The seminar usually take place on **Wednesday
Morning**.

**Guohuan Qiu**(Academy of Mathematics and Systems Science)**Liming Sun**(Academy of Mathematics and Systems Science)**Jinping Zhuge**(Morningside Center of Mathematics, Academy of Mathematics and Systems Science)

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

Sep 6, 10:30-11:20am | Longjuan Xu (Capital Normal) | Stress analysis and higher derivative estimates for Stokes flow |

Sep 13, 10-11am | Feida Jiang (Southeast) | Keller-Osserman conditions for k-Hessian type equations with gradient terms |

Sep 20, 10:30-11:20am | Khanh Nguyen (AMSS) | Existence and uniqueness of limits at infinity for homogeneous Sobolev functions |

Sep 27, 10:30-11:20am | Han Hong (Tsinghua) | Stability of capillary surfaces in three-dimensional Riemannian manifolds |

Oct 10(Tu), 1:30pm-2:30pm | Jiguang Bao(Beijing Normal) | Optimal Solvability of Exterior Dirichlet Problems for Monge-Ampère Equations |

Oct 12(Th), 10am-11am | Kyeongsu Choi (KIAS) | Ancient finite-entropy curve shortening flows |

Oct 12(Th), 4:30pm-5:30pm | Alexander Logunov (U. Geneva) | Landis’ conjecture on exponential localization |

Oct 17(Tu), 1:30-2:20pm | Haigang Li (Beijing Normal) | Babuska Problem in Composite Materials and Suspension Problem in Stokes Flow |

Oct 18, 10:30-11:30am | Yeyao Hu(Central South) | Mean field type equations and some related problems |

Oct 25, 10-11am | Qing Han (Notre Dame) | Kelvin transforms and the asymptotic analysis |

Oct 25, 11-12am | Yong Liu (USTC) | Multiple-end solutions of the Allen-Cahn equation on the plane |

Nov 2 (Tu)， 5-6pm | Melanie Rupflin (Oxford) | Sharp quantitative rigidity results for maps between spheres of general degre |

Nov 14 (Tu), 10-11am | Xuezhang Chen (Nanjing) | Prescribing \(\sigma_2\) curvature and boundary mean curvature on compact manifolds |

Nov 21, 10:30-11:30 | Lingwei Ma (Tianjin Normal) | Gibbons’ conjecture and Liouville theorem for master equations |

Dec 4, 10:00-11:00 | Bo Guan (Ohio State) | On concavity and subsolutions for fully nonlinear elliptic equations |

Dec 4, 11:00-12:00 | Qiang Xu (Lanzhou) | Annealed Calderon-Zygmund estimates for elliptic operator with random coefficients on C2 domains |

Concerned with elliptic operators with stationary random coefficients of integrable correlations and bounded C2 domains, this paper mainly studies annealed Calder'on-Zygmund estimates, which is new even for constant coefficients. Stronger than some classical results derived by a perturbation argument in the deterministic case, our result owns a scaling-invariant property, which additionally requires a non-perturbation argument recently developed by M. Jeosen and F. Otto [JFA,22’]. To handle boundary estimates, we have to introduce boundary correctors to treat for an optimal estimate from suboptimal. For the weighted estimates, we hand over the proof to Shen’s real arguments. The most attractive part of the paper is to show how these two powerful real methods work together to make the result clean and robust.

Both subsolutions and concavity play important roles in a priori estimates and classical solvability of fully nonlinear elliptic equations. In this talk we explore the possibility to weaken the concavity condition. We shall also discuss generalized notions of subsolution introduced by Szekelyhidi and myself, respectively, to treat equations on closed manifolds, and clarify their relations. More precisely, we show that these weak notions of subsolutions are equivalent for equations defined on convex cones of type 1 in the sense defined by Caffarelli, Nirenberg and Spruck.

In this talk, we will discuss some of our recent work on the master equation involving the fully fractional heat operator. Specifically, we will present two main results. The first one pertains to a generalized version of Gibbons’ conjecture in the context of the master equation. The other one deals with a Liouville theorem for the homogeneous master equation.

On a smooth compact manifold of dimensions three and four with totally non-umbilic boundary，imposing non-negativity assumptions on curvatures of the background metric, we establish that there exists a conformal metric having a positive \(σ_2\) curvature and a non-negative boundary mean curvature, which is necessary as shown by counter-examples. Two obstructions to the existence result are well-known to experts in this field: One is local C^2 estimates near generic boundary; the other is to build the blow-up analysis based on local estimates. In this talk, we are only enough to outline the proof of latter part, i.e. blow-up analysis, assuming the local C^2 estimates. This is joint work with Wei Wei.

As the energy of any map v of degree k>0 from \(S^2\) to \(S^2\) is at least \(4\pi k\) with equality if and only if v is a rational map, it is natural to ask whether maps with small energy defect \(E (v)-4\pi k\) are necessarily close to a rational map. While such a rigidity statement turns out to be false for maps of general degree, we will explain in this talk that any map with small energy defect is essentially given by a collection of rational maps that describe the behaviour of the map at very different scales and that the distance of any map to such collections is controlled by a sharp quantitative rigidity estimate.

Allen-Cahn equation is a classical model arising from phase transition and closely related to the minimal surface theory. In dimension two, an important class of solutions to this equation is the so called multiple-end solutions. They have finite Morse index. While there already exist some results on the construction of these solutions near the boundary of the whole moduli space of multiple-end solutions, a general variational construction is still missing. In this talk we prove that there exists a family of 6-end solutions with prescribed slopes using minimax arguments. This is joint work with Jun Wang and Wen Yang.

It is well-known that the Kelvin transform plays an important role in studying harmonic functions. With the Kelvin transform, the study of harmonic functions near infinity is equivalent to studying the transformed harmonic functions near the origin. In this talk, we will demonstrate that the Kelvin transform also plays an important role in studying asymptotic behaviors of solutions of nonlinear elliptic equations near infinity. We will study solutions of the minimal surface equation, the Monge-Ampere equation, and the special Lagrange equation and prove an optimal decomposition of solutions near infinity.

In this talk, we will first review the resolution of the celebrated Chang-Yang’s conjecture on a sharp Moser-Trudinger type inequality on the two-dimensional sphere. Then we recall the complete theory in dimension one and emphasize the key role played by mean field type equations. Moreover, higher dimensional analogues will be presented and some very recent progress will be introduced. We will also mention some related open problems that the audience might be interested in.

In a high-contrast elastic composite media, when inclusions are spaced closely, the stress always concentrates in between inclusions and causes damage initiation. For Babuska problem in linear elasticity, we obtain the blow-up asymptotic expressions of the gradients of solutions to the Lame system with partially infinite coefficients in the narrow region when the distance between inclusions tends to zero. An iteration method for the energy is developed for elliptic equations and systems. By using this method, recently, the stress concentration problem between two adjacent rigid inclusions suspending in the Stokes flow is solved.

The curve shortening flow is a heat equation of curves, and therefore we can characterize its ancient solutions as parabolic Liouville theorems. For example, it is known that the only convex ancient flows are shrinking circles, translating Grim Reaper curves, Angenent ovals, and static lines. A natural question would be to find a condition, weaker than convexity, under which the ancient CSFs are classified. In this talk, we will discuss the classification of ancient curve shortening flows with finite entropy. In particular, we put emphasis on the uniqueness of tangent flow at negative infinity, which is the round circle or a straight line with multiplicity. This is a joint work with Dong-Hwi Seo, Wei-Bo Su, and Kai-Wei Zhao.

Landis’ conjecture states that if \(u\) is a non-zero solution to \(\Delta u + V u =0\) in the Euclidean space, where \(V\) is a real, bounded function, then \(u\) cannot decay faster than exponentially at infinity. We will discuss the subtleties of the problem and a two-dimensional method, which prohibits the decay \(|u(x)| \leq exp(-|x|^{1+\epsilon})\). Based on a joint work with E.Malinnikova, N.Nadirashvili and F. Nazarov.

The talk provides two necessary and sufffcient conditions for the existence of solutions to the exterior Dirichlet problem of the Monge–Ampère equation with prescribed asymptotic behavior at inffnity. We remove the C2 regularity assumptions on the boundary value and the inner boundary, which are required in the proofs of the corresponding existence theorems in the recent literatures.

In this talk, we will discuss stability results for noncompact capillary surfaces. A classical result in minimal surface theory by Fischer-Colbrie and Schoen (independently by do Carmo-Peng) says that a stable complete minimal surface in \(\mathbb{R}^3\) must be a plane. We show that, under certain curvature assumptions, a weakly stable capillary surface in a 3-manifold with boundary has only three possible topological configurations. In particular, we prove that a weakly stable capillary surface in a half-space of \(\mathbb{R}^3\) which is minimal or has the contact angle less than or equal to \(\pi/2\) must be a half-plane. A natural application of the rigidity result is curvature estimates for capillary surfaces in 3-manifolds. This is a joint work with Artur Saturnino.

We establish the existence and uniqueness of limits at infinity along infinite curves outside a zero modulus family for functions in a homogeneous Sobolev space under the assumption that the underlying space is equipped with a doubling measure which supports a Poincaré inequality. We also characterize the settings where this conclusion is nontrivial. Secondly, we introduce notions of weak polar coordinate systems and radial curves on metric measure spaces. Then sufficient and necessary conditions for existence of radial limits are given. As a consequence, we characterize the existence of radial limits in certain concrete settings.

In this talk, we consider a k-Hessian type equations with gradient terms and provide a necessary and sufficient condition for the solvability of entire admissible subsolutions, which can be regarded as a generalized Keller-Osserman condition. The existence and nonexistence results are proved in different ranges of the parameter, which embrace the standard Hessian equation case as a typical example. The difference between the semilinear case (k = 1) and the fully nonlinear case (\(k\ge 2\)) is also concerned.

This talk concerns the estimates for stress concentration between two adjacent rigid inclusions in Stokes flow. We establish the pointwise upper bounds of the gradient and the second-order partial derivatives for Stokes flow in the presence of two closely located strictly convex inclusions in dimensions two and three. Moreover, the lower bounds of the gradient estimates at the narrowest place of the narrow region show the optimality of the blow-up rate. We also show the optimal blow-up rate of Cauchy stress tensor. In dimensions greater than three, the upper bounds of the gradient are established. These results answer the questions raised by Hyeonbae Kang in ICM (2022).