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Current contact: Gerardo Mendoza
The seminar takes place Mondays 2:40 - 3:30 pm in Wachman 617. Click on title for abstract.
Thomas Krainer, PennState Altoona
TBA
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)
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.
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.
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)
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.
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.
Eric Stachura, Kennesaw State University
Thomas Krainer, PennSate-Altoona
Postponed to Fall semester.
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 $U$.
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).