Grad Analysis $\nabla$ Seminar is a graduate seminar at Rutgers University, Department of Mathematics, Orgnized by Qi Ma since Nov. 2024.
Those who wish to receive e-mails for latest information of the seminar may subscribe to our mail-list by sending “Subscribe” to grad_analysis-join@email.rutgers.edu or you could contact Qi Ma at qi.ma@rutgers.edu.
Title：Minimizers for the Area Functional
Abstract：The area functional is of linear growth in the gradient, so a natural space in which to pose the corresponding minimization problem is the W^1,1 Sobolev space. As it turns out, this space is too small, and we should instead consider the larger space of BV functions, and then extend our area functional to this space. I’ll explain what this means, and state some results about structure of BV functions and their singularities that help us study this extended problem.
Title：Infinitely many bubble solutions for the prescribed scalar curvature problem on $\mathbb{S}^N$
Abstract：The prescribed scalar curvature problem on $\mathbb{S}^n$ asks if one can find a conformally invariant metric such that the curvature becomes a fixed smooth function. In the case that the given function is positive, rotationally symmetric and has a local maximum point between the poles, one can construct infinitely many non-radial positive solutions to this problem by applying Lyapunov-Schmidt reduction arguments. The energy of these solutions can be made arbitrarily large.
Title: Regularity Theory of Energy Minimizing Maps between Riemannian Manifolds
Abstract: By analysing the Dirichlet energy of harmonic functions, one can apply variational methods to get regularity of solutions even from weak solutions. However, int he case of harmonic maps, i.e. maps onto Riemannian manifolds, the situation is decidedly different as one has to contend with the geometry of the target. In this talk, we examine harmonic maps and elucidate how a result due to Schoen and Uhlenbeck gives us analogous regularity results.
Title: Pseudo-differential operators and regularity for elliptic PDEs with smooth coefficients.
Abstract: We will start by introducing the Kohn-Nirenberg symbols. We will then introduce Pseudodifferential operators (PsiDO) which enlarges the class of differential operators. We will then study the action of the PsiDOs on Schwartz functions and tempered distributions which allow us to understand certain smoothing properties. Then we will construct elliptic parametrix for elliptic PsiDOs which will in particular show regularity for any elliptic PDE with smooth coefficients, including Sobolev regularity for PDEs in the same class.
Title: Liouville theorems for second order conformally invariant equations and their applications
Abstract: I will present optimal Liouville-type theorems for second order conformally invariant equations. A crucial new ingredient in proving these theorems is our enhanced understanding of solution behaviors near isolated singularities of such equations. These Liouville-type theorems lead to optimal local gradient estimates for a wide class of fully nonlinear elliptic equations involving Schouten (Ricci) tensors. As an application of these Liouville-type theorems and gradient estimates, we establish new existence and compactness results for conformal metrics on a closed Riemannian manifold with prescribed symmetric functions of the Schouten (Ricci) tensor, allowing the scalar curvature of the conformal metrics to have varying signs.