Talks

In this talk, we discuss non-local operators like elliptic integrodifferential operators of fractional type
\[
Au := p.v. \int_{\mathbb{R}^d} \frac{u(x) - u(y)}{|x-y|^{d+2\alpha}}dy \quad \quad (1)
\]
or the Stokes operator with bounded measurable coefficients $\mu$, formally given by
\[
Au := -div( \mu \nabla u ) + \nabla \phi, \quad div(u) = 0 \; in \; \mathbb{R}^d. \quad \quad (2)
\]
These operators satisfy $L^{2}$-resolvent estimates of the form
\[
|| \lambda ( \lambda + A )^{-1} f ||_{L^2} \leq C || f ||_{L^2} \quad (f \in L^2(\mathbb{R^d}))
\]
for $\lambda$ in some complex sector $\left\{z \in \mathbb{C} \smallsetminus {0} : | arg(z) | < \theta \right\}$. We describe how analogues of such a resolvent estimate can be established in $L^{p}$ by virtue of certain non-local Caccioppoli inequalities. Such estimates build the foundation for many important functional analytic properties of these operators like maximal $L^{q}$-regularity.

More precisely, we establish resolvent estimates in $L^{p}$ for $p$ satisfying
\[
\left|\frac{1}{p} - \frac{1}{2} \right| < \frac{\alpha}{d}
\]
in the case (1) and
\[
\left|\frac{1}{p} - \frac{1}{2}\right| < \frac{1}{d} \quad \quad (3)
\]
in the case (2). This resembles a well-known situation for elliptic systems in divergence form with $L^{\infty}$-coefficients. Here, important estimates like Gaussian upper bounds for the semigroup cease to exist and the $L^{p}$-extrapolation has be concluded by other means. In particular, for elliptic systems one can establish resolvent bounds for numbers p that satisfy (3) and if $d \geq 3$, Davies constructed examples which show that corresponding resolvent bounds do not hold for numbers $1 < p < \infty$ that satisfy
\[
\left|\frac{1}{p} - \frac{1}{2} \right| > \frac{1}{d}.
\]
These elliptic results give an indication that the result for the Stokes operator with $L^{\infty}$-coefficients is optimal as well.

We study integral input-to-state stability of bilinear systems with
unbounded control operators and derive natural sufficient conditions. The
results are applied to a bilinearly controlled Fokker-Planck equation.

Gauss quadrature rules are nowadays not only a powerful tool to compute integrals in many scientific applications, but also a numerical method that most people in the scientific community at least heard of at some point in there life.
Even if they are not the only tool to compute integral numerically, they provide the possibility to integrate any function multiplied by a given weight function (or measure), by estimating the integral of the product using a weighted sum of the function evaluations at given values (nodes).
Classical measures are well known (e.g Legendre, Chebyshev, Laguerre, Hermite), and their associated quadrature rules are well studied and documented in the literature.
While some measures allow to estimate integrals over infinite intervals (e.g Laguerre or Hermite), others also allow to integrate a function with singularities (e.g Chebyshev of the first, third and fourth kind).
However, the use of non-classical measures for specific applications can also be considered, and even this is not often used in the community, many algorithms exist to compute the nodes and weights of those quadrature rules.
In this talk we will give a quick overview of those algorithms, their efficiency, numerical stability, and some current challenge that still need to be solved.
Furthermore, under some conditions, all Gauss quadrature rules share some common properties, in particular when considering a large number of nodes.
We will give a quick overview of those common asymptotic properties, and show how they can be generalized to other applications (e.g barycentric Lagrange interpolation).
While some of those properties have been proven in particular cases, we will present some situations where they have not been proved theoretically yet, or still need to be verified.