sum $\sigma_3(G)\geq n+2m$ and $\kappa(G)\geq 2m+1$.

math.CO↗

arXiv2023

Sign-changing solution for an overdetermined elliptic problem on unbounded domain

We prove the existence of two smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} admits a sign-changing solution. The domains bifurcate from the straight cylinder $B_1\times \mathbb{R}$, where $B_1$ is the unit ball in $\mathbb{R}^N$. These results can be regarded as counterexamples to the Berenstein conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall-Rabinowitz type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal.

math.AP↗

arXiv2023

Interval bifurcation theorems for Fredholm operator and its application to an elliptic overdetermined problem in bounded domains

We establish local interval bifurcation theorem and global interval bifurcation theorem for Fredholm operator with index $0$ via $0$-group. As one of applications, we investigate the existence of a family of nontrivial domains $\Omega_{\rho}\subset \mathbb{R}^N$ ($N=2,3$ or $4$), bifurcating from a small ball, such that the problem \begin{equation} -\Delta u=u-\left(u^+\right)^3\,\, \text{in}\,\,\Omega_{\rho}, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega_{\rho} \nonumber \end{equation} has a sign-changing bounded solution. Compared with the recent result \cite[Theorem 2.1]{Ruiz}, here we obtain a family of domains $\Omega_{\rho}$ instead of a sequence of domains.

math.AP↗

arXiv2022

Global bifurcation structure and geometric properties for steady periodic water waves with vorticity

This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work \cite{ConstantinStrauss}, we first obtain two continuous bifurcation curves which meet the laminar flow only one time by using modified analytic bifurcation theorem. They are symmetric waves whose profiles are monotone between each crest and trough. Furthermore, we find that there is at least one inflection point on the wave profile between successive crests and troughs and the free surface is strictly concave at any crest and strictly convex at any trough. In addition, for favorable vorticity, we prove that the vertical displacement of water waves decreases with depth.

math.AP↗

arXiv2022

Existence of solutions for singular double phase problems via the Nehari manifold method

In this paper we study quasilinear elliptic equations driven by the double phase operator and a right-hand side which has the combined effect of a singular and of a parametric term. Based on the fibering method by using the Nehari manifold we are going to prove the existence of at least two weak solutions for such problems when the parameter is sufficiently small.

math.AP↗

arXiv2020

Component factors in $K_{1,r}$-free graphs

A graph is said to be $K_{1,r}$-free if it does not contain an induced subgraph isomorphic to $K_{1,r}$. An $\mathcal{F}$-factor is a spanning subgraph $H$ such that each connected component of $H$ is isomorphic to some graph in $\mathcal{F}$. In particular, $H$ is called an $\{P_2,P_3\}$-factor of $G$ if $\mathcal{F}=\{P_2,P_3\}$; $H$ is called an $\mathcal{S}_n$-factor of $G$ if $\mathcal{F}=\{K_{1,1},K_{1,2},K_{1,3},...,K_{1,n}\}$, where $n\geq2$. A spanning subgraph of a graph $G$ is called a $\mathcal{P}_{\geq k}$-factor of $G$ if its each component is isomorphic to a path of order at least $k$, where $k\geq2$. A graph $G$ is called a $\mathcal{F}$-factor covered graph if there is a $\mathcal{F}$-factor of $G$ including $e$ for any $e\in E(G)$. In this paper, we give a minimum degree condition for a $K_{1,r}$-free graph to have an $\mathcal{S}_n$-factor and a $\mathcal{P}_{\geq 3}$-factor, respectively. Further, we obtain sufficient conditions for $K_{1,r}$-free graphs to be $\mathcal{P}_{\geq 2}$-factor, $\mathcal{P}_{\geq 3}$-factor or $\{P_2,P_3\}$-factor covered graphs. In addition, examples show that our results are sharp.

math.CO↗

arXiv2016

Spectrum of Navier $p$-biharmonic problem with sign-changing weight

In this paper, we consider the following eigenvalue problem {{l} (|u"|^{p-2}u")"=λm(x)|u|^{p-2}u, x\in (0,1), u(0)=u(1)=u"(0)=u"(1)=0, where $1<p<+\infty$, $λ$ is a real parameter and $m$ is sign-changing weight. We prove there exists a unique sequence of eigenvalues for above problem. Each eigenvalue is simple and continuous with respect to $p$, the $k$-th eigenfunction, corresponding to the $k$-th positive or negative eigenvalue, has exactly $k-1$ generalized simple zeros in $(0,1)$.

math.CA↗

arXiv2015

Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros

In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.

math.AP↗

arXiv2014

Two Whyburn type topological theorems and its applications to Monge-Ampère equations

In this paper we correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge-Ampère equation \begin{eqnarray} \left\{ \begin{array}{lll} \det\left(D^2u\right)=λ^N a(x)f(-u)\,\, &\text{in}\,\, Ω,\\ u=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial Ω. \end{array} \right.\nonumber \end{eqnarray} We establish global bifurcation results for the problem. We find intervals of $λ$ for the existence, multiplicity and nonexistence of strictly convex solutions for this problem.

math.FA↗