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We consider time-harmonic acoustic scattering by a compact sound-soft obstacle $\Gamma\subset \mathbb{R}^n$ ($n\geq 2$) that has connected complement $\Omega := \mathbb{R}^n\setminus \Gamma$. This scattering problem is modelled by the…

In this paper we consider the minimization of the functional \[ J[u]:=\int_\Omega |\Delta u|^2+\chi_{\{u>0\}} \] in the admissible class of functions \[ \mathcal A:= \left\{u\in W^{2, 2}(\Omega) {\mbox{ s.t. }} u-u_0\in W^{1,2}_0(\Omega)…

We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $\Omega \Subset \mathbb R^d$ with general -- strictly positive and temporally constant -- Dirichlet boundary…

Robotic motion optimization often focuses on task-specific solutions, overlooking fundamental motion principles. Building on Riemannian geometry and the calculus of variations (often appearing as indirect methods of optimal control), we…

We consider the first eigenvalue $\lambda_1(\Omega,\sigma)$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $\Omega$ with smooth boundary, $\sigma\in\bf R$ being the Robin boundary parameter. When $\sigma>0$…

The paper concerns the necessary maximum principle for robust optimal control problems of quadratic BSDEs. The coefficient of the systems depends on the parameter $\theta$, and the generator of BSDEs is of quadratic growth in $z$. Since the…

We consider the evolution operator exp(-it(-Delta+V)) associated with a Schr\"odinger operator on a Riemannian manifold (M,g). We are interested in the dependence of this operator on V running in L^p(M). Under a geometrical hypothetis, we…

Let $\Omega$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_{\Omega}$ denote the torsion function for $\Omega$. It is known that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian…

This article is concerned in the first place with the short-time observability constant of the heat equation from a subdomain $\omega$ of a bounded domain $M$. The constant is of the form $e^{\frac{K}{T}}$, where $K$ depends only on the…

In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schr\"odinger equation. After presenting the basic properties of the equation, we give a self contained proof of…

We prove a sharp quantitative version of the Faber--Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit $\delta(f;\Omega)$ which measures by how much the STFT of a function $f\in L^2(\mathbb R)$…

In the recent years the Schr\"odinger problem has gained a lot of attention because of the connection, in the small-noise regime, with the Monge-Kantorovich optimal transport problem. Its optimal value, the \emph{entropic cost}…

We consider a shape optimization problem for the persistence threshold of a biological species dispersing in a periodically fragmented environment, the unknown shape corresponding to the portion of the habitat which is favorable to the…

In this note we analyze how perturbations of a ball $\mathfrak{B}_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty-$eigenvalues when a volume constraint $\\mathscr{L}^n(\Omega) =…

We describe the isometry group of $L^2(\Omega, M)$ for Riemannian manifolds $M$ of dimension at least two with irreducible universal cover. We establish a rigidity result for the isometries of these spaces: any isometry arises from an…

We investigate a linearised Calder\'on problem in a two-dimensional bounded simply connected $C^{1,\alpha}$ domain $\Omega$. After extending the linearised problem for $L^2(\Omega)$ perturbations, we orthogonally decompose $L^2(\Omega) =…

We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator $A_B$ with a…

Weakly harmonic maps from a domain $\Omega$ (the upper half-space $\Rd$ or a bounded $C^{1,\alpha}$ domain, $\alpha\in (0,1]$) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes…

We consider heat operators on a bounded domain $\Omega \subseteq \mathbb{R}^n$, with a critically singular potential diverging as the inverse square of the distance to $\partial \Omega$. While null boundary controllability for such…

We consider the vorticity gradient growth of solutions to the two-dimensional Euler equations in domains without boundary, namely in the torus $\mathbb{T}^{2}$ and the whole plane $\mathbb{R}^{2}$. In the torus, whenever we have a steady…