Boundary Control of PDEs via Curvature Flows: the View from the Boundary, II

We describe some results on the exact boundary controllability of the wave equation on an orientable two-dimensional Riemannian manifold with nonempty boundary. If the boundary has positive geodesic curvature, we show that the problem is controllable in finite time if (and only if) there are no closed geodesics in the interior of the manifold. This is done by solving a parabolic problem to construct a convex function. We exhibit an example for which control from a subset of the boundary is possible, but cannot be proved by means of convex functions. We also describe a numerical implementation of this method.     

Boundary Optimal Control of Time-Periodic Stokes-Oseen Flows

This work deals with the existence of optimal solution and the maximum principle for the optimal control problem governed by time-periodic Stokes–Oseen equations with boundary control. An example of a laminar flow is given, and the general unique continuation hypothesis for the Stokes–Oseen operator is checked in this case.     

On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

This paper deals with the following prescribed boundary mean curvature problem in B^N ■ where ■ is a bounded nonnegative function with ■ .Combining the finite-dimensional reduction method and local Pohozaev type of identities,we prove that N≥5 and ■ has a stable critical point ■ with r₀> d and ■>0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily large.Here our result fill the gap that the above critical points may inlude the sad...     

Examples of Ricci-Mean Curvature Flows

Let \(\pi :{\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\rightarrow {\mathbb {P}}^{n-1}\) be a projective bundle over \({\mathbb {P}}^{n-1}\) with \(1\le k \le n-1\). We denote \({\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\) by \(N_{k}^{n}\) and endow it with the U(n)-invariant gradient shrinking Kähler Ricci soliton structure constructed by Cao (Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, 1996) and Koiso (Recent topics in differential and analytic geometry. Advanced studies in pure mathematics, Boston, 1990). In this paper, we show that lens space \(L(k\, ;1)(r)\) with radius r embedded in \(N_{k}^{n}\) is a self-similar solution. We also prove that there exists a pair of critical radii \(r_{1}<r_{2}\), which satisfies the following. The lens space \(L(k\, ;1)(r)\) is a self-shrinker if \(r<r_{2}\) and self-expander if \(r_{2}<r\), and the Ricci-mean curvature flow emanating from \(L(k\, ;1)(r)\) collapses to the 0-section of \(\pi \) if \(r<r_{1}\) and to the \(\infty \)-section of \(\pi \) if \(r_{1}<r\). This paper gives explicit examples of Ricci-mean curvature flows.     

Non-parametric Inverse Curvature Flows in the AdS-Schwarzschild Manifold

We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and, and the principle curvatures of the evolving hypersurface converge to 1 exponentially fast as time tends to infinity.     

Nonexistence of Blow-up Flows for Symplectic and Lagrangian Mean Curvature Flows

In this paper we mainly study the relation between $|A|^2, |H|^2$ and cosα (α is the Kähler angle) of the blow up flow around the type II singularities of a symplectic mean curvature flow. We also study similar property of an almost calibrated Lagrangian mean curvature flow. We show the nonexistence of type II blow-up flows for a symplectic mean curvature flow satisfying $|A|^2≤λ|H|^2$ and $cosα≥δ>1-\frac{1}{2λ}(½≤α≤ 2)$, or for an almost calibrated Lagrangian mean curvature flow satisfying $|A|^2≤λ|H|^2$ and $cosθ≥δ>max\ {0,1-\frac{1}{λ}}(\frac34≤λ≤ 2)$, where θ is the Lagrangian angle.     

(Co)bordism groups in quantum super PDEs. II: Quantum super PDEs

Following our previous works on the integral (co)bordism groups of quantum PDEs [A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publishing Co., Singapore, 1996, 760pp; A. Prástaro, (Co)bordisms in PDE's and quantum PDE's, Rep. Math. Phys. 38(3) (1996) 443–455; A. Prástaro, Quantum and integral (co)bordism groups in partial differential equations, Acta Appl. Math. 51(3) (1998) 243–302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59(2) (1999) 111–202; A. Prástaro, (Co)bordism groups in quantum PDE's, Acta Appl. Math. 64(2/3) (2000) 111–217; A. Prástaro, Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. 47/4 (2001) 2609–2620; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co., Singapore, 2004, 500pp; A. Prástaro, Quantum super Yang-Mills equations: global existence and mass-gap, Dynamic Syst. Appl. 4 (2004) 227–234; A. Prástaro, Th.M. Rassias, A geometric approach to a noncommutative generalized d’Alembert equation, C. R. Acad. Sci. Paris 330(I-7) (2000) 545–550; A. Prástaro, Th.M. Rassias, Results on the J.D’Alembert equation, Ann Acad. Paed. Crac. Stud. Math. 1 (2001) 117–128.], we specialize, now, on quantum super partial differential equations, i.e., partial differential equations built in the category of quantum supermanifolds. These are manifolds modeled on locally convex topological vector spaces built starting from quantum algebras endowed also with a ${\mathbb{Z}}_2$-gradiation, and a ${\mathbb{Z}}_2$-graded Lie algebra structure, (quantum superalgebra). Then, we extend to these manifolds, with such richer structure, our previous results, and build a geometric theory of quantum super PDEs, that allows us to obtain theorems of existence of (smooth) local and global solutions in the category of quantum supermanifolds. Some quantum (super) PDEs, arising from the Dirac quantization of some classical (super) PDEs, are considered in some details.     

Simulation of Mean Curvature Flows on Surfaces of Revolution

Computational Mathematics and Mathematical Physics - This paper is devoted to the simulation of the mean curvature flow on a surface of revolution. The surface is discretized using the icosahedral...     

流动中边界面的识别

将流场的边界面定义为流动表面,在该表面上剪切率为0,并利用所建议的程序求解.该方法是基于速度向量场的计算,与坐标系的选择无关.     

State-Constrained Optimization of PDEs via Infinite Penalization Methods

We consider optimization problems constrained by partial differential equations (PDEs) with additional constraints placed on the solution of the PDEs. Specifically, we consider problems involving constraints on the average value of the state in subdomains. We develop a general framework using infinite-valued penalization functions and Clarke generalized gradients to obtain optimality conditions. A numerical example involving a linear elliptic PDE is presented. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary Amenability of Groups and Positive Scalar Curvature

We show that if a finitely presented discrete group acts amenably on the boundary of a metric compactification of itself, then the Gromov–Lawson–Rosenberg conjecture holds for . Thus, if M is a compact aspherical Spin manifold with fundamental group , then M does not admit a Riemannian metric with positive scalar curvature. The class of groups having such a property includes hyperbolic groups and amenable groups and is closed under semidirect product.     

Qualitative Properties of Separating Flows in the Prandtl Boundary Layer

We study the behavior of solutions to the system of Prandtl boundary layer equations beyond the separation point of the boundary layer. We obtain conditions on the positive pressure gradient which guarantee the attachment of the boundary layer to the streamlined surface after separation. We prove the possibility of controlling the boundary layer by alternating suction and injection.     

Mayer brackets and solvability of PDEs - II

For the Spencer -cohomologies of a symbolic system we construct a spectral sequence associated with a subspace. We calculate the sequence for the systems of Cohen-Macaulay type and obtain a reduction theorem, which facilitates computation of -cohomologies by reducing dimension of the system. Using this algebraic result we prove an efficient compatibility criterion for a system of two scalar non-linear PDEs on a manifold of any dimension in terms of (generalized) Mayer brackets.

     

The Christoffel-Minkowski Problem Ⅱ： Weingarten Curvature Equations

In this paper the authors discuss the existence and convexity of hypersurfaces with prescribed Weingarten curvature.     

The Second Type Singularities of Symplectic and Lagrangian Mean Curvature Flows

This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a Kähler surface. The relation between the maximum of the Kähler angle and the maximum of |H|² on the limit flow is studied. The authors also show the nonexistence of type II blow-up flow of a symplectic mean curvature flow which is normal flat or of an almost calibrated Lagrangian mean curvature flow which is flat.     

Classical Solutions of Nonautonomous Riccati Equations Arising in Parabolic Boundary Control Problems, II

An abstract linear-quadratic regulator problem over finite time horizon is considered; it covers a large class of linear nonautonomous parabolic systems in bounded domains, with boundary control of Dirichlet or Neumann type. We give the proof of some result stated in [AT5], and in addition we prove uniqueness of the Riccati operator, provided its final datum is suitably regular. Accepted 14 October 1998