Analysis and Partial Differential Equations

Abstract

The fractional Laplacian in Euclidean space is a non-local operator. We establish Liouville type theorems, non-existence of positive solutions of Dirichlet problem involving the fractional Laplacian. We obtain the equivalence between a partial differential equation and an integral equation from the uniqueness of harmonic functions on a half space. Applying the method of moving planes in integral forms, we prove the non-existence of positive solutions of an integral equation.

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Abstract

For smooth bounded Euclidean domains, we prove upper and lower $L^2$ bounds on the boundary data of Neumann eigenfunctions, and prove quasi-orthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of eigenvalue; this is achieved by working with an appropriate norm for boundary functions, which includes a ‘spectral weight’, that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for ‘whispering gallery’ type eigenfunctions.

As an application, we give bounds on the distance from an arbitrary positive real number $E>0$ to the nearest Neumann eigenvalue, in terms of boundary normal-derivative data of a trial function $u$ solving the Helmholtz equation $\left(\Delta -E\right)u=0$. These improve over previously known bounds by a factor of $E^{5∕6}$. This can be used to design an improved “method of particular solutions” for finding Neumann eigenfunctions numerically.

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Abstract

We study the least gradient problem in two special cases:

1. the natural boundary conditions are imposed on a part of the stricly convex domain;
2. the Dirichlet type data are imposed on a boundary of a rectangle.

We show the existence of solutions and study properties of solutions for special cases of the data.

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Abstract

Let $M$ be a von Neumann algebra and let $\left(L_p\left(M\right),\parallel .{\parallel }_p\right)$, $1\le p<\infty$ be Haagerup’s $L_p$-space on $M$. The main ideas of the proof that the differentiability properties of $\parallel .{\parallel }_p$ are precisely the same as those of classical (commutative) $L_p$-spaces are presented. Our main instruments are the theories of multiple operator integrals and singular traces.

This is joint work with Fedor Sukochev, Denis Potapov and Dimitry Zanin.

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Abstract

Maximal regularity can often be used to obtain a priori estimates which give global existence results. In this talk I will explain a new approach to maximal $L^p$-regularity for parabolic PDEs with time dependent generator $A\left(t\right)$. Here we do not assume any continuity properties of $A\left(t\right)$ as a function of time. We show that there is an abstract operator theoretic condition on A(t) which is sufficient to obtain maximal $L^p$-regularity. As an application I will obtain an optimal $L^p\left(L^q\right)$ regularity result in the case each $A\left(t\right)$ is a system of $2m$-th order elliptic differential operator on ${ℝ}^d$ in non-divergence form. The main novelty is that the coefficients are merely measurable in time and we allow the full range $1<p,q<\infty$. This talk is based on joint work with Chiara Gallarati.

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Abstract

We obtain the boundedness of the $n$-th dimensional Calderón-Coifman-Journé multicommutator from $L^{p_0}\left({ℝ}^n\right)\times L^{p_1}\left({ℝ}^n\right)\times \cdots \times L^{p_k}\left({ℝ}^n\right)$ to $L^p\left({ℝ}^n\right)$ in the largest possible open set of indices $\left(\frac{1}{p_0},\frac{1}{p_1},\dots ,\frac{1}{p_k}\right)$ with $\frac{1}{p_0}+\frac{1}{p_1}+\cdots +\frac{1}{p_k}=\frac{1}{p}$, which is the range $\frac{1}{k+1}<p<\infty$. The proof exploits the limited smoothness of the symbol of the multicommutator via a new multilinear multiplier theorem for symbols of restricted smoothness which lie locally in certain Sobolev spaces. Our multiplier approach to this problem is a new contribution in the understanding of Calderón’s commutator program.

This is a joint work with Loukas Grafakos, Danqing He and Hanh Van Nguyen.

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