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Guowei Dai

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math.AP math.CA Computer Vision eess.IV math.FA

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preprint2026arXiv

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.

preprint2022arXiv

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.

preprint2022arXiv

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.

preprint2016arXiv

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 $(0,1)$ .

preprint2015arXiv

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.

preprint2014arXiv

Eigenvalue, global bifurcation and positive solutions for a class of fully nonlinear problems

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_Ω\vert \nabla u\vert^2\,dx\right)Δu=λu+h(x,u,λ)\,\,\text{in}\,\, Ω,\\ u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}\,\,Ω. \end{array} \right.\nonumber \end{equation} Under some natural hypotheses on $h$ , we show that $\left(aλ_1,0\right)$ is a bifurcation point of the above problem. As applications of the above result, we shall determine the interval of $λ$ , in which there exist positive solutions for the above problem with $h(x,u;λ)=λf(x,u)-λu$ , where $f$ is asymptotically linear at zero and is asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we also provide a positive answer to an open problem involving the case of $a=0$ .

preprint2014arXiv

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.

preprint2012arXiv

Eigenvalue, bifurcation, existence and nonexistence of solutions for Monge-Ampère equations

In this paper we study the following eigenvalue boundary value problem for Monge-Ampère equations: {equation} \{{array}{l} \det(D^2u)=λ^N f(-u)\,\, \text{in}\,\, Ω, u=0,\,\text{on}\,\, \partial Ω. {array}. {equation} We establish the unilateral global bifurcation results for the problem with $f(u)=u^N+g(u)$ and $Ω$ being the unit ball of $\mathbb{R}^N$ . More precisely, under some natural hypotheses on the perturbation function $g:\mathbb{R}\rightarrow\mathbb{R}$ , we show that $(λ_1,0)$ is a bifurcation point of the problem and there are two distinct unbounded continua of one-sign solutions, where $λ_1$ is the first eigenvalue of the problem with $f(u)=u^N$ . As the applications of the above results, we consider with determining interval of $λ$ , in which there exist solutions for this problem in unit ball. Moreover, we also get some results on the existence and nonexistence of convex solutions for this problem in general domain by domain comparison method.

preprint2012arXiv

Eigenvalues, bifurcation and one-sign solutions for the periodic $p$ -Laplacian

In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$ -Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues $λ_0^+$ and $λ_0^-$ . Furthermore, under some natural hypotheses on perturbation function, we show that $(λ_0^ν,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $\mathscr{C}_ν^{+}$ and $\mathscr{C}_ν^{-}$ , consisting of the continuum $\mathscr{C}_ν$ emanating from $(λ_0^ν, 0)$ , where $ν\in\{+,-\}$ . As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $λ$ are also studied.

preprint2012arXiv

Maximum principle and one-sign solutions for the elliptic $p$ -Laplacian

In this paper, we prove a maximum principle for the $p$ -Laplacian with a sign-changing weight. As an application of this maximum principle, we study the existence of one-sign solutions for a class of quasilinear elliptic problems.

preprint2012arXiv

Unilateral global bifurcation and nodal solutions for the $p$ -Laplacian with sign-changing weight

In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(μ_k^ν(p),0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^ν)^+$ and $(\mathcal{C}_{k}^ν)^-$ , consisting of the bifurcation branch $\mathcal{C}_{k}^ν$ from $(μ_k^ν(p), 0)$ , where $μ_k^ν(p)$ is the $k$ -th positive or negative eigenvalue of the linear problem corresponding to the above problems, $ν\in\{+,-\}$ . As the applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Drábek and Huang (1997) [\ref{DH}], we study the existence of one-sign solutions for a class of high dimensional quasilinear elliptic problems with sign-changing weight.

preprint2012arXiv

Unilateral global bifurcation for fourth-order eigenvalue problems with sign-changing weight

In this paper, we shall establish the unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(μ_k^ν,0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^ν)^+$ and $(\mathcal{C}_{k}^ν)^-$ , consisting of the bifurcation branch $\mathcal{C}_{k}^ν$ from $(μ_k^ν, 0)$ , where $μ_k^ν$ is the $k$ -th positive or negative eigenvalue of the linear problem corresponding to the above problems, $ν\in{+,-}$ . As the applications of the above result, we study the existence of nodal solutions for a class of fourth-order eigenvalue problems with sign-changing weight. Moreover, we also establish the Sturm type comparison theorem for fourth-order problems with sign-changing weight.

preprint2012arXiv

Unilateral global interval bifurcation theorem for $p$ -Laplacian and its applications

In this paper, we establish a unilateral global bifurcation result from interval for a class of $p$ -Laplacian problems. By applying the above result, we study the spectrum of a class of half-quasilinear problems. Moreover, we also investigate the existence of nodal solutions for a class of half-quasilinear eigenvalue problems.

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Guowei Dai

Research overview

Is this your work?

13 publications

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

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

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

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

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

Eigenvalue, global bifurcation and positive solutions for a class of fully nonlinear problems

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

Eigenvalue, bifurcation, existence and nonexistence of solutions for Monge-Ampère equations

Eigenvalues, bifurcation and one-sign solutions for the periodic $p$ -Laplacian

Maximum principle and one-sign solutions for the elliptic $p$ -Laplacian

Unilateral global bifurcation and nodal solutions for the $p$ -Laplacian with sign-changing weight

Unilateral global bifurcation for fourth-order eigenvalue problems with sign-changing weight

Unilateral global interval bifurcation theorem for $p$ -Laplacian and its applications

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Guowei Dai

Research overview

Is this your work?

13 publications

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

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

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

Spectrum of Navier ppp-biharmonic problem with sign-changing weight

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

Eigenvalue, global bifurcation and positive solutions for a class of fully nonlinear problems

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

Eigenvalue, bifurcation, existence and nonexistence of solutions for Monge-Ampère equations

Eigenvalues, bifurcation and one-sign solutions for the periodic ppp-Laplacian

Maximum principle and one-sign solutions for the elliptic ppp-Laplacian

Unilateral global bifurcation and nodal solutions for the ppp-Laplacian with sign-changing weight

Unilateral global bifurcation for fourth-order eigenvalue problems with sign-changing weight

Unilateral global interval bifurcation theorem for ppp-Laplacian and its applications

See the researcher in context

Building this map preview

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

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

Eigenvalues, bifurcation and one-sign solutions for the periodic $p$ -Laplacian

Maximum principle and one-sign solutions for the elliptic $p$ -Laplacian

Unilateral global bifurcation and nodal solutions for the $p$ -Laplacian with sign-changing weight

Unilateral global interval bifurcation theorem for $p$ -Laplacian and its applications