Analysis and Partial Differential Equations

Abstract

I will prove a strong rigidity theorem for convex ancient hypersurface flows in the sphere, showing that for any geometric, parabolic flow the only possibilities are shrinking spherical caps.

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Abstract

Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^2\left(X\right)$ satisfying Gaussian upper bounds on its heat kernels. In this talk , we will present the development if the theory of Besov spaces ${Ḃ}_{p,q}^{\alpha ,L}\left(X\right)$ and Triebel–Lizorkin spaces ${Ḟ}_{p,q}^{\alpha ,L}\left(X\right)$ associated to the operator $L$ for the full range $0<p,q\le \infty$ and $\alpha \in ℝ$. Some applications will be discussed.

The talk is based on joint work with H-Q. Bui, P. D’Ancona, X. T. Duong and D. Müller.

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Abstract

We obtain some existence results for semilinear elliptic equations with a Hardy potential and gradient dependent nonlinearities. Furthermore, we also analyse the behaviour near zero for the positive solutions of such equations.

This is joint work with Florica Cîrstea.

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Abstract

A classical result by Greiner from 1982 asserts that a positive contraction semigroup on $L_p$ converges if it has a fixed vector and one of the semigroup operators is a kernel operator. In my talk, I shall explain how this result can be obtained as a special case of a much more general convergence theorem for positive semigroups. This general result, in turn, rests on the interplay of three things: 1) a certain compactness property, 2) the fact that the real numbers are a divisible group and 3) an analogue of the classical Peter-Weyl theorem for positive representations of compact groups on atomic Banach lattices.

This is joint work with Jochen Glück.

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Abstract

I will show how generalized convexity can be used to prove the classical Toponogov triangle comparison theorem and a sharp isoperimetric type inequality involving the cut distance of a bounded domain. More precisely, I will show that among all domains with cut distance $l$ and with a Ricci curvature lower bound $\left(n-1\right)k$, the ball of radius $l$ in the space form of curvature $k$ has the largest area-to-volume ratio. I will also present a closely related Heintze-Karcher type inequality which relates the volume of the domain with a boundary integral involving the mean curvature. If time allows, I will give some applications of these results and another isoperimetric-type inequality involving the extrinsic radius of the domain.

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Abstract

In this talk, we first give a brief introduction to the optimal transport problem, and then its extension to nonlinear case and applications in geometric optics. Last, we introduce some recent results on optimal partial transport problem, which is based on joint work with Shibing Chen (USTC) and Xu-Jia Wang (ANU).

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Abstract

We discuss some recent developments on extending the known results for linear Calderón problems to semilinear cases. Traditionally, the Calderón problem concerns the recovery of a potential $V$ in the Schrodinger equation from boundary measurements based on the Dirichlet-Neumann map. For the semilinear case, we consider instead a nonlinear perturbation of the Laplacian

Assuming that the nonlinearity $V\left(x,u\right)$ satisfies some nice properties, we can reduce the problem to a set of integral identities, from which the method of complex geometric optics applies.

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Abstract

he density of states is a non-negative measure associated to a Schrödinger operator $H$ which is supported on its essential spectrum. Theoretical questions concerning the existence and properties of the density of states are of interest in solid state physics. We have recently found that quite generally the density of states measure can be computed by a formula involving a Dixmier trace. This is a surprising new application of singular traces to mathematical physics which uses recently developed techniques in operator integration theory.

Joint work with N. Azamov, F. Sukochev and D. Zanin.

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Abstract

It is well known that the solution operators $cos\left(t\sqrt{-\Delta }\right)$ and $sin\left(t\sqrt{-\Delta }\right)$ to the classical wave equation ${\partial }_t^{2}u=\Delta u$ are not bounded on $L^p\left({ℝ}^n\right)$, for $n\ge 2$ and $1\le p\le \infty$, unless $p=2$ or $t=0$. In fact, for $1<p<\infty$ these wave operators are bounded from $W^{s_p,p}\left({ℝ}^n\right)$ to $L^p\left({ℝ}^n\right)$ for $s_p=\left(n-1\right)|\frac{1}{p}-\frac{1}{2}|$, and this exponent cannot be improved. The same boundedness property holds for the solution operators to more general wave equations with smooth coefficients, as follows from the theory of Fourier integral operators.

In this talk, I will introduce a class of Hardy spaces , for $1\le p\le \infty$, on which suitable Fourier integral operators are bounded. These spaces also satisfy Sobolev embeddings that allow one to recover the boundedness properties of Fourier integral operators on the $L^p$ scale.

The solution operators to wave equations with rough coefficients are typically not Fourier integral operators, and little is known about their $L^p$ mapping properties. I will also indicate how one can combine the Hardy spaces for Fourier integral operators with iterative parametrix constructions to extend the $L^p$ regularity theory for wave equations with smooth coefficients to wave equations with rough coefficients.

This talk is based on joint work with Andrew Hassell and Pierre Portal (Australian National University), and Zhijie Fan, Naijia Liu and Liang Song (Sun Yat-Sen University, China).

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Abstract

We will begin with a brief overview of Fourier decoupling inequalities, highlighting connections to PDEs and number theory. We will then turn to some more recent work, about decoupling on quadratic 3-folds in ${ℝ}^5$, and a bilinear proof of decoupling for the moment curve on ${ℝ}^n$. The former is joint work with Shaoming Guo, Changkeun Oh, Joris Roos and Pavel Zorin-Kranich; the latter is joint work with Shaoming Guo, Zane Kun Li and Pavel Zorin-Kranich.

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