Let $\Omega \subset \mathbb{R}^n$, $n \geq 2$, be a bounded, connected, open set with Lipschitz boundary.

Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either a Dirichlet, Neumann or Robin boundary condition.

If an eigenfunction $u$ associated with the $k$--th eigenvalue has exactly $k$ nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we call the corresponding eigenvalue a Courant-sharp eigenvalue.

We first discuss some known results for the Courant-sharp Dirichlet and Neumann eigenvalues of the Laplacian on Euclidean domains.

We then discuss whether the Robin eigenvalues of the Laplacian on the square are Courant-sharp.

This is based on joint work with B. Helffer (Université de Nantes).

# Past PDE CDT Lunchtime Seminar

The Camassa-Holm (CH) equation is a nonlinear nonlocal dispersive equation which arises as a model for the propagation of unidirectional shallow water waves over a flat bottom. One of the most important features of the CH equation is the existence of peaked travelling waves, also called peakons. The aim of this talk is to review some asymptotic stability result for peakon solutions for CH-type equations as well as to present some new result for higher-order generalization of the CH equation.

Do classical solutions of the compressible Navier-Stokes equations converge to an entropy solution of their inviscid counterparts, the Euler equations? In this talk we present a result which answers this question affirmatively, in the one-dimensional case, for a particular class of fluids. Specifically, we consider gases that exhibit approximately polytropic behaviour in the vicinity of the vacuum, and that are isothermal for larger values of the density (which we call approximately isothermal gases). Our approach makes use of methods from the theory of compensated compactness of Tartar and Murat, and is inspired by the earlier works of Chen and Perepelitsa, Lions, Perthame and Tadmor, and Lions, Perthame and Souganidis. This is joint work with Matthew Schrecker.

We present some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient a = a(·, ·). The model case is driven by the following nonlocal double phase operator,

$$\int \frac{|u(x) − u(y)|^{p−2} (u(x) − u(y))} {|x − y|^{n+sp}} dy+ \int a(x, y) \frac{|u(x) − u(y)|^{ q−2} (u(x) − u(y))} {|x − y|^{n+tq}} dy$$

where $q ≥ p$ and $a(·, ·) = 0$. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are Hölder continuous, whereas similar sharp results for the classical local case do require a to be Hölder continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.

Formation of singularities for the relativistic Euler equations (N. Athanasiou): An archetypal phenomenon in the study of hyperbolic systems of conservation laws is the development of singularities (in particular shocks) in finite time, no matter how smooth or small the initial data are. A series of works by Lax, John et al confirmed that for some important systems, when the initial data is a smooth small perturbation of a constant state, singularity formation in finite time is equivalent to the existence of compression in the initial data. Our talk will address the question of whether this dichotomy persists for large data problems, at least for the system of the Relativistic Euler equations in (1+1) dimensions. We shall also give some interesting studies in (3+1) dimensions. This is joint work with Dr. Shengguo Zhu.

Global Well-Posedness for a Class of Stochastic McKean-Vlasov Equations in One Dimension (A. Mayorcas): We show global well-posedness for a family of parabolic McKean--Vlasov SPDEs with additive space-time white noise. The family of interactions we consider are those given by convolution with kernels that are at least integrable. We show that global well-posedness holds in both the repulsive/defocussing and attractive/focussing cases. Our strategy relies on both pathwise and probabilistic techniques which leverage the Gaussian structure of the noise and well known properties of the deterministic PDEs.

I will present recent results concerning a class of nonlinear parabolic systems of partial differential equations with small cross-diffusion (see doi.org/10.1051/m2an/2018036 and arXiv:1906.08060). Such systems can be interpreted as a perturbation of a linear problem and they have been proposed to describe the dynamics of a variety of large systems of interacting particles. I will discuss well-posedness, regularity, stability and convergence to the stationary state for (strong) solutions in an appropriate Banach space. I will also present some applications and refinements of the above-mentioned results for specific models.

In the seminar, I will talk about a parabolic toy-model for the incompressible Navier-Stokes equations, that satisfies the same energy inequality, same scaling symmetry and which is also super-critical in dimension 3. I will present some partial regularity results that this model shares with the incompressible model and other results that occur only for our model.

Federer’s characterization, which is a central result in the theory of functions of bounded variation, states that a set is of finite perimeter if and only if n−1-dimensional Hausdorff measure of the set's measure-theoretic boundary is finite. The measure-theoretic boundary consists of those points where both the set and its complement have positive upper density. I show that the characterization remains true if the measure-theoretic boundary is replaced by a smaller boundary consisting of those points where the lower densities of both the set and its complement are at least a given positive constant.

The Anderson Hamiltonian is used to model particles moving in

disordered media, it can be thought of as a Schrödiger operator with an

extremely irregular random potential. Using the recently developed theory of

"Paracontrolled Distributions" we are able to define the Anderson

Hamiltonian as a self-adjoint non-positive operator on the 2- and

3-dimensional torus and give an explicit description of its domain.

Then we use these results to solve some semi-linear PDEs whose linear part

is given by the Anderson Hamiltonian, more precisely the multiplicative

stochastic NLS and nonlinear Wave equation.

This is joint work with M. Gubinelli and B. Ugurcan.

The aim of this talk is to give an overview about the structure theory of finite dimensional RCD metric measure spaces. I will first focus on rectifiability, existence, uniqueness and constancy of the dimension of tangents up to negligible sets.

Then I will motivate why boundaries of sets of finite perimeter are natural codimension one objects to look at in this framework and present some recent structure results obtained in their study.

This is based on joint works with Luigi Ambrosio, Elia Bruè and Enrico Pasqualetto.