**Guohuan Qiu**(Academy of Mathematics and Systems Science)**Liming Sun**(Academy of Mathematics and Systems Science)**Tencent meeting**: 882-6602-5608

Date | Speaker | Title |
---|---|---|

Sep 8st, 9-10am | Nam Q. Le (Indiana U.) | Convergence of an inverse iterative scheme for the Hessian eigenvalue |

Sep 15th, 9-10am | Xukai Yan (OSU) | On interaction energy and steady states of aggregation-diffusion equations |

Sep 22nd, 9-10am | Cheikh Ndiaye (Howard U.) | Exact bubbling rate formula for the resonant Q-curvature equation and applications |

Sep 29th, 9-10am | Jingbo Dou (Shaanxi Normal U.) | Sharp Hardy-Littlewood-Sobolev inequalities and related inequalities |

Oct 7th, 4-5pm | Shengwen Wang (Queen Mary UL) | A Brakke type regularity theorem for the Allen-Cahn flow |

Oct 13th, 9-10am | Jiakun Liu (U. Wollongong) | Some applications of optimal transportation |

Oct 20th, 9-10am | Yannick Sire (JHU) | Harmonic maps with free boundary and beyond |

Oct 27th, 9-10am | Heming Jiao (Harbin IT) | A Second Order Estimate for Convex Solutions of Degenerate k-Hessian Equations |

Nov 3rd, 9-10am | Ling Xiao (UCONN) | Entire \(\sigma_k\) curvature flow in Minkowski space |

Nov 3rd, 4-5pm | Fengrui Yang (Freiburg) | Prescribed curvature measure problem in hyperbolic space |

Nov 10th | Junichi Harada (Akita U.) | Type II blowup solutions to the 5D semilinear heat equation with double power nonlinearity |

Nov 17th, 2-3pm | Changwei Xiong (Sichuan) | Some estimates on an exterior Steklov eigenvalue problem |

Nov 17th, 3-4pm | Qianzhong Ou (Guangxi Normal) | On the classification of entire solutions to the critical p-Laplace equation |

Nov 24th, 9-10am | Lu Xu (Hunan) | The constant rank theorem and its applications |

Dec 1st, 9-10am | Zhihan Wang (Princeton) | Generic Regularity for All Minimal Hypersurfaces in 8-Manifolds |

Dec 8th, 9-10am | Rong Tang (JHU) | Nonlocal Filtration Equations on the Heisenberg group |

Since the work by Caffarelli-Chan-Silvestre-Vasseur, integro-differential equations naturally arise from models in physics, engineering, and finance, attracting an increasing level of interest. They are also a natural generalization of fractional differential equations. In this talk, we will discuss the nonlocal integro-differential equations on the Heisenberg group. First, we will talk about the motivations to study the nonlocal filtration equations on the Heisenberg group. Then I will establish the existence, uniqueness and some qualitative properties of the solutions. Moreover, we will discuss some regularity results. In particular, the Hölder regularity will hold for the porous medium type equations under consideration.

The well-known Simons cone suggests that singularities may exist in a stable minimal hypersurface in Riemannian manifolds of dimension greater than 7, locally modeled on stable minimal hypercones. It was conjectured that generically they can be perturbed away. In this talk, we present a way to eliminate these singularities by perturbing metric in an 8-manifold. By combining with a Sard-Type Theorem for space of singular minimal hypersurfaces of dimension 7, joint with Yangyang Li, we proved that in an 8-manifold with generic metric, every locally stable minimal hypersurface has no singularity. In particular, this proves the existence of infinitely many SMOOTH minimal hypersurfaces in a generic 8-manifold.

The constant rank theorem was initially developed by Caffarelli-Friedman in 1985 in two-dimensions for convex solutions of semilinear equations. Later, Korevaar-Lewis extended the result to higher dimensions. The theory was generalized to fully nonlinear case twenty years ago, and it had became an important ingredient in the study of prescribed curvature problems, such as the Christoffel-Minkowski problem and the prescribed Weingarten curvature problem.

In this talk, I will give some applications of the constant rank theorem. These work are jointly finished with my co-authors.

In this talk, we will focus on the classification of positive solutions to the critical p-Laplace equation. It is well known that such issue is crucial in many applications such as a priori estimates, blow-up analysis and asymptotic analysis. Note that for the subcritical case, the equations have no positive solutions by the well known works of Gidas-Spruck [CPAM1981] and Serrin-Zou [ACTA2002]. While for the critical case, there are nontrivial 2-parameters family of solutions and which were classified by Caffarelli-Gidas-Spruck [CPAM1989] for p=2 , and by J. Vetois [JDE2016] (for 1<p<2) and B. Sciunzi Adv.Math.2016 under the additional assumption of finite energy, via the method of moving planes. Then by exploiting the method of integral estimate, we obtain the same classification results for (n+1)/3<p<n without any further assumption.

In this talk we will discuss a Steklov eigenvalue problem on an exterior Euclidean domain. We will present sharp lower and upper bounds for its first eigenvalue under various conditions on the domain. Time permitting, we shall discuss an upper bound for its second eigenvalue.

We will discuss the existence of blowup solutions to \(u_t=\Delta u+|u|^\frac{4}{n-2}u-|u|^{q-1}u\) with \(0<q<\frac{n+2}{n-2}\). We show that there are three kinds of blowup solutions in this equation. In particular, for the case \(q\in(0,1)\), we construct blowup solutions by gluing a specific blowup solutions of \(u_t=\Delta u+|u|^\frac{4}{n-2}u\) and a specific solution of \(u_t=\Delta u-|u|^{q-1}u\) with an extinction property.

The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider prescribed curvature measure problem in hyperbolic space.We obtain the existence of star-shaped k-convex bodies with prescribed (n-k)-th curvature measures (k<n) by establishing crucial C^2 regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.

In this talk, we will discuss the \(\sigma_k\) curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We show that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling, the flow converges to a self-expander. This is a joint work with Zhizhang Wang.

In this talk, I will report a recent work with Zhizhang Wang about the second order estimates for degenerate k-Hessian equations. We establish the a priori \(C^{1,1}\) estimates for convex solutions of degenerate k-Hessian equations in a strictly convex domain with non-homogenous boundary conditions under the condition that the right hand side function f is only assumed to satisfy \(f^{1/(k-1)} \in C^{1,1}\).

I will introduce a new heat flow for harmonic maps with free boundary. After giving some motivations to study such maps in relation with extremal metrics in spectral geometry, I will construct weak solutions for the flow and derive their partial regularity. The introduction of this new flow is motivated by the so-called half-harmonic maps introduced by Da Lio and Riviere, which provide a new approach to the old topic of harmonic maps with free boundary. I will also state some open problems and possible generalizations.

In this talk, we will introduce some interesting applications of optimal transportation in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric inequality in geometry; and an application in image recognition relating to a transport between hypercubes. This talk is based on a series of joint work with Shibing Chen, Xu-Jia Wang, and with Gregoire Loeper.

We will talk about an analogue of the Brakke’s local regularity theorem for the \(\epsilon\) parabolic Allen-Cahn equation. In particular, we show uniform \(C^{2,\alpha}\) regularity for the transition layers converging to smooth mean curvature flows as \(\epsilon\) tend to 0 under the almost unit-density assumption. This can be viewed as a diffused version of the Brakke regularity for the limit mean curvature flow. This talk is based on joint work with Huy Nguyen.

In this talk, I will summarize some sharp Hardy-Littlewood-Sobolev(HLS) inequalities, reversed HLS inequalities and logarithmic HLS inequality on whole space and on the upper half space respectively, which includes our past decade work. As applications, we also present sharp Carleman inequality on the unit disk and weighted Sobolev inequalities on the upper half space, which are closely related to HLS inequality. These are mainly joint works with Meijun Zhu.

In this talk, we will discuss a Chen-Li exact bubbling rate formula for the Q-curvature equation in dimension 4. As applications, we will present some existence and compactness results, and a Leray-Schauder degree formula for the resonant Q-curvature problem.

In this talk, I will talk about two results related to the interaction energy \(E[f]= \int f(x)f(y)W(x-y) dxdy\) for a nonnegative density f and radially decreasing interaction potential W. The celebrated Riesz rearrangement shows that \(E[f] \le E[f^*]\), where \(f^*\) is the radially decreasing rearrangement of f. It is a natural question to make quantitative stability estimate of \(E[f^*]-E[f]\) in terms of some distance between \(f\) and \(f^*\). Previously such stability estimates have been obtained for characteristic functions. In this talk, I will discuss a joint work with Yao Yao about the stability for general densities.

I will also discuss another joint work with Matias Delgadino and Yao Yao about the uniqueness and non-uniqueness of steady states of aggregation equations with degenerate diffusion, where the convexity of the interaction energy plays an important role. For the diffusion power \(m\geq 2\), we constructed a novel interpolation curve between any two radially decreasing densities with the same mass, and showed that the interaction energy is convex along this interpolation. This lead to the uniqueness of the steady state, and the threshold is sharp.

In this talk, we will first introduce a non-degenerate inverse iterative scheme to solve the eigenvalue problem for the k-Hessian operator on a smooth, bounded domain in Euclidean spaces. We show that the scheme converges, with a rate, to the k-Hessian eigenvalue for all k. We also prove a local L^1 convergence of the Hessian of solutions of the scheme. Hyperbolic polynomials play an important role in our analysis.