The APDE seminar on Monday, 8/24, will be given by Ugur G. Abdulla (Okinawa Institute of Science and Technology) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Kolmogorov Problem and Wiener-type Criteria in Potential Theory

Abstract: The central problem in the Analysis of PDEs is understanding the nature of singular-
ities that reflect natural phenomena. This talk will present a full characterization of the fundamental boundary singularity, and equivalently, the unique solvability of the singular
Dirichlet problem for the elliptic and parabolic PDEs. The results are threefold. We prove
a new Wiener-type criterion for the ”geometric” characterization of the removability of the
fundamental singularity for arbitrary open sets in terms of the fine-topological thinness of
the complementary set near the singularity point. In the special case when the surface
of revolution forms the boundary of the open set near the singularity point, we establish
a Kolmogorov-Petrovsky-type test to characterize the removability of the singularity and
uniqueness. Finally, in the special case when a continuous graph locally represents the boundary of the open set, the minimal thinness criterion for the removability of the singularity is expressed in terms of the minimal regularity of the boundary manifold at the singularity point. From the probabilistic point of view, the criteria present an asymptotic
law for conditional Brownian motion. In the topological context, the criteria present a full characterization of the neighborhood base of the boundary singularity point in the minimal-
fine topology. In the more general framework, the talk will outline a program for the full characterization of the singularities formed by the elliptic and parabolic PDEs.

The APDE seminar on Monday, 5/4, will be given by Claude Warnick (University of Cambridge) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Small data global well posedness of the vacuum Einstein equations in centred Newman-Unti gauge

Abstract:The global stability of Minkowski spacetime is a foundational question in mathematical relativity, and the proof of this result by Christodoulou and Klainerman in their `94 monograph is a landmark achievement in the field. The result has been re-proved in several ways, most notably by Lindblad-Rodnianski `04. In this talk I shall present a new proof with Jonathan Luk and Sun-Jin Oh based on a single coordinate system constructed from outgoing lightcones emanating from a singe central geodesic. In this coordinate system the Einstein equations have a remarkably elegant structure, as noted by Newman-Unti in `62, which can be exploited to give an efficient proof of a wide range of results in the literature.

The APDE seminar on Monday, 4/27, will be given by Juhi Jang (USC) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Stable Larson-Penston collapse

Abstract: I will discuss recent progress on mathematical construction of self-similar solutions to the Euler-Poisson system describing gravitational collapse and nonlinear stability of the Larson-Penston collapse against radially symmetric perturbations. At the heart of the latter stability result is the ground state character of the Larson-Penston solution featuring important monotonicity properties. The talk is based on joint works with Yan Guo, Mahir Hadzic and Matthew Schrecker.

The APDE seminar on Monday, 4/20, will be given by Josef Greilhuber (Stanford) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Non-density of nodal lines in the clamped plate problem

Abstract: It is well known that the nodal set (i.e., zero set) of an eigenfunction of the Laplacian – modelling a fundamental mode of vibration of an elastic membrane – is dense at the scale of its characteristic wave-length.

In contrast, we show that the nodal set of high energy eigenfunctions of the clamped plate problem – a fourth order PDE modeling a vibrating metal plate – is not necessarily dense and can in fact exhibit macroscopic “nodal voids”.

Specifically, we construct small deformations of the unit disk admitting a clamped plate eigenfunction of arbitrarily high frequency that does not vanish in a disk of radius ~0.44.

Remarkably, this radius is sharp, simultaneously providing the asymptotic upper bound for the size of such circular nodal voids among small perturbations of the disk.

The APDE seminar on Monday, 4/13, will be given by Ioan Bejenaru (UCSD) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: An effective resolution space for the Schroedinger equation

Abstract: In the analysis of nonlinear dispersive PDEs it is important to design resolution spaces which replicate key estimates available for free solutions. While most resolution spaces transfer the linear estimates, also known as Strichartz estimates, this is not always the case for bilinear/multilinear restriction estimates. In this talk we propose a new structure which transfers the classical bilinear L^2 estimate without loss, among other desirable properties. This is done in the context of the Schroedinger equation.

The APDE seminar on Monday, 4/6, will be given by our own Daniel Tataru in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Global solutions for 3D gravity water waves

Abstract: The aim of the talk is to describe work in progress on the problem of local and global well-posedness in the small data, low regularity regime for gravity waves in a fluid of infinite depth, and infinite width, in spatial dimension n ≥ 3 and higher. This is joint work with Mihaela Ifrim and Ben Pineau.

The APDE seminar on Monday, 3/16, will be given by Yuefeng Song (Stanford) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Weak null singularity for the Einstein–Euler system

Abstract: We study the behavior of a self-gravitating perfect relativistic fluid satisfying the Einstein–Euler system in the presence of a weak null terminal spacetime singularity. This type of singularities is expected in the interior of generic dynamical black holes. In the vacuum case, weak null singularities have been constructed locally by Luk, where the metrics extend continuously to the singularities while the Christoffel symbols fail to be square integrable in any neighborhood of any point on the singular boundaries. We prove that this type of singularities persists in the presence of a self-gravitating fluid. Moreover, using the fact that the speed of sound is strictly less than the speed of light, we prove that the fluid variables also extend continuously to the singularity.

The APDE seminar on Monday, 3/9, will be given by Jesús Oliver (Cal State East Bay) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Global existence for a Fritz John equation in expanding FLRW spacetimes

Abstract: We study the semilinear wave equation

\[\square_{\mathbf g_p}\phi = (\partial_t \phi)^2\]

on expanding FLRW spacetimes with spatial slices $\mathbb{R}^3$ and power-law scale factor $a(t)=t^p$, where $0 < p \le 1$. This equation extends the classical Fritz John blow-up model on Minkowski space (the case $p=0$) to a non-stationary cosmological background.

In Minkowski space, nontrivial solutions arising from smooth, compactly supported data blow up in finite time. In contrast, we prove that for $0 < p \le 1$, sufficiently small, smooth, compactly supported initial data generate global-in-time solutions toward the future.

Earlier joint work with Costa and Franzen treated the accelerated regime $p>1$, where global existence follows from the integrability of the inverse scale factor. In the present setting, this mechanism is unavailable. Instead, we develop a vector field method adapted to FLRW geometry that exploits the interaction between dispersion and spacetime expansion to suppress the nonlinear blow-up mechanism. The argument relies on commuting the Laplace–Beltrami operator with a boosts-free subset of the Poincaré algebra and establishing Klainerman–Sideris type estimates adapted to the non-stationary background.

The approach provides a robust framework for quantifying the regularizing effect of cosmological expansion and is expected to extend to a broader class of nonlinear wave equations.

The APDE seminar on Monday, 2/23, will be given by Richard Bamler (Berkeley) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

Abstract: We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons.

Our approach relies on two new techniques. The first, called the PDE-ODI principle, converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This framework bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. The second combines a new leading mode condition combined with an “induction over thresholds” argument” to obtain even finer asymptotic estimates.

This is joint work with Yi Lai.

The APDE seminar on Monday, 2/9, will be given by Federico Franceschini (Stanford) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: The dimension and behaviour of singularities of stable solutions to semilinear elliptic equations

Abstract: Let f(t) be a convex, positive, increasing nonlinearity. It is known that stable solutions of -\Delta u =f(u) can be singular (i.e., unbounded) if the dimension n>9.

Brezis asked wether, if x=0 is such a singular point, then in general f'(u(x)) blows-up like ~|x|^{2-n}, as it happens in the model cases f(u)=u^p, f(u)=e^u.

In this talk I will show the answer to this question and the interesting consequences it entails. This is a joint work with Alessio Figalli.