Juncheng Wei

Professor

Research Classification

Mathematics

Research Interests

Nonlinear partial differential equations
applied and geometric analysis
reaction-diffusion systems
singular perturbations and concentration phenomena
Mathematical biology
singularity formations in fluids

Relevant Degree Programs

 

Research Methodology

qualitative analysis

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Master's students
Doctoral students
Any time / year round

De Giorgi conjectures, fractional equations, blow-ups in nonlinear parabolic and elliptic equations

I support public scholarship, e.g. through the Public Scholars Initiative, and am available to supervise students and Postdocs interested in collaborating with external partners as part of their research.
I am open to hosting Visiting International Research Students (non-degree, up to 12 months).

Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
New solutions to local and non-local elliptic equations (2018)

We obtain a few existence results for elliptic equations.We develop in Chapter 2 a new infinite dimensional gluing scheme for fractionalelliptic equations in the mildly non-local setting. Here it is applied to the catenoid. As a consequence of this method, a counter-example to a fractional analogue of De Giorgi conjecture can be obtained [51].Then, in Chapter 3, we construct singular solutions to the fractional Yamabe problem using conformal geometry. Fractional order ordinary differential equations are studied.Finally, in Chapter 4, we obtain the existence to a suitably perturbed doublycritical Hardy–Schr¨odinger equation in a bounded domain in the hyperbolic space.

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Some new results on the SU(3) Toda system and Lin-Ni problem (2015)

In this thesis, we mainly consider two problems. First, we study the SU(3)Toda system. Let (M,g) be a compact Riemann surface with volume 1, h₁and h₂ be a C¹ positive function on M and p1; p2 ∈ ℝ⁺. The SU(3) Todasystem is the following one on the compact surface M[Formula and equation omitted]where ∆ is the Beltrami-Laplace operator, αq ≥ 0 for every q ∈ S₁, S₁ ⊂ M,Bq ≥ 0 for every q ∈ S₂,S₂ ⊂ M and q is the Dirac measure at q ∈ M. Weinitiate the program for computing the Leray-Schauder topological degreeof SU(3) Toda system and succeed in obtaining the degree formula for p1 ∈(0,4π)(4π,8π), p2 ∉ 4πℕ when S₁ = S₂ = 0.Second, we consider the following nonlinear elliptic Neumann problem{∆u-μu +uq =0 in Ω,u > 0 in Ω,au/av=0 on aΩ. where q=n+2/n-2, μ > 0 and Ω is a smooth and bounded domain in ℝn. Linand Ni (1986) conjectured that for μ small, all solutions are constants. Inthe second part of this thesis, we will show that this conjecture is false fora general domain in n = 4, 6 by constructing a nonconstant solution.

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Master's Student Supervision (2010 - 2018)
Effect of geometry on the behavior of steady Newtonian fluid in a multiply connected domain (2014)

We start with the prototype problem of flow of a Newtonian fluid in the annular region between two infinitely long circular cylinders, with the givenvelocity and temperature on the boundaries of the domain.Then we will try to find out how does geometry affect the behavior of the flow inside of the domain. We will explore two invariant mappingsK_T and K_psi , such that under appropriate conditions on the boundary, the mapping K_T would preserve solution of temperature field from one domain to another and the mapping k_psi would preserve solution of velocity field.We will prove that if a mapping is conformal, it would preserve the convection-diffusion equation in both domains. After that, we will find which subsets of the conformals would also preserve the velocity field as well. In order to answer that question, we will obtain the required condition for the mapping , such that it would preserve both velocity and temperature fields, from one domain to another.

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