Sur toute variété compacte de dimension $n\ge 3$ à bord, on prescrit toute partie finie du spectre de Steklov dans une classe conforme donnée. En particulier, on prescrit la multiplicité des valeurs propres. Sur une surface compacte à bord donnée, on montre que la multiplicité de la $k$-ième valeur propre est bornée indépendamment de la métrique. Sur le disque, on donne des résultats plus précis : la multiplicité de la première et la deuxième valeurs propres non nulles sont au plus 2 et 3 respectivement. Pour le problème de Steklov–Neumann sur le disque, on montre que la multiplicité de la $k$-ième valeur propre non nulle est au plus $k+1$.

On any compact manifold of dimension $n\ge 3$ with boundary, we prescribe any finite part of the Steklov spectrum within a given conformal class. In particular, we prescribe the multiplicity of the first eigenvalues. On a compact surface with boundary, we show that the multiplicity of the $k$-th eigenvalue is bounded independently of the metric. On the disk, we give more precise results: the multiplicity of the first and second positive eigenvalues are at most 2 and 3 respectively. For the Steklov–Neumann problem on the disk, we prove that the multiplicity of the $k$-th positive eigenvalue is at most $k+1$.

We study weakly stable semilinear hyperbolic boundary value problems with highly oscillatory data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency $\beta$ in the hyperbolic region. As a consequence of this degeneracy there is an amplification phenomenon: outgoing waves of amplitude $O\left({\epsilon }^2\right)$ and wavelength $\epsilon$ give rise to reflected waves of amplitude $O\left(\epsilon \right)$, so the overall solution has amplitude $O\left(\epsilon \right)$. Moreover, the reflecting waves emanate from a radiating wave that propagates in the boundary along a characteristic of the Lopatinskii determinant.

An approximate solution that displays the qualitative behavior just described is constructed by solving suitable profile equations that exhibit a loss of derivatives, so we solve the profile equations by a Nash–Moser iteration. The exact solution is constructed by solving an associated singular problem involving singular derivatives of the form ${\partial }_{x^{\prime }}+\beta {\partial }_{{\theta }_0}∕\epsilon$, $x^{\prime }$ being the tangential variables with respect to the boundary. Tame estimates for the linearization of that problem are proved using a first-order (wavetrain) calculus of singular pseudodifferential operators constructed in a companion article (“Singular pseudodifferential calculus for wavetrains and pulses”, arXiv 1201.6202, 2012). These estimates exhibit a loss of one singular derivative and force us to construct the exact solution by a separate Nash–Moser iteration.

The same estimates are used in the error analysis, which shows that the exact and approximate solutions are close in $L^{\infty }$ on a fixed time interval independent of the (small) wavelength $\epsilon$. The approach using singular systems allows us to avoid constructing high-order expansions and making small divisor assumptions. Our analysis of the exact singular system applies with no change to the case of pulses, provided one substitutes the pulse calculus from the companion paper for the wavetrain calculus.

We prove the existence of solutions to the 1-harmonic flow — that is, the formal gradient flow of the total variation of a vector field with respect to the $L^2$-distance — from a domain of ${ℝ}^m$ into a hyperoctant of the $N$-dimensional unit sphere, ${\mathbb{S}}_{+}^{N-1}$, under homogeneous Neumann boundary conditions. In particular, we characterize the lower-order term appearing in the Euler–Lagrange formulation in terms of the “geodesic representative” of a BV-director field on its jump set. Such characterization relies on a lower semicontinuity argument which leads to a nontrivial and nonconvex minimization problem: to find a shortest path between two points on ${\mathbb{S}}_{+}^{N-1}$ with respect to a metric which penalizes the closeness to their geodesic midpoint.

We show that $C^{\ast }$-algebras of the form $C\left(X\right)\otimes \mathcal{Z}$, where $X$ is compact and Hausdorff and $\mathcal{Z}$ denotes the Jiang–Su algebra, have decomposition rank at most $2$. This amounts to a dimension reduction result for $C^{\ast }$-bundles with sufficiently regular fibres. It establishes an important case of a conjecture on the fine structure of nuclear $C^{\ast }$-algebras of Toms and Winter, even in a nonsimple setting, and gives evidence that the topological dimension of noncommutative spaces is governed by fibres rather than base spaces.

We prove scattering for a massless wave equation which is critical in two space dimensions. Our method combines conformal inversion with decay estimates from Struwe’s previous work on global existence of a similar equation.

We consider the following Hamiltonian equation on a special manifold of rational functions:

$$i{\partial }_{t}u=\Pi \left(|u{|}^{2}u\right)+\alpha \left(u|1\right),\alpha \in ℝ,$$

where $\Pi$ denotes the Szegő projector on the Hardy space of the circle ${\mathbb{S}}^1$. The equation with $\alpha =0$ was first introduced by Gérard and Grellier as a toy model for totally nondispersive evolution equations. We establish the following properties for this equation. For $\alpha <0$, any compact subset of initial data leads to a relatively compact subset of trajectories. For $\alpha >0$, there exist trajectories on which high Sobolev norms exponentially grow in time.

That the weak solutions of degenerate parabolic PDEs modelled on the inhomogeneous $p$-Laplace equation

$$u_t-÷\left(|\nabla u{|}^{p-2}\nabla u\right)=f\in L^{q,r},p>2$$

are $C^{0,\alpha }$, for some $\alpha \in \left(0,1\right)$, has been known for almost 30 years. What was hitherto missing from the literature was a precise and sharp knowledge of the Hölder exponent $\alpha$ in terms of $p,q,r$ and the space dimension $n$. We show in this paper that

$$\alpha =\frac{\left(pq-n\right)r-pq}{q\left[\left(p-1\right)r-\left(p-2\right)\right]}$$

using a method based on the notion of geometric tangential equations and the intrinsic scaling of the $p$-parabolic operator. The proofs are flexible enough to be of use in a number of other nonlinear evolution problems.

We introduce a class of functions near zero on the logarithmic cover of the complex plane that have convergent expansions into generalized power series. The construction covers cases where noninteger powers of $z$ and also terms containing $logz$ can appear. We show that, under natural assumptions, some important theorems from complex analysis carry over to this class of functions. In particular, it is possible to define a field of functions that generalize meromorphic functions, and one can formulate an analytic Fredholm theorem in this class. We show that this modified analytic Fredholm theorem can be applied in spectral theory to prove convergent expansions of the resolvent for Bessel type operators and Laplace–Beltrami operators for manifolds that are Euclidean at infinity. These results are important in scattering theory, as they are the key step in establishing analyticity of the scattering matrix and the existence of generalized eigenfunctions at points in the spectrum.