We study long-time existence and asymptotic behavior for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even $L_p$-Minkowski problems in ${ℝ}^{n+1}$ for $p>-n-1$.

We consider the Muskat problem describing the motion of two unbounded immiscible fluid layers with equal viscosities in vertical or horizontal two-dimensional geometries. We first prove that the mathematical model can be formulated as an evolution problem for the sharp interface separating the two fluids, which turns out to be, in a suitable functional-analytic setting, quasilinear and of parabolic type. Based upon these properties, we then establish the local well-posedness of the problem for arbitrary large initial data and show that the solutions become instantly real-analytic in time and space. Our method allows us to choose the initial data in the class $H^s$, $s\in \left(\frac{3}{2},2\right)$, when neglecting surface tension, respectively in $H^s$, $s\in \left(2,3\right)$, when surface-tension effects are included. Besides, we provide new criteria for the global existence of solutions.

We investigate new settings of velocity averaging lemmas in kinetic theory where a maximal gain of half a derivative is obtained. Specifically, we show that if the densities $f$ and $g$ in the transport equation $v\cdot {\nabla }_{x}f=g$ belong to $L_x^r{L}_v^{r^{\prime }}$, where $2n∕\left(n+1\right)<r\le 2$ and $n\ge 1$ is the dimension, then the velocity averages belong to $H_x^{1∕2}$.

We further explore the setting where the densities belong to $L_x^{4∕3}{L}_v^2$ and show, by completing the work initiated by Pierre-Emmanuel Jabin and Luis Vega on the subject, that velocity averages almost belong to $W_x^{n∕\left(4\left(n-1\right)\right),4∕3}$ in this case, in any dimension $n\ge 2$, which strongly indicates that velocity averages should almost belong to $W_x^{1∕2,2n∕\left(n+1\right)}$ whenever the densities belong to $L_x^{2n∕\left(n+1\right)}{L}_v^2$.

These results and their proofs bear a strong resemblance to the famous and notoriously difficult problems of boundedness of Bochner–Riesz multipliers and Fourier restriction operators, and to smoothing conjectures for Schrödinger and wave equations, which suggests interesting links between kinetic theory, dispersive equations and harmonic analysis.

We consider wave maps on $\left(1+d\right)$-dimensional Minkowski space. For each dimension $d\ge 8$ we construct a negatively curved, $d$-dimensional target manifold that allows for the existence of a self-similar wave map which provides a stable blowup mechanism for the corresponding Cauchy problem.

Recent experimental evidence on rubber has revealed that the internal cracks that arise out of the process, often referred to as cavitation, can actually heal.

We demonstrate that crack healing can be incorporated into the variational framework for quasistatic brittle fracture evolution that has been developed in the last twenty years. This will be achieved for two-dimensional linearized elasticity in a topological setting, that is, when the putative cracks are closed sets with a preset maximum number of connected components.

Other important features of cavitation in rubber, such as near incompressibility and the evolution of the fracture toughness as a function of the cumulative history of fracture and healing, have yet to be addressed even in the proposed topological setting.

All unitary (contractive) perturbations of a given unitary operator $U$ by finite-rank-$d$ operators with fixed range can be parametrized by $d\times d$ unitary (contractive) matrices $\mathrm{\Gamma }$; this generalizes unitary rank-one ($d=1$) perturbations, where the Aleksandrov–Clark family of unitary perturbations is parametrized by the scalars on the unit circle $\mathbb{T}\subset ℂ$.

For a strict contraction $\mathrm{\Gamma }$ the resulting perturbed operator $T_{\mathrm{\Gamma }}$ is (under the natural assumption about star cyclicity of the range) a completely nonunitary contraction, so it admits the functional model.

We investigate the Clark operator, i.e., a unitary operator that intertwines $T_{\mathrm{\Gamma }}$ (written in the spectral representation of the nonperturbed operator $U$) and its model. We make no assumptions on the spectral type of the unitary operator $U$; an absolutely continuous spectrum may be present.

We first find a universal representation of the adjoint Clark operator in the coordinate-free Nikolski–Vasyunin functional model; the word “universal” means that it is valid in any transcription of the model. This representation can be considered to be a special version of the vector-valued Cauchy integral operator.

Combining the theory of singular integral operators with the theory of functional models, we derive from this abstract representation a concrete formula for the adjoint of the Clark operator in the Sz.-Nagy–Foiaş transcription. As in the scalar case, the adjoint Clark operator is given by a sum of two terms: one is given by the boundary values of the vector-valued Cauchy transform (postmultiplied by a matrix-valued function) and the second one is just the multiplication operator by a matrix-valued function.

Finally, we present formulas for the direct Clark operator in the Sz.-Nagy–Foiaş transcription.

We give examples of regular boundary data for the Dirichlet problem for the complex homogeneous Monge–Ampère equation over the unit disc, whose solution is completely degenerate on a nonempty open set and thus fails to have maximal rank.

A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian-type equations on domains in ${ℂ}^n$. The arguments are modeled on the theory of viscosity solutions for real Hessian-type equations developed by Trudinger (1990). As a consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.

We show that the resolvent grows at most exponentially with frequency for the wave equation on a class of stationary spacetimes which are bounded by nondegenerate Killing horizons, without any assumptions on the trapped set. Correspondingly, there exists an exponentially small resonance-free region, and solutions of the Cauchy problem exhibit logarithmic energy decay.

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to {ℝ}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the values of the generalized Gauss map. Furthermore, the interpolating immersions can be chosen to be complete, proper into ${ℝ}^n$ if the prescription of values is proper, and injective if $n\ge 5$ and the prescription of values is injective. We may also prescribe the flux map of the examples.

We also show analogous results for a large family of directed holomorphic immersions $M\to {ℂ}^n$, including null curves.