Mathematics Department - Nonlinear Analysis - Spring 2017

# Nonlinear Analysis - Spring 2017

### Organizer(s)

Yanyan Li, Haim Brezis

## Tuesday, April 25th

### " Extremal functions for Morrey’s inequality in convex domains"

Time: 1:40 PM
Location: Hill 705
Abstract: A celebrated result in the theory of Sobolev spaces is Morrey's inequality, which establishes the continuous embedding of the continuous functions in certain Sobolev spaces. Interestingly enough the equality case of this inequality has not been thoroughly investigated (unless the underlying domain is R^n). We show that if the underlying domain is a bounded convex domain, then the extremal functions are determined up to a multiplicative factor. We will explain why the assertion is false if convexity is dropped and why convexity is not necessary for this result to hold.

This is joint work with Erik Lindgren of KTH.

## Tuesday, February 28th

### "Characterization of f-extremal disks"

Time: 1:40 PM
Location: Hill 705
Abstract: In this talk we show uniqueness for overdetermined elliptic problems defined on topological disks with regular boundary, i.e., positive solutions $u$ to $Delta u + f(u)=0$ in $Omega subset (M^2,g)$ so that $u = 0$ and $frac{partial u}{partial veceta} = cte$ along $partial Omega$, $veceta$ the unit outward normal along $partialOmega$ under the assumption of the existence of a candidate family. To do so, we adapt the G'alvez-Mira generalized Hopf-type Theorem to the realm of overdetermined elliptic problem. In particular, this gives a positive answer to the Schiffer conjecture for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains, also called {it Serrin Problem}) in $mathbb S ^2$.

This is a joint work with L. Mazet.

## Tuesday, February 21st

### "Memorial for Felix Browder"

Time: 1:40 PM
Location: Hill 705
Abstract: go to: http://www.math.rutgers.edu/events/BrowderMemorial_Feb2017.pdf

## Tuesday, February 14th

### "Borderline Sobolev inequalities after Bourgain-Brezis and applications"

Time: 1:40 PM
Location: Hill 705
Abstract: About 15 years ago, Bourgain and Brezis discovered astonishing Sobolev style inequalities at the end-point where the classical Sobolev embedding theorem fails. In this talk we will extend these inequalities to Riemannian symmetric spaces of non-compact type of any rank and also present applications to Strichartz inequalities for wave and Schrodinger equations, incompressible Navier-Stokes flow in 2D with prescribed vorticity and the Maxwell equations for Electromagnetism. These results have been obtained with Jean Van Schaftingen and Po-lam Yung.

## Tuesday, February 14th

Special Nonlinear Analysis

### "Boundaries with almost-constant mean curvature"

Time: 10:30 AM
Location: Hill 705
Abstract: We describe joint works with Giulio Ciraolo (U Palermo) Brian Krummel (U Berkeley) and Matias Delgadino (ICTP Trieste) concerning the quantitative description of boundaries whose mean curvature is close to a constant. These appear in the study of capillarity theory as well as in problems in Geometry.

## Friday, February 10th

### "Global regularities of 2-D density patch for viscous inhomogeneous incompressible flow with general density"

Time: 1:40 PM
Location: Hill 525
Abstract: Toward the open problem proposed by P.-L. Lions in [Mathematical topics in fluid mechanics. Vol. 1. Incompressible models, 1998] concerning the propagation of regularities of density patch for viscous inhomogeneous incompressible flow, we first establish the global in time well-posedness of two-dimensional inhomogeneous incompressible Navier-Stokes system with initial density being the summation of the characteristic function on a bounded, simply connected $W^{k+2,p}(R^2)$ domain $Om_0$ multiplied by a positive constant $eta_1$ and the characteristic function on the completement of $Om_0$ multiplied by a positive constant $eta_2$ for any pair of positive constants $(eta_1,eta_2).$ We then prove that the time evolved domain $Om(t)$ also belongs to the class of $W^{k+2,p}$ for any $t>0.$ Thus in some sense, we have solved the aforementioned open problem of Lions in the two-dimensional case.

## Wednesday, February 8th

### "On Green functions of second-order elliptic operators on Riemannian Manifolds: the critical case"

Time: 1:40 PM
Location: Hill 425
Abstract: Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation Pu=0 in M admits a positive supersolution which is not a solution if and only if P admits a unique positive minimal Green function on M, and in this case, P is said to be subcritical in M. If P does not admit a positive Green function but admits a global positive solution, then such a solution is called a ground state of P in M, and P is said to be critical in M.

We prove for a critical operator P in M, the existence of a Green function which is dominated above by the ground state of P away from the singularity. Moreover, in a certain class of Green functions, such a Green function is unique, up to an addition of a product of the ground states of P and P^*. This result extends and sharpens the celebrated result of Peter Li and Luen-Fai Tam concerning the existence of a symmetric Green function for the Laplace-Beltrami operator on a smooth and complete Riemannian manifold M.

This is a joint work with Debdip Ganguly.

## Tuesday, February 7th

### "Localization of nonlocal continuum models"

Time: 1:40 PM
Location: Hill 705
Abstract: Recent development of nonlocal vector calculus and nonlocal calculus of variations provides a systematic mathematical framework for the analysis of nonlocal continuum models in the form of partial-integral equations. In this lecture, we discuss the localization of some nonlocal models and associated nonlocal function spaces in order to study connections with traditional local models given by partial differential equations and Sobolev spaces. In particular, we present some recent results on heterogeneous localization of nonlocal space including an extension of classical trace theorems to nonlocal spaces of functions with significantly weaker regularity. We also discuss their implications in nonlocal modeling of multiscale processes.

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