We investigate the behaviour of rotating incompressible flows near a nonflat horizontal bottom. In the flat case, the velocity profile is given explicitly by a simple linear ODE. When bottom variations are taken into account, it is governed by a nonlinear PDE system, with far less obvious mathematical properties. We establish the well-posedness of this system and the asymptotic behaviour of the solution away from the boundary. In the course of the proof, we investigate in particular the action of pseudodifferential operators in nonlocalized Sobolev spaces. Our results extend an older paper of Gérard-Varet (J. Math. Pures Appl. $\left(9\right)$ 82:11 (2003), 1453–1498), restricted to periodic variations of the bottom, using the recent linear analysis of Dalibard and Prange (Anal. & PDE 7:6 (2014), 1253–1315).

We prove uniqueness results for a Calderón-type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth-order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometrical optics solutions which reduce the Calderón-type problem to a tomography problem for 2-tensors. The arguments in this paper allow us to establish partial data results for elliptic systems that generalize the scalar results due to Kenig, Sjöstrand and Uhlmann.

The Euclidean mixed isoperimetric-isodiametric inequality states that the round ball maximizes the volume under constraint on the product between boundary area and radius. The goal of the paper is to investigate such mixed isoperimetric-isodiametric inequalities in Riemannian manifolds. We first prove that the same inequality, with the sharp Euclidean constants, holds on Cartan–Hadamard spaces as well as on minimal submanifolds of ${ℝ}^n$. The equality cases are also studied and completely characterized; in particular, the latter gives a new link with free-boundary minimal submanifolds in a Euclidean ball. We also consider the case of manifolds with nonnegative Ricci curvature and prove a new comparison result stating that metric balls in the manifold have product of boundary area and radius bounded by the Euclidean counterpart and equality holds if and only if the ball is actually Euclidean.

We then consider the problem of the existence of optimal shapes (i.e., regions minimizing the product of boundary area and radius under the constraint of having fixed enclosed volume), called here isoperimetric-isodiametric regions. While it is not difficult to show existence if the ambient manifold is compact, the situation changes dramatically if the manifold is not compact: indeed we give examples of spaces where there exists no isoperimetric-isodiametric region (e.g., minimal surfaces with planar ends and more generally $C^0$-locally asymptotic Euclidean Cartan–Hadamard manifolds), and we prove that on the other hand on $C^0$-locally asymptotic Euclidean manifolds with nonnegative Ricci curvature there exists an isoperimetric-isodiametric region for every positive volume (this class of spaces includes a large family of metrics playing a key role in general relativity and Ricci flow: the so-called Hawking gravitational instantons and the Bryant-type Ricci solitons).

Finally we prove the optimal regularity of the boundary of isoperimetric-isodiametric regions: in the part which does not touch a minimal enclosing ball, the boundary is a smooth hypersurface outside of a closed subset of Hausdorff codimension $8$, and in a neighborhood of the contact region, the boundary is a $C^{1,1}$ hypersurface with explicit estimates on the $L^{\infty }$ norm of the mean curvature.

We consider the semilinear heat equation in large dimension $d\ge 11$

$${\partial }_{t}u=\Delta u+|u{|}^{p-1}u,p=2q+1,q\in \mathbb{N},$$

on a smooth bounded domain $\Omega \subset {ℝ}^d$ with Dirichlet boundary condition. In the supercritical range $p\ge p\left(d\right)>1+\frac{4}{d-2}$, we prove the existence of a countable family ${\left(u_{\ell }\right)}_{\ell \in \mathbb{N}}$ of solutions blowing up at time $T>0$ with type II blow up:

$$\parallel u_{\ell }\left(t\right){\parallel }_{L^{\infty }}\sim C{\left(T-t\right)}^{-c_{\ell }}$$

with blow-up speed $c_{\ell }>\frac{1}{p-1}$. The blow up is caused by the concentration of a profile $Q$ which is a radially symmetric stationary solution:

$$u\left(x,t\right)\sim \frac{1}{\lambda {\left(t\right)}^{\frac{2}{p-1}}}Q\left(\frac{x-x_0}{\lambda \left(t\right)}\right),\lambda \sim C\left(u_n\right){\left(T-t\right)}^{\frac{c_{\ell }\left(p-1\right)}{2}},$$

at some point $x_0\in \Omega$. The result generalizes previous works on the existence of type II blow-up solutions, which only existed in the radial setting. The present proof uses robust nonlinear analysis tools instead, based on energy methods and modulation techniques. This is the first nonradial construction of a solution blowing up by concentration of a stationary state in the supercritical regime, and it provides a general strategy to prove similar results for dispersive equations or parabolic systems and to extend it to multiple blow ups.