# Analysis Seminar 2020

Current contact: Gerardo Mendoza

The seminar takes place Mondays 2:30 - 3:30 pm either in Wachman 617 or virtually via Zoom. Please see the announcment of the specific meeting. For virtual meetings, please write if you wish to join. Click on title for abstract.

• Monday February 3, 2020 at 14:40, Wachman 617

Cecilia Freire Mondaini, Drexel University

This talk focuses on the study of convergence/mixing rates for stochastic dynamical systems towards statistical equilibrium. Our approach uses the weak Harris theorem combined with a generalized coupling technique to obtain such rates for infinite-dimensional stochastic systems in a suitable Wasserstein distance. In particular, we show two scenarios where this approach is applied in the context of stochastic fluid flows. First, to show that Markov kernels constructed from a suitable numerical discretization of the 2D stochastic Navier-Stokes equations converge towards the invariant measure of the continuous system. This depends crucially on a spectral gap result for the discrete Markov kernel that is independent of the level of discretization. Second, to approximate the posterior measure obtained via a Bayesian approach to inverse PDE problems, particularly when applied to advection-diffusion type PDEs. In this latter case, the Markov transition kernel is constructed with an exact preconditioned Hamiltonian Monte Carlo algorithm in infinite dimensions. A rigorous proof of mixing rates for such algorithm was an open problem until quite recently. Our approach provides an alternative and flexible methodology to establish mixing rates for other Markov Chain Monte Carlo algorithms. This is a joint work with Nathan Glatt-Holtz (Tulane U).

• Monday February 17, 2020 at 14:40, Wachman 617

Xiaojun Huang, Rutgers University

Let $\Omega$ be a Stein space (of complex dimension at least two) with possibly isolated singularities and a connected compact strongly pseudoconvex smooth boundary $M = \partial \Omega$. Let $(f,D)$ be a non-constant CR mapping, where $D$ is an open connected subset of $M$. Suppose that $(f,D)$ admits a CR continuation along any curve in $M$ and for each CR mapping element $(g,D^*)$ with $D^*\subset M$ obtained by continuing $(f,D)$ along a curve in $M$, it holds that $\|g\|\leq C$ for a certain fixed constant $C$. Then $(f,D)$ admits a holomorphic continuation along any curve $\gamma$ with $\gamma(0) \in D$ and $\gamma(t) \in \mathrm{Reg}(\Omega)$ for $t \in (0, 1]$. Moreover, for any holomorphic mapping element $(h,U)$ with $U \subset \mathrm{Reg}(\Omega)$ obtained from continuation of $(f,D)$, we have $\|h\| < C$ on $U$.

• Monday March 9, 2020 at 14:30, Wachman 617
(Cancelled)

Thomas Krainer, PennSate-Altoona

Postponed to Fall semester.

• Monday March 30, 2020 at 14:40, Wachman 617
TBA

Eric Stachura, Kennesaw State University

• Monday April 6, 2020 at 14:40, Wachman 617

Martin Dindos, The University of Edinburgh

Abstract: The notion of an elliptic partial differential equation (PDE) goes back at least to 1908, when it appeared in a paper J. Hadamard.

The essence of ellipticity is described by L. Evans in his classic textbook as follows: "The following calculations are often technically difficult but eventually yield extremely powerful and useful assertions concerning the smoothness of weak solutions. As always, the heart of each computation is the invocation of ellipticity: the point is to derive analytic estimates from the structural, algebraic assumption of ellipticity."

In this talk we present a recently discovered structural condition, called $p$-ellipticity, which generalizes classical ellipticity and plays a fundamental role in many seemingly mutually unrelated aspects of the $L^p$ theory of elliptic complex valued PDE. So far, $p$-ellipticity has proven to be the key condition for:

(i) convexity of power functions (Bellman functions) (ii) dimension-free bilinear embeddings, (iii) $L^p$-contractivity and boundedness of semigroups $(P_t^A)_{t>0}$ associated with elliptic operators, (iv) holomorphic functional calculus, (v) multilinear analysis, (vi) regularity theory of elliptic PDE with complex coefficients.

During the talk I will describe my contribution to this development in particularly to (vi). It is of note that the $p$-ellipticity was co-discovered independently by Carbonaro and Dragicevic on one side (from the perspective of (i) and (ii)), and Pipher and myself on the other.

• Monday October 12, 2020 at 14:30, Zoom meeting

Shif Berhanu, Temple University

In $1993$, Baouendi and Rothschild proved the following boundary unique continuation result: Let $B^+$ be a half ball in the upper half space in $\mathbb R^n$, $u$ continuous on $\overline{B^+}$, harmonic in $B^+$, and $u(x',0)\geq 0$ on the flat piece of $\partial B^+$. If $u$ vanishes to infinite order at the origin in the sense that $u(x)=O(|x|^N)$ for all $N$, then $u\equiv 0.$

They conjectured that a similar result holds for more general domains and more general second order elliptic operators. We will present a positive solution of the conjecture for second order elliptic operators with real analytic coefficients with data given on a real analytic hypersurface. Our result will be a special case of a more general theorem for real analytic elliptic differential operators of any order. Our results have applications to unique continuation for CR functions which was the original inspiration for Baouendi and Rothschild.

• Monday November 2, 2020 at 14:30, Zoom meeting

Wolfram Bauer, Leibniz University

We give a short introduction to subriemannian geometry.  Based on the Popp measure construction for an equiregular distribution an intrinsic sub-Laplacian can be defined. Generalizing the tangent space,

nilpotent Lie groups $G$ serve as local models for a subriemannian manifold and themselves are equipped with a left-invariant subriemannian structure. We introduce pseudo-$H$-type groups $G$ which form a class of step-2-nilpotent Lie groups and consider their quotients by a lattice $\Gamma \subset G$ (pseudo-$H$-type nilmanifolds). Based on a well-known expression

of the heat kernel of the sub-Laplacian on the compact left-coset space $\Gamma \backslash G$ we can perform an explicit spectral analysis. In a natural way a pseudo-$H$-type group also carries a pseudo-subriemannian structure which from an analytic viewpoint  induces an ultra-hyperbolic operator $\Delta_{\textrm{UH}}$. We aim to discuss the following questions:

-- Can we explicitely construct and classify isospectral (in the subriemannian sense) non-homeomorphic nilmanifolds $\Gamma \backslash G$?

-- Is the operator $\Delta_{\textrm{UH}}$ locally solvable? Can we explicitly construct its inverse in the case of existence?

The talk is based on the (joint) papers:

-- W. Bauer, A. Froehly, I. Markina, Fundamental solutions of a class of ultra-hyperbolic operators on pseudo-$H$-type groups}, Adv. Math. 369, (2020), 1-46.

-- W. Bauer, K. Furutani, C. Iwasaki, A. Laaroussi,  Spectral theory of a class of nilmanifolds attached to Clifford modules}, Math. Z. (2020)

• Monday November 9, 2020 at 14:30, Zoom meeting

Yuan Yuan, Syracuse University

The regularity of Bergman projection is one of the classical problems in several complex variables. Some $L^p$ and Sobolev regularities on some domains with nonsmooth boundary (e.g. Hartgos triangle, quotient domains) have been studied intensively recently. The symmetrized bidisk is another interesting model of non-smooth domains. In this talk, I will discuss the regularity of the Bergman projection on the weighted Sobolev space on the symmetrized bidisk. This is a joint work with Chen and Jin.

• Monday November 16, 2020 at 14:30, Zoom meeting

Paulo Cordaro, University of Sao Paulo

In this talk I will consider the problem of local (and also global) solvability for the differential complexes associated to a locally integrable structure. I will survey the known results and describe in some detail the case in which the structure is hypocomplex. Some of the recent results were obtained  in collaboration with M.R. Janhke.

• Monday November 30, 2020 at 14:30, Zoom meeting

Elmar Schrohe, Leibniz University

Let $X$ be a manifold with boundary and bounded geometry. On $X$ we consider a uniformly strongly elliptic second order operator $A$ that locally is of the form

$A=-\sum_{j,k} a_{jk} \partial_{x_j}\partial_{x_k}+ \sum_{j} b_j\partial_{x_j} +c.$

$A$ is endowed with a boundary operator $T$ of the form

$T=\varphi_0\gamma_0 + \varphi_1\gamma_1,$

where $\gamma_0$ and $\gamma_1$ denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary, and $\varphi_0$, $\varphi_1$ are non-negative $C^\infty_b$ functions on the boundary with $\varphi_0+\varphi_1\ge c_0>0$. This problem is not elliptic in the sense of Lopatinskij and Shapiro, unless either  $\varphi_1\not=0$ everywhere or $\varphi_1=0$ everywhere.

We show that the realization $A_T$ of $A$ in $L^p(\Omega)$ has a bounded $H^\infty$-calculus of arbitrarily small angle whenever the $a_{jk}$ are H\"older continuous and $b_j$ as well as $c$ are $L^\infty$.

For the proof we first treat the operator with smooth coefficients on $\mathbb R^n_+$. Here we rely on an extension of Boutet de Monvel's calculus to operator-valued symbols of H\"ormander type $(1,\delta)$. We then use   $H^\infty$-perturbation techniques in order to treat the non-smooth case.

The existence of a bounded $H^\infty$-calculus allows us to apply maximal regularity techniques. We show how a theorem of Cl\'ement and Li can be used to establish the existence of a short time solution to the porous medium equation on $X$ with boundary condition $T$.

(Joint work with Thorben Krietenstein, Hannover)

• Monday December 7, 2020 at 14:30, Zoom meeting

Thomas Krainer, PennState Altoona

A common theme in PDEs is to exploit invariance properties with respect to scaling of equations. This leads to fundamental solutions, the heat kernel, as well as the notion of principal symbol. Perturbation theory is then used to derive qualitative results for more general equations, where the dominant scaling-invariant pieces are the principal parts on which invertibility assumptions (ellipticity conditions) are placed. While invertibility of the principal symbol of an elliptic operator governs certain qualitative properties of the equation locally, additional conditions are required to determine well-posedness and regularity on spaces with noncompact ends, and especially on manifolds with incomplete geometry such as boundaries and singularities (i.e. one needs to impose boundary conditions). There are operator-valued analogues of the principal symbol of the operator that are associated with the boundaries and singularities that govern the behavior of solutions and the conditions to be placed on them for the equation. These dominant terms again exhibit certain top-order homogeneity properties, i.e., scaling invariance in a suitable sense, and are sometimes referred to as model operators.

In this talk I will speak about model operators from a purely functional analytic perspective. We will obtain several results, some previously known in special cases, as well as new ones as consequences of generic functional analytic properties.