Seminar

### FALL 2021 Nonlinear Analysis Special Lecture Part 2 of 3

2021-12-01
Online via Zoom
Associate Professor Kabe Moen , The University of Alabama Title: Fractional Integrals and weights Part II Abstract:

In this talk we will cover the one weight inequalities for the fractional integral operator and related fractional maximal operator. We will discuss the background of A_p weights and A_{p,q} weights and go over the dyadic decomposition of the fractional integral operator. We will also cover auxiliary results like sharp constants and.

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Seminar

### FALL 2021 Nonlinear Analysis Seminar Series

2021-12-07
Online via Zoom
★DISTINGUISHED LECTURE Professor Yoshikazu Giga , The University of Tokyo Title: On a singular limit of a single-well Modica-Mortola functional and its applications Abstract:

It is important to describe the motion of phase boundaries by macroscopic energy in the process of phase transitions. Typical energy describing the phenomena is the van der Waals energy, which is also called a Modica-Mortola functional with a double-well potential or the Allen-Cahn functional. It turns out that it is also important to consider the Modica-Mortola functional with a single-well potential since it is often used in various settings including the Kobayashi-Warren-Carter energy, which is popular in materials science. It is very fundamental to understand the singular limit of such a type of energies as the thickness parameter of a diffuse interface tends to zero. In the case of double-well potentials, such a problem is well-studied and it is formulated, for example, as the Gamma limit under L 1

convergence.

However, if one considers the Modica-Mortola functional, it turns out that L 1

convergence is too rough even in the one-dimensional problem.

We characterize the Gamma limit of a single-well Modica-Mortola functional under the topology which is finer than L 1

topology. In a one-dimensional case, we take the graph convergence. In higher-dimensional cases, it is more involved. As an application, we give an explicit representation of a singular limit of the Kobayashi-Warren-Carter energy. Since the higher-dimensional cases can be reduced to the one-dimensional case by a slicing argument, studying the one-dimensional case is very fundamental. A key idea to study the one-dimensional case is to introduce “an unfolding of a function” by changing an independent variable by the arc-length parameter of its graph. This is based on a joint work with Jun Okamoto (The University of Tokyo), Masaaki Uesaka (The University of Tokyo, Arithmer Inc.), and Koya Sakakibara (Okayama University of Science, RIKEN).

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Seminar

### FALL 2021 Nonlinear Analysis Special Lecture Part 3 of 3

2021-12-08
Online via Zoom
Associate Professor Kabe Moen , The University of Alabama Title: Fractional Integrals and weights Part III Abstract:

In this talk we will cover the two weight inequalities for the fractional integral operator and related fractional maximal operator. We will discuss the background of two-weight inequalities and Sawyer’s testing conditions and two weight characterization. We will also discuss bump conditions and some open questions.

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Seminar

### FALL 2021 Nonlinear Analysis Special Lecture Part 1 of 2

2021-12-13
Online via Zoom
Mr. Julian Weigt , Aalto University Title: Higher dimensional techniques for the regularity of maximal functions Abstract:

It has been an open question if maximal operators M satisfy the endpoint regularity bound m a t h o p v a r ( M f ) ≤ C v a r ( f )

. So far the majority of the known results has been in one dimension. I give an overview of the progress on this question with a focus on the techniques. Next I present the techniques used in the recent proofs of m a t h o p v a r ( M f ) ≤ C v a r ( f )

for several maximal operators in higher dimensions. They are mostly geometric measure theoretic in the spirit of the relative isoperimetric inequality and involve a stopping time and various covering arguments.

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Seminar

### FALL 2021 Nonlinear Analysis Special Lecture Part 2 of 2

2021-12-15
Online via Zoom
Mr. Julian Weigt , Aalto University Title: Higher dimensional techniques for the regularity of maximal functions Abstract:

Two weight inequalities. Sawyer’s testing conditions, bump conditions, and further work.

It has been an open question if maximal operators M satisfy the endpoint regularity bound $$mathop{\mathrm{var}}(Mf) \leq C \mathop{\mathrm{var}}(f)$$. So far the majority of the known results has been in one dimension. I give an overview of the progress on this question with a focus on the techniques. Next I present the techniques used in the recent proofs of $$mathop{\mathrm{var}}(Mf) \leq C \mathop{\mathrm{var}}(f)$$ for several maximal operators in higher dimensions. They are mostly geometric measure theoretic in the spirit of the relative isoperimetric inequality and involve a stopping time and various covering arguments.

Please click here to register *After registering, you will receive a confirmation email containing information about joining the meeting.