Geometric Hypotheses Research Articles

Optimal Control of a Parabolic Equation with Time-Dependent State Constraints

In this paper we study the optimal control problem of the heat equation with a cubic nonlinearity by a distributed control over a subset of the domain, in the presence of a state constraint. The latter is integral over the space and has to be satisfied each time. Using for the first time the technique of alternative optimality systems in the context of optimal control of partial differential equations, we show that both the control and multiplier are continuous in time. Under some natural geometric hypotheses, we can prove that extended polyhedricity holds, allowing us to obtain no-gap second-order optimality conditions, that characterize quadratic growth. An expansion of the value function and of approximate solutions can be computed for a directional perturbation of the right-hand side of the state equation.

Spectral geometry and asymptotically conic convergence

In this paper we define asymptotically conic convergence in which a family of smooth Riemannian metrics degenerates to have an isolated conic singularity. For a conic metric (M0,g0) and an asymptotically conic (scattering) metric (Z,gz) we define a nonstandard blowup, the resolution blowup, in which the conic singularity in M0 is resolved by Z. Equivalently, the resolution blowup resolves the boundary of the scattering metric using the conic metric; the resolution space is a smooth compact manifold. This blowup induces a smooth family of metrics {g } on the compact resolution space M, and we say (M,g ) converges asymptotically conically to (M0,g0) as ! 0. Let and 0 be geometric Laplacians on (M,g ) and (M0,g0), respectively. Our first result is convergence of the spectrum of to the spectrum of 0 as ! 0. Note that this result implies spectral convergence for the k-form Laplacian under certain geometric hypotheses. This theorem is proven using rescaling arguments, standard elliptic techniques, and the b-calculus of [26]. Our second result is technical: we construct a parameter ( ) dependent heat operator calculus which contains, and hence describes precisely, the heat kernel for as ! 0. The consequences of this result include: the existence of a polyhomogeneous asymptotic expansion for H as ! 0, with uniform convergence down to t = 0. To prove this result we construct manifolds with corners (heat spaces) using both standard and non-standard blowups, on which we construct corresponding heat operator calculi. A parametrix construction modeled after the heat kernel construction of [26] and a maximum principle type argument complete this proof.

Satellite Constellations for Continuous and Early Warning Observation: A Correlation-Based Approach

Recently, telecommunication and navigation corporations have shown a growing interest in low and medium Earth orbit constellations due to several operational advantages, such as reduced power requirements and signal time delays with respect to geostationary platforms. An increased imaging resolution of Earth's surface is also associated with the use of low orbits rather than high altitude orbits. A large set of parameters is involved in defining constellation configurations, so usually some basic assumptions are taken. For instance, this occurs for Walker constellations, exhibiting a high degree of symmetry and suitability for coverage of large areas. Yet, for special purposes, such as continuous observation or early warning monitoring of a specific location, geometric hypotheses often seem inadequate or oversimplified. In this paper all satellites are placed in circular, repeating orbits with semimajor axis and inclination yielding maximum visibility of a preselected target location. Then, a correlation function is introduced as an effective tool to find all allowed time delays between two consecutive passes over the target. Some optimal constellation configurations (of 2, 4, 8 satellites) are deduced and discussed with reference to different kinds of local observation.

Soliton solutions for quasilinear Schrödinger equations: The critical exponential case

Quasilinear elliptic equations in R 2 of second order with critical exponential growth are considered. By using a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H 1 ( R 2 ) and satisfy the geometric hypotheses of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v . In the proof that v is nontrivial, the main tool is the concentration–compactness principle [P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. H. Poincaré Anal. Non. Linéaire 1 (1984) 109–145, 223–283] combined with test functions connected with optimal Trudinger–Moser inequality.

Vertical forces in labial and lingual orthodontics applied on maxillary incisors--a theoretical approach.

Theoretical and experimental biomechanical analyses explain most labial orthodontics (LaO); however, lingual orthodontic (LiO) biomechanical principles are rarely introduced. The objective of this study was to apply basic biomechanical considerations in understanding the influence of maxillary incisor inclination and to compare the effect of labial vs lingual intrusive/extrusive forces on tooth movement. Basic anatomic and geometric hypotheses were assumed, ie, tooth length (crown and root), location of the center of resistance, and crown thickness. Incisor inclination as related to a perpendicular line to the occlusal plane (OP) varied between -35 degrees (retroclination) and 45 degrees (proclination). A 0 degrees inclination was defined as a tooth position with its long axis perpendicular to the OP. The buccolingual moment for characterizing root movement was calculated for an applied force perpendicular to the OP. The results showed that when using LaO, an extrusion force resulted in labial root movement from a retroclination of 20 degrees up to a proclination of 45 degrees. In LiO, labial root movement occurred only when the tooth was proclined more than 20 degrees. In all other tooth inclinations, lingual root movement occurred. The opposite tooth movement occurred when an intrusive force was applied. Application of a vertical force has different clinical effects on tooth movement with labial and lingual appliances. Application of a lingual force is more complicated, and its effect on tooth movement depends on bracket position and initial tooth inclination.

From local to global behavior in competitive Lotka-Volterra systems

In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics. The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point p ∈ int ⁡ R + n p\in \operatorname {int}{{\mathbf R}^n_+} and the carrying simplex of the system lies to one side of its tangent hyperplane at p p , then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.

A momentum construction for circle-invariant Kähler metrics

Examples of Kahler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kahler metrics of constant scalar curvature by ODE methods. One fruitful idea-the Calabi ansatz-is to begin with an Hermitian line bundle p: (L, h) → (M, g M ) over a Kahler manifold, and to search for Kahler forms ω = p*ωM + dd c f(t) in some disk subbundle, where t is the logarithm of the norm function and f is a function of one variable. Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kahler metric g arising from the Calabi ansatz. This suggests geometric hypotheses (which we call a-constancy) to impose upon the base metric g M and Hermitian structure h in order that the scalar curvature of g be specified by solving an ODE. We show that σ-constancy is necessary and sufficient for the Calabi ansatz to work in the following sense. Under the assumption of σ-constancy, the disk bundle admits a one-parameter family of complete Kahler metrics of constant scalar curvature that restrict to g M on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of σ-constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E). Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kahler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kahler metric on C 2 .

The signature of a toric variety

We identify a combinatorial quantity (the alternating sum of the h-vector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by R. Charney and M. Davis in their work, which in particular showed that its nonnegativity is closely related to a conjecture of H. Hopf on the Euler characteristic of a nonpositively curved manifold. We prove positive (or nonnegative) lower bounds for this quantity under geometric hypotheses on the polytope and, in particular, resolve a special case of their conjecture. These hypotheses lead to ampleness (or weaker conditions) for certain line bundles on toric divisors, and then the lower bounds follow from calculations using the Hirzebruch signature formula. Moreover, we show that under these hypotheses on the polytope, the ith L-class of the corresponding toric variety is (−1)i times an effective class for any i.

The signature of a toric variety

We identify a combinatorial quantity (the alternating sum of the h-vector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by R. Charney and M. Davis in their work, which in particular showed that its nonnegativity is closely related to a conjecture of H. Hopf on the Euler characteristic of a nonpositively curved manifold. We prove positive (or nonnegative) lower bounds for this quantity under geometric hypotheses on the polytope and, in particular, resolve a special case of their conjecture. These hypotheses lead to ampleness (or weaker conditions) for certain line bundles on toric divisors, and then the lower bounds follow from calculations using the Hirzebruch signature formula. Moreover, we show that under these hypotheses on the polytope, the ith L-class of the corresponding toric variety is (−1)i times an effective class for any i.

Singular Sturm–Liouville Theory on Manifolds

In this paper we investigate Schrödinger operators L=−Δg+a(x) on a compact Riemannian manifold (M, g), where the potential function a(x) is assumed to be continuous, but not necessarily bounded, outside of some closed set Σ⊂M of measure zero. Under certain geometric hypotheses on Σ and growth conditions on a(x) as x→Σ, we prove that the Dirichlet extension of L is bounded from below with discrete spectrum; in many cases, a(x) is allowed to approach −∞ as x→Σ. We also consider conditions on Σ and a(x) under which the Sturm–Liouville theory of L is “singular” in that no boundary conditions are needed to specify the eigenvalues and eigenfunctions of L; in particular, this occurs when the domain of L does not depend on boundary conditions, for example, when L is essentially self-adjoint or more generally “essentially Dirichlet” (a new property that we define). The behavior of L on weighted Sobolev spaces is also discussed. In most of the paper we assume that Σ is a k-dimensional submanifold without boundary, but in the last few sections we generalize our results to stratified sets.

A variational approach to mean curvature flow with Dirichlet conditions

We study a variational approach, called Generalized Minimizing Movements (GMM), to evolution of hypersurfaces by mean curvature in the case of a Dirichlet boundary datum. We prove an existence theorem of a GMM when on the initial solid are made suitable geometric hypotheses.

Étude symbolique de la perte de régularité C ∞ par morceaux dans l'interaction d'ondes semi-linéaires

We consider the interaction of two piecewise smooth waves for a first order semilinear system. We know that under geometric hypotheses the piecewise smooth regularity can be not conserved after the interaction. A symbolic study allows us to prove that this loss does not appear on principal singularities. We then show the microlocal mechanism of the creation and the propagation of a logarithmic singularity.

Generalized minimizing movements for the mean curvature flow with Dirichlet boundary condition

We study a variational approach, called Generalized Minimizing Movemenents (GMM) and proposed by E. De Giorgi, to evolution of hypersurfaces by mean curvature in the case of a Dirichlet boundary datum. We prove an existence theorem of a GMM when on the initial solid are made suitable geometric hypotheses.

The Minimal Free Resolution of the Migliore–Peterson Rings in the Case That the Reflexive Sheaf Has Even Rank

LetRbe a commutative noetherian ring and ϕ:F→Gbe a homomorphism of freeR-modules where rankF=fand rankG=g. Fix an elementbg+1∈⋀g+1Fand a generator ωG*for ⋀gG*. The module action of ⋀·F* on ⋀·Fproduces the elementb1=[(⋀gϕ*)(ωG*)](bg+1) inF. LetJdenote the image ofb1:F*→R. Assume that gradeJ=f−g, which is the largest grade possible and is attained in the generic case. The idealJmay be interpreted as the defining ideal of the degeneracy locus of a regular section of a rankf−greflexive sheaf. It may also be interpreted as the order ideal of an element in a second syzygy module of rankf−g. Also,Jmay be interpreted as the defining ideal for the symmetric algebra of a module of projective dimension two. Migliore and Peterson have studied the idealJunm, which is the unmixed part ofJ. Under geometric hypotheses, they have shown thatR/Junmis a Cohen–Macaulay ring and they have resolved this ring. Furthermore, iff−gis odd, thenJunmis a Gorenstein ideal and is not equal toJ. On the other hand, iff−gis even, thenJunm=J. In the present paper, we produce the resolution ofR/Jby freeR-modules in the case thatf−gis even and (f−g−2)! is a unit inR. Our resolution is minimal whenever the data are local or homogeneous. Our resolution is built from the differential graded algebra (⋀·F*〈X1,…,Xg〉,d), where the restriction ofdto ⋀·F* is the Koszul complex associated tob1:F*→Rand the degree two divided power variablesX1,…,Xghave been adjointed in order to kill the cycles ϕ*(G*)⊆⋀1F*. The acyclicity lemma is used to prove exactness. Ifg=1, then the idealJis equal to the Huneke–Ulrich almost complete intersection idealI1(yX), whereyis a 1×fmatrix andXis anf×falternating matrix. The resolution of this ideal is already known.

Uniform Stabilizability of a Full Von Karman System with Nonlinear Boundary Feedback

Full von Karman system accounting for in-plane accelerations and describing the transient deformations of a thin, elastic plate subject to edge loading is considered. The energy dissipation is introduced via the nonlinear velocity feedback acting on a part of the edge of the plate. It is known [J. Puel and M. Tucsnak, SIAM J. Control Optim., 33 (1995), pp. 255--273] that in the case of linear dissipation and star-shaped domains, boundary velocity feedback with the tangential derivatives of horizontal displacements leads to the exponential decay rates for the energy of the resulting closed loop system. The main goal of the paper is to derive the uniform energy decay rates valid for the model without the above-mentioned restrictions. In particular, it is shown that simple, monotone nonlinear feedback (without the tangential derivatives of the horizontal displacements) provides the uniform decay rates for the energy in the absence of geometric hypotheses imposed on the controlled part of the boundary. This is accomplished by establishing, among other things, sharp regularity results valid for the boundary traces of solutions corresponding to this nonlinear model and by employing a Holmgren-type uniqueness result proved recently in [V. Isakov, J. Differential Equations, 97 (1997), pp. 134--147] for the dynamical systems of elasticity which are overdetermined on the boundary.

Embedded minimal annuli inR 3 bounded by a pair of straight lines

The subject o f this paper is embedded minimal annuli bounded by two straight lines. The only known examples o f such surfaces are given by subdomains o f the singly periodic Riemann examples, ~ . There is a 1-parameter family o f these surfaces. A fundamental domain o f a Riemann example consists o f a minimal annulus bounded by two straight lines, and a copy of that surface produced by Schwarz reflection about one o f the boundary lines. (See Figure 1 and the analytic description o f these surfaces in Section 2.) We will prove that the examples o f Riemann constitute all of the examples, under certain geometric hypotheses. 1

Geometric and kinematic modelling of a human costal slice

More and more powerful calculation methods are being used in the modelization of the human thorax, and considering the progress made in the domain of numerical analysis, this modelization is naturally being oriented toward the utilization of finite element methods. However, thoracic models are usually based on extremely simple geometric hypotheses, due mostly to the lack of dependable experimental data. Hence, the exploitation of sophisticated software is far from optimal. This study is based on experimental observations which allow the capabilities of the current means of calculation to be exploited to a maximum. The objectives of the study are the geometric and kinematic representations of a typical costal slice. A precise topographical measurement, performed by a robot, allows description of the costal geometry. The exploitation of these measurements then allows the identification of the costo-vertebral articulation.

Movement of hot spots in Riemannian manifolds

of "hot spots" is compact for all t > 0. The collection of such Riemannian manifolds is reasonably large. Besides the compact ones, the class contains those noncompact Riemannian manifolds for which for all t > 0, r EL~(M), we have (PtqO(x) ~ 0 as x --oo (see [ 12], [5], [7] for results on the geometric hypotheses guaranteeing this condition). Here we initiate a study of the location of H(t) relative to the support of ~a, S,, and, more precisely, the behavior of H(t) a s t t +oo.

Holonomic measurables in geodesy

The geometric hypotheses on which geodesy currently rests are not consonant with available instrumental techniques. Euclidean, and even Riemannian, geometry is unsatisfactory since the fundamental parallelisms are not physically realizable. Nonmetrical affine connections arising from physical, physiological, and psychological bases give a foundation for geodesy in harmony with available instrumentation. These connections are curvature free, and all information of geodetic interest is contained in their torsions. Nonvanishing torsion accounts for misclosure of measurements in closed traverses. The role of coordinates in measurement is clarified, as is the relation of level to equipotential. The geometrical aspects of measurement flow from an axiom much weaker than the free mobility tacit in the usual geodetic hypotheses. The analysis of measurement shows coordinates, hence tensor analysis, to be unnatural. Geodetic measuring instruments establish reference frames, not coordinates. Therefore the exterior calculus of E. Cartan is used throughout, simplified calculations being a bonus.