#### RG Analysis and Partial Differential Equations

# Graduate seminar on Advanced topics in PDE

## Organizers

## Schedule

UPDATE: On the following dates, the seminar will take place in room N 0.007 (Neubau, Endenicher Allee 60) instead: 6.12., 13.12., 10.1., 17.1., 31.1.

##### January 31st - Dimitrije Cicmilovic

##### Title:

Symplectic nonsqueezing and nonlinear Schrodinger equation##### Abstract:

In this talk we shall discuss infinite dimensional generalization of Gromov's sympelctic nonsqueezing result. As an application we will present mass subcritical and critical nonlinear Schrodinger equation. Nonsqueezing property of the said flows was already known, however the techniques used are based on finite dimensional Gromov's result, while ours presents a more natural way of looking at the Hamiltonian structure of the equations. Joint work with Herbert Koch.

##### January 24th

The seminar is combined with the workshop 11th Itinerant Workshop in PDEs: Analyse des équations aux dérivées partielles.

##### January 17th - Frédéric Rousset

##### Title:

Asymptotic stability of equilibria for Vlasov-Poisson systems##### Abstract:

After an introduction to the problem, the aim of the talk will be to present a recent approach to the stability of homogeneous equilibria for Vlasov-Poisson systems in the whole space which is alternative to the one of Bedrossian, Masmoudi and Mouhot. This approach is based on the use of sharp dispersive estimates in the physical space.

##### January 10th - Gennady Uraltsev

##### Title:

Uniform bounds for the Bilinear Hilbert Transform##### Abstract:

The Bilinear Hilbert Transform is a prototypical modulation invariant multi-linear singular operator.$ BHT_\alpha(f_1,f_2)(x):=\int_{\mathbb R} \int_{\mathbb R} \widehat{f_1}(\xi_1) \widehat{f_2}(\xi_2) \chi_{[0,+\infty)}(\xi_1-\alpha \xi_2) e^{2\pi i (\xi_1+\xi_2)x}d\xi_1 d\xi_2. $

It arises in many contexts including Cauchy integrals along Lipschitz curves and in the study of Calderón commutators. BHT satisfies the bounds

$ \|BHT_\alpha(f_1,f_2)(x)\|_{L^{p_3'}(\mathbb R)}\leq C_{p_1,p_2,\alpha} \|f_1\|_{L^{p_1}(\mathbb R)} \|f_2\|_{L^{p_2}(\mathbb R)} $

for any $p_{1,2,3} \in (1,+\infty)$ with $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=1$. A longstanding open problem is that of bounds for the Bilinear Hilbert Transform uniform in the parameter $\alpha$. The dyadic analog of this problem has been solved by Oberlin and Thiele (2010).

We will give a general overview of the topic. We will then sketch how the framework of outer measure $L^p$ spaces in time-scale-frequency space plays a crucial role in proving the result.

##### December 20th - No talk

##### December 13th - Leonardo Tolomeo

##### Title:

Ergodicity for stochastic dispersive equations

##### Abstract:

In this talk, we study the long time behaviour of some stochastic partial differential equations (SPDEs). We will introduce the notions of ergodicity, unique ergodicity and convergence to equilibrium, and we will discuss how these have been proven for a very large class of parabolic SPDEs. We will then shift our attention to nonlinear dispersive SPDEs, and show that (unexpectedly) this general strategy is bound to fail for most dispersive equations. We will describe this failure in detail for the wave equation on the 1-dimensional torus, and show what can be done to recover unique ergodicity for this particular model.

##### December 6th - Tobias König

##### Title:

Energy asymptotics in the three-dimensional Brezis-Nirenberg problem

##### Abstract:

For a bounded open set $\Omega\subset\R^3$ we consider the minimization problem $$ S(a+\epsilon V) = \inf_{0\not\equiv u\in H^1_0(\Omega)} \frac{\int_\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int_\Omega u^6\,dx)^{1/3}} $$ involving the critical Sobolev exponent. Under certain assumptions on $a$ and $V$ we compute the asymptotics of $S(a+\epsilon V)-S$ as $\epsilon\to 0+$, where $S$ is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to $a$ and we determine the location of the concentration point within that set. We will also discuss analogous energy asymptotics for space dimension $N \geq 4$ as well as the relation of these results to a conjecture by Brezis and Peletier. This is joint work with Rupert Frank and Hynek Kovarik.

##### November 29th - David de Laat

##### Title:

Optimization and harmonic analysis techniques for problems in discrete geometry

##### Abstract:

A finite subset of the unit sphere is said to be a spherical D-code if the inner product between any two distinct points in the code lies in D. An important problem in discrete geometry is to find the maximum cardinality of a spherical D-code on the n-dimensional unit sphere. For D=[-1,cos(pi/3)] this is the kissing number problem (the maximum number of pairwise nonoverlapping unit spheres that can touch a central unit sphere). We are interested in computing upper bounds on these quantities in order to prove that a given configuration is optimal. In this talk I want to explain what kind of optimization, harmonic analysis, and real algebraic techniques go into computing such upper bounds. I will start with the classical Delsarte bound and work up to the most recent techniques.

##### November 22nd - Marco Fraccaroli

##### Title:

Reverse Hölder inequality for outer L^p spaces

##### Abstract:

The L^p theory of outer measure spaces was introduced by Do and Thiele to formalize a paradigma for proofs in time-frequency analysis. According to this motivation, it was developed in the direction of real interpolation properties. In this talk we discuss instead previously untouched aspects of the outer L^p spaces. In particular, we prove the expected reverse Hölder inequality and countable triangular inequality for the outer L^p quasi-norms up to constants independent of the outer measure space. Focusing on the case of the upper half space, these results allow us to conclude the equivalence between outer L^p spaces in this setting and classical tent spaces.

##### November 15th - Lashi Bandara

##### Title:

Boundary value problems for general first-order elliptic differential operators

##### Abstract:

The Bär-Ballmann framework is a comprehensive machine useful in studying elliptic boundary value problems (as well as their index theory) for first-order elliptic operators on manifolds with compact and smooth boundary. A fundamental assumption in their work is that an induced operator on the boundary can be chosen self-adjoint. Many operators, including all Dirac type operators, satisfy this requirement. In particular, this includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general first-order elliptic operators, with the quintessential example being the Rarita-Schwinger operator on 3/2-spinors. In general dimensions, every induced boundary operator for the Rarita-Schwinger operator is non self-adjoint. In this talk, I will present recent work with Bär where we and consider general first-order elliptic operators by dispensing with the self-adjointness requirement for induced boundary operators. The ellipticity of the operator allows us to understand the structure of the induced operator on the boundary, modulo a lower order additive perturbation, as bi-sectorial operator. We use a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory as well as methods arising from the resolution of the Kato square root problem to extend the Bär-Ballman framework. If time permits, I will also touch on the non-compact boundary case, and potential extensions of this to the L^p setting and Lipschitz boundary.

##### November 8th - Lucrezia Cossetti

##### Title:

Unique Continuation for the Zakharov-Kuznetsov equation

##### Abstract:

In this talk we analyze uniqueness properties of solutions to the Zakharov-Kuznetsov (ZK) equation $$ \partial_t u + \partial_{x}^3u + \partial_{x}\partial_{y}^2u + u \partial_x u=0, \qquad (x,y)\in \mathbb{R}^2,\quad t\in [0,1]. $$ Mainly motivated by the very well known PDE's counterpart of the Hardy uncertainty principle, we provide a two times unique continuation result. More precisely, we prove that given $u_1, u_2$ two solutions to ZK, as soon as the difference $u1-u2$ decays (spatially) fast enough at two different instants of time, then $u1 \equiv u2.$ As expected, it turns out that the decay rate needed to get uniqueness reflects the asymptotic behavior of the fundamental solution of the associated linear problem. Encouraged by this fact we also prove optimality of the result. Some recent results concerning the $(3+1)$- dimensional ZK equation will be also presented. The seminar is based on a recent paper in collaboration with L. Fanelli and F. Linares.

##### November 1st - No talk (Allerheiligen)

##### October 25th - Lenka Slavíková

##### Title:

Boundedness criteria for bilinear Fourier multipliers

##### Abstract:

In this talk we will discuss criteria for the L² × L² → L¹ boundedness of bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Results of this type have applications for proving boundedness of various operators in harmonic analysis, including rough bilinear singular integrals and bilinear spherical maximal functions. Our main focus will be on the question of optimality of these bilinear multiplier theorems. Some of the results presented in this talk were obtained in collaboration with Loukas Grafakos and Danqing He.

##### October 18th - Alexandra Antoniouk

##### Title:

Multidimensional nonlinear pseudo-differential equation with p-adic spatial variables

##### Abstract:

We discuss the Cauchy problem for $p$-adic non-linear evolutionary pseudo-differential equations for complex-valued functions. Among the equations under consideration there is the $p$-adic analog of the porous medium equation (or more generally, the nonlinear filtration equation) which arise in numerous application in mathematical physics and mathematical biology. Our approach is based on the construction of a linear Markov semigroup on a $p$-adic ball and the proof of m-accretivity of the appropriate nonlinear operator. The latter result is equivalent to the existence and uniqueness of a mild solution of the Cauchy problem of a nonlinear equation of the porous medium type.## News

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