Suitable Geometric Conditions Research Articles

A variational method for the sheath potential of hypersonic leading edges with space-charge limitations

Electron transpiration cooling for the leading edges (LE) of hypersonic aircraft utilizes thermionic emission; however, space-charge effects limit the electron emission rate, potentially diminishing the efficiency of this cooling mechanism. We develop a variational weak form of the Poisson equation that describes the sheath potential and then numerically solve it using the finite element method. This formulation has two main benefits: (1) the space-charge limit condition can be incorporated as a constraint and (2) it allows for the analysis of three-dimensional geometries with complex boundary conditions. We demonstrate that the current emitted from the surface of an LE is generally a small fraction of the Child–Langmuir limit due to space charge. We then propose several methods to enhance the emitted current from the surface and to boost the cooling effect of thermionic emission. These include increasing the plasma density, applying a negative surface potential, and using fringe fields under suitable geometric conditions. For a LaB6 emitting LE, the total emitted current is shown to be minimal and independent of the temperature of a surface with floating potential. However, when a negative potential is applied and the surface is heated, the emitted current follows the Richardson–Dushman relationship up to a critical temperature, beyond which it remains constant. At an applied surface potential of −5 V, the critical temperature is around 1700 K.

Matrix weights and regularity for degenerate elliptic equations

We prove local boundedness, Harnack inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form. Degeneracy is via a non negative, symmetric, measurable matrix-valued function Q(x) and two suitable non negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev–Poincaré inequalities. Data integrability is close to L1 and it is exploited in terms of suitable version of Stummel-Kato class that in some cases is also necessary to the regularity.

Optimal control of the Fokker-Planck equation under state constraints in the Wasserstein space

We analyze a problem of optimal control of the Fokker-Planck equation with state constraints in the Wasserstein space of probability measures. We give first-order necessary conditions for optimality in the form of a mean-field game system of partial differential equations associated with an exclusion condition. Under suitable geometric conditions on the constraint we prove that optimal controls are Lipschitz continuous.

Convergence of the forward-backward algorithm: beyond the worst-case with the help of geometry

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or Łojasiewicz properties. These geometrical notions are usually local by nature, and may fail to describe the fine geometry of objective functions relevant in inverse problems and signal processing, that have a nice behaviour on manifolds, or sets open with respect to a weak topology. Motivated by this observation, we revisit those geometric notions over arbitrary sets. In turn, this allows us to present several new results as well as collect in a unified view a variety of results scattered in the literature. Our contributions include the analysis of infinite dimensional convex minimization problems, showing the first Łojasiewicz inequality for a quadratic function associated to a compact operator, and the derivation of new linear rates for problems arising from inverse problems with low-complexity priors. Our approach allows to establish unexpected connections between geometry and a priori conditions in inverse problems, such as source conditions, or restricted isometry properties.

On asymptotic rigidity and continuity problems in nonlinear elasticity on manifolds and hypersurfaces

Intrinsic nonlinear elasticity deals with the deformations of elastic bodies as isometric immersions of Riemannian manifolds into the Euclidean spaces (see Ciarlet [9,10]). In this paper, we study the rigidity and continuity properties of elastic bodies for the intrinsic approach to nonlinear elasticity. We first establish a geometric rigidity estimate for mappings from Riemannian manifolds to spheres (in the spirit of Friesecke–James–Müller [23]), which is the first result of this type for the non-Euclidean case as far as we know. Then we prove the asymptotic rigidity of elastic membranes under suitable geometric conditions. Finally, we provide a simplified geometric proof of the continuous dependence of deformations of elastic bodies on the Cauchy–Green tensors and second fundamental forms, which extends the Ciarlet–Mardare theorem in [18] to arbitrary dimensions and co-dimensions.

Mass propagation for electromagnetic Schrödinger evolutions

We investigate the validity of Gaussian lower bounds for solutions to an electromagnetic Schrödinger equation with a bounded time-dependent complex electric potential and a time-independent vector magnetic potential. We prove that, if a suitable geometric condition is satisfied by the vector potential, then positive masses inside of a bounded region at a single time propagate outside the region, provided a suitable average in space–time cylinders is taken.

Uniform null controllability for parabolic equations with discontinuous diffusion coefficients

In this article, we study the null-controllability of a heat equation in a domain composed of two media of different constant conductivities. In particular, we are interested in the behavior of the system when the conductivity of the medium on which the control does not act goes to infinity, corresponding at the limit to a perfectly conductive medium. In that case, and under suitable geometric conditions, we obtain a uniform null-controllability result. Our strategy is based on the analysis of the controllability of the corresponding wave operators and the transmutation technique, which explains the geometric conditions.

Transfer maps in generalized group homology via submanifolds

Let N \subset M be a submanifold embedding of spin manifolds of some codimension k \geq 1 . A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that M does not admit a metric of positive scalar curvature if k = 2 and the Dirac operator of N has non-trivial index, provided that suitable geometric conditions on N \subset M are satisfied. In the cases k=1 and k=2 , Zeidler and Kubota, respectively, established more systematic results: There exists a transfer \text{KO}_\ast(\text{C}^{\ast} \pi_1 M)\to \text{KO}_{\ast - k}(\text{C}^\ast \pi_1 N) which maps the index class of M to the index class of N . The main goal of this article is to construct analogous transfer maps E_\ast(\text{B}\pi_1M) \to E_{\ast-k}(\text{B}\pi_1N) for different generalized homology theories E and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer E_\ast(M) \to E_{\ast-k}(N) induced by the inclusion N \subset M for a chosen orientation on the normal bundle. Under varying restrictions on homotopy groups and the normal bundle, we construct transfers in the following cases in particular: In ordinary homology, it works for all codimensions. This slightly generalizes a result of Engel and simplifies his proof. In complex K-homology, we achieve it for k \leq 3 . For k \leq 2 , we have a transfer on the equivariant KO-homology of the classifying space for proper actions.

Stabilization of the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n $

In this article, we consider the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n(n\ge 3) $. By virtue of the Morawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments, we obtain some stability result of the transmission wave/plate system under suitable geometric conditions.

The evolution problem associated with eigenvalues of the Hessian

In this paper we study the evolution problem \[ \left\lbrace\begin{array}{ll} u_t (x,t)- \lambda_j(D^2 u(x,t)) = 0, & \text{in } \Omega\times (0,+\infty), \\ u(x,t) = g(x,t), & \text{on } \partial \Omega \times (0,+\infty), \\ u(x,0) = u_0(x), & \text{in } \Omega, \end{array}\right. \] where $\Omega$ is a bounded domain in $\mathbb{R}^N$ (that verifies a suitable geometric condition on its boundary) and $\lambda_j(D^2 u)$ stands for the $j-$st eigenvalue of the Hessian matrix $D^2u$. We assume that $u_0 $ and $g$ are continuous functions with the compatibility condition $u_0(x) = g(x,0)$, $x\in \partial \Omega$. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, $g(x,t) =g(x)$, we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as $t\to \infty$. For $j=1$ the limit profile is just the convex envelope inside $\Omega$ of the boundary datum $g$, while for $j=N$ it is the concave envelope. We obtain this result with two different techniques: with PDE tools and and with game theoretical arguments. Moreover, in some special cases (for affine boundary data) we can show that solutions coincide with the stationary solution in finite time (that depends only on $\Omega$ and not on the initial condition $u_0$).

On the blow-up of GSBV functions under suitable geometric properties of the jump set

Abstract In this paper, we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given an open set Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} and given p > 1 {p>1} , we study the blow-up of functions u ∈ GSBV ⁢ ( Ω ) {u\in\mathrm{GSBV}(\Omega)} , whose jump sets belong to an appropriate class 𝒥 p {\mathcal{J}_{p}} and whose approximate gradients are p-th power summable. In analogy with the theory of p-capacity in the context of Sobolev spaces, we prove that the blow-up of u converges up to a set of Hausdorff dimension less than or equal to n - p {n-p} . Moreover, we are able to prove the following result which in the case of W 1 , p ⁢ ( Ω ) {W^{1,p}(\Omega)} functions can be stated as follows: whenever u k {u_{k}} strongly converges to u, then, up to subsequences, u k {u_{k}} pointwise converges to u except on a set whose Hausdorff dimension is at most n - p {n-p} .

The Decoupling of Elastic Waves from a Weak Formulation Perspective

Elastic scattering governed by the Lame system associated with the third-type or fourth-type boundary condition is considered. It was shown in [8] by two of the authors that under suitable geometric conditions on the boundary surface of the elastic inclusion, the longitudinal and shear waves can be decoupled. The decoupling result in [8] was derived based on analyzing the local boundary behaviours of the elastic fields. In this article, we provide a different argument from a variational perspective in proving the decoupling result.

On the geometry of geodesics in discrete optimal transport

We consider the space of probability measures on a discrete set mathcal {X}, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset mathcal {Y}subseteq mathcal {X}, it is natural to ask whether they can be connected by a constant speed geodesic with support in mathcal {Y} at all times. Our main result answers this question affirmatively, under a suitable geometric condition on mathcal {Y} introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton–Jacobi equations, which is of independent interest.

On the eigenmodes of periodic orbits for multiple scattering problems in 2D

Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the oscillatory properties of the solution. However, the high-frequency wave pattern becomes very complicated in the presence of multiple scattering obstacles. We consider a boundary integral equation formulation of the Helmholtz equation in two dimensions involving several obstacles, for which ray tracing schemes have been previously proposed. The existing analysis of ray tracing schemes focuses on periodic orbits between a subset of the obstacles. One observes that the densities on each of the obstacles approach an equilibrium with an increasing number of reflections. In this paper we present an asymptotic approximation of the phases of those densities in equilibrium, in the form of a Taylor series. The densities represent a full cycle of reflections in a periodic orbit. We initially exploit symmetry in the case of two circular scatterers, but also provide an explicit algorithm for a finite number of 2D obstacles satisfying suitable geometrical conditions. The coefficients, as well as the time to compute them, are independent of the wavenumber and of the incident wave. After further research, the results may be used to accelerate ray tracing schemes after a small number of initial iterations.

Continuity of minimizers to weighted least gradient problems

We revisit the question of existence and regularity of minimizers to the weighted least gradient problem with Dirichlet boundary condition inf∫Ωa(x)|Du|:u∈BV(Ω),u|∂Ω=g,where g∈C(∂Ω), and a∈C2(Ω̄) is a weight function that is bounded away from zero. Under suitable geometric conditions on the domain Ω⊂Rn, we construct continuous solutions of the above problem for any dimension n≥2, by extending the Sternberg–Williams–Ziemer technique (Sternberg et al., 1992) to this setting of inhomogeneous variations. We show that the level sets of the constructed minimizer are minimal surfaces in the conformal metric a2∕(n−1)In. This result complements the approach in Jerrard et al. (2018) since it provides a continuous solution even in high dimensions where the possibility exists for level sets to develop singularities. The proof relies on an application of a strict maximum principle for sets with area minimizing boundary established by Leon Simon in Simon (1987).

Comments on: Continuous approximation models in freight distribution management

Any time a group of authors of this caliber gets together, the result is sure to be a great read, and this article is no exception. This paper is a survey of the continuous approximation paradigm for logistics systems analysis, which is a reductionist philosophy that seeks to isolate the factors that influence a geospatial optimization problem most significantly. The defining feature of continuous approximation models is that “detailed data are replaced by concise summaries, and numerical methods are replaced by analytic models” (Daganzo 2005); this means that rather than finding an algorithm to solve a particular problem instance efficiently, one seeks a simple, algebraic expression that approximates the true cost of that problem under suitable probabilistic or geometric conditions (e.g. a limiting condition under uniformly distributed demand in a convex planar region).

Exact Controllability for Wave Equations with Switching Controls

In this paper, we analyze the exact controllability problem for wave equations endowed with switching controls. The goal is to control the dynamics of the system by switching among different actuators such that, in each instant of time, there are as few active actuators as possible. We prove that the system is exactly controllable under suitable geometric control conditions.

Hypocoercive estimates on foliations and velocity spherical Brownian motion

By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic Hörmander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the convergence to equilibrium of the diffusion in $H^1$ and in $L^2$.

Stabilization of the transmission wave/plate equation with variable coefficients

We consider the stabilization of the transmission wave/plate equation with variable coefficients and a general transmission boundary. By virtue of the Riemannian geometric approach, we obtain some stability results for nonlinear feedbacks acting on the regions under suitable geometric conditions.

The $$r^{p}$$ r p -Weighted Energy Method of Dafermos and Rodnianski in General Asymptotically Flat Spacetimes and Applications

In [11], Dafermos and Rodnianski presented a novel approach to establish uniform decay rates for solutions $${\upvarphi }$$ to the scalar wave equation $$\square _{g}{\upvarphi }=0$$ on Minkowski, Schwarzschild and other asymptotically flat backgrounds. This paper generalises the methods and results of [11] to a broad class of asymptotically flat spacetimes $$(\mathcal {M},g)$$ , including Kerr spacetimes in the full subextremal range $$|a|<M$$ , but also radiating spacetimes with no exact symmetries in general dimension $$d+1$$ , $$d\ge 3$$ . As a soft corollary, it is shown that the Friedlander radiation field for $${\upvarphi }$$ is well defined on future null infinity. Moreover, polynomial decay rates are established for $${\upvarphi }$$ , provided that an integrated local energy decay statement (possibly with a finite loss of derivatives) holds and the near region of $$(\mathcal {M},g)$$ satisfies some mild geometric conditions. The latter conditions allow for $$(\mathcal {M},g)$$ to be the exterior of a black hole spacetime with a non-degenerate event horizon (having possibly complicated topology) or the exterior of a compact moving obstacle in an ambient globally hyperbolic spacetime satisfying suitable geometric conditions.