Non-degeneracy Condition Research Articles

Existence and nonexistence theorems of global weak solutions to degenerate quasilinear wave equations for the elasticity

We prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations vtt=c(|v|s−1v)xx with the nonnegative initial data v0(x) and s>1. These results are extensions of the results in the second author’s paper (Sugiyama, 2016), where the existence and the nonexistence of the unique global classical solution were studied with a threshold on ∫−∞∞v1(x)dx and the non-degeneracy condition v0(x)≥c0>0 on the initial data.

Codimension one and two bifurcations of a discrete stage-structured population model with self-limitation

ABSTRACTIn this paper, the bifurcations of a discrete stage-structured population model with self-limitation between the two subgroups are investigated. We explore all possible codimension-one bifurcations associated with transcritical, flip (period doubling) and Neimark-Sacker bifurcations and discuss the stabilities of the fixed points in these non-hyperbolic cases. Meanwhile, we give the explicit approximate expression of the closed invariant curve which is caused by the Neimark-Sacker bifurcation. After that, through the theory of approximation by a flow, we explore the codimension two bifurcations associated with 1:3 strong resonance. We convert the nondegenerate condition of 1:3 resonance into a parametric polynomial, and determine its sign by the theory of complete discrimination system. We introduce new parameters and utilize some variable substitutions to obtain the bifurcation curves around 1:3 resonance, which are returned to the original variables and parameters to express for easy verification. By using a series of complicated approximate identity transformations and polar coordinate transformation, we explore 1:6 weak resonance. Moreover, we calculate the two boundaries of Arnold tongue which are caused by 1:6 weak resonance and defined as the resonance region. Numerical simulations and numerical bifurcation analyzes are made to demonstrate the effective of the theoretical analyzes and to present the relations between these bifurcations. Furthermore, our theoretical analyzes and numerical simulations are explained from the biological point of view.

On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

We consider the problem $$\begin{aligned} \epsilon ^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad \text{ in }\quad \Omega , \quad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\quad \partial \Omega , \end{aligned}$$ where $$\Omega $$ is a bounded domain in $${\mathbb {R}}^2$$ with smooth boundary, the exponent p is greater than 1, $$\epsilon >0$$ is a small parameter, V is a uniformly positive, smooth potential on $$\bar{\Omega }$$ , and $$\nu $$ denotes the outward unit normal of $$\partial \Omega $$ . Let $$\Gamma $$ be a curve intersecting orthogonally $$\partial \Omega $$ at exactly two points and dividing $$\Omega $$ into two parts. Moreover, $$\Gamma $$ satisfies stationary and non-degeneracy conditions with respect to the functional $$\int _{\Gamma }V^{\sigma }$$ , where $$\sigma =\frac{p+1}{p-1}-\frac{1}{2}$$ . We prove the existence of a solution $$u_\epsilon $$ concentrating along the whole of $$\Gamma $$ , exponentially small in $$\epsilon $$ at any fixed distance from it, provided that $$\epsilon $$ is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by Ambrosetti et al. (Indiana Univ Math J 53(2), 297–329, 2004).

An Investigation on Optimal Initial Self-Stress Design of Tensegrity Grid Structures

To achieve the optimal feasible force density vector of a given geometry configuration tensegrity grid structure, an efficient procedure is presented for optimal initial self-stress design of tensegrity grid structures by consecutively solving two linear homogeneous systems in conjunction with a minimization problem. The nonlinear constrained optimization algorithm, known as the Interior-Point Method (I-PM), is utilized to obtain the minimal solution, leading to a set of force densities which guarantee the non-degeneracy condition of the force density matrix. The evaluation of the eigenvalues of tangent stiffness matrix is also introduced to check the geometric stability of the tensegrity grid structures. Finally, three numerical examples have been investigated comprehensively to prove the capability of the proposed method in optimal initial self-stress design of tensegrities. Furthermore, division of number of member group has been discussed in detail for the purpose of demonstrating the efficiency of the proposed method in seeking initial force densities of tensegrity grid structures.

Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations

We consider a mechanical system consisting of $n$ penduli and a $d$-dimensional generalized rotator subject to a time-dependent perturbation. The perturbation is not assumed to be either Hamiltonian, or periodic or quasi-periodic. The strength of the perturbation is given by a parameter $\epsilon\in\mathbb{R}$. For all $|\epsilon|$ sufficiently small, the augmented flow has a $(2d + 1)$-dimensional normally hyperbolic locally invariant manifold $\tilde\Lambda_\epsilon$. We define a Melnikov vector, which gives the first order expansion of the displacement of the stable and unstable manifolds of $\tilde\Lambda_0$ under the perturbation. We provide an explicit formula for the Melnikov vector in terms of convergent improper integrals of the perturbation along homoclinic orbits of the unperturbed system. We show that if the perturbation satisfies some explicit non-degeneracy conditions, then the stable and unstable manifolds of $\tilde\Lambda_\epsilon$, $W^s(\tilde\Lambda_\epsilon)$ and $W^u(\tilde\Lambda_\epsilon)$, respectively, intersect along a transverse homoclinic manifold, and, moreover, the splitting of $W^s(\tilde\Lambda_\epsilon)$ and $W^u(\tilde\Lambda_\epsilon)$ can be explicitly computed, up to the first order, in terms of the Melnikov vector. This implies that the excursions along some homoclinic trajectories yield a non-trivial increase of order $O(\epsilon)$ in the action variables of the rotator, for all sufficiently small perturbations. The formulas that we obtain are independent of the unperturbed motions and give, at the same time, the effects on periodic, quasi-periodic, or general orbits. When the perturbation is Hamiltonian, we express the effects of the perturbation, up to the first order, in terms of a Melnikov potential. In addition, if the perturbation is periodic, we obtain that the non-degeneracy conditions on the Melnikov potential are generic.

Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in Bandini et al. (2018), we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton–Jacobi–Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear–quadratic model.

Cubic Model CR-Manifolds without the Assumption of Complete Nondegeneracy

In the paper, we study model surfaces with Hormander numbers (2, 3) that do not satisfy the condition of complete nondegeneracy. Simple criteria are given for the finiteness of the type, holomorphic nondegeneracy, and holomorphic homogeneity. It is proved that the dimension of the automorphism group of the model surface is maximal in the class of germs subordinated to the surface, and also that this group is a Lie subgroup of the Cremona group. We consider the case of a surface with a unique Hormander number 3 separately.

Tangent measures and absolute continuity of harmonic measure

We show that for uniform domains \Omega\subseteq \mathbb R^{d+1} whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to \alpha -dimensional Hausdorff measure unless \alpha\leq d . We employ a lemma that shows that, at almost every non-degenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.

Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

Large deviations and mixing for dissipative PDEs with unbounded random kicks

We study the problem of exponential mixing and large deviations for discrete-time Markov processes associated with a class of random dynamical systems. Under some dissipativity and regularisation hypotheses for the underlying deterministic dynamics and a non-degeneracy condition for the driving random force, we discuss the existence and uniqueness of a stationary measure and its exponential stability in the Kantorovich–Wasserstein metric. We next turn to the large deviations principle (LDP) and establish its validity for the occupation measures of the Markov processes in question. The proof is based on Kifer’s criterion for non-compact spaces, a result on large-time asymptotics for generalised Markov semigroup, and a coupling argument. These tools combined together constitute a new approach to LDP for infinite-dimensional processes without strong Feller property in a non-compact space. The results obtained can be applied to the two-dimensional Navier–Stokes system in a bounded domain and to the complex Ginzburg–Landau equation.

Donaldson–Thomas theory of $$[\mathbb {C}^2/\mathbb {Z}_{n+1}]\times \mathbb {P}^1$$[C2/Zn+1]×P1

We study the relative orbifold Donaldson-Thomas theory of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$. We establish a correspondence between the DT theory relative to 3 fibers to quantum multiplication by divisors in the Hilbert scheme of points on $[\mathbb{C}^2/\mathbb{Z}_{n+1}]$. This determines the whole theory if a further nondegeneracy condition is assumed. The result can also be viewed as a crepant resolution correspondence to the DT theory of $\mathcal{A}_n\times \mathbb{P}^1$.

Sensitivity Analysis for Mirror-Stratifiable Convex Functions

This paper provides a set of sensitivity analysis and activity identification results for a class of convex functions with a strong geometric structure, that we coined mirror-stratifiable . These functions are such that there is a bijection between a primal and a dual stratification of the space into partitioning sets, called strata. This pairing is crucial to track the strata that are identifiable by solutions of parametrized optimization problems or by iterates of optimization algorithms. This class of functions encompasses all regularizers routinely used in signal and image processing, machine learning, and statistics. We show that this mirror-stratifiable structure enjoys a nice sensitivity theory, allowing us to study stability of solutions of optimization problems to small perturbations , as well as activity identification of first-order proximal splitting-type algorithms. Existing results in the literature typically assume that, under a non-degeneracy condition, the active set associated to a minimizer is stable to small perturbations and is identified in finite time by optimization schemes. In contrast, our results do not require any non-degeneracy assumption: in consequence, the optimal active set is not necessarily stable anymore, but we are able to track precisely the set of identifiable strata.We show that these results have crucial implications when solving challenging ill-posed inverse problems via regularization, a typical scenario where the non-degeneracy condition is not fulfilled. Our theoretical results, illustrated by numerical simulations, allow to characterize the instability behaviour of the regularized solutions, by locating the set of all low-dimensional strata that can be potentially identified by these solutions. Key words. convex analysis, inverse problems, sensitivity, active sets, first-order algorithms, splitting algorithm, applications in imaging and machine learning

Global regularity for the free boundary in the obstacle problem for the fractional Laplacian

We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $\varphi$ satisfies $\Delta\varphi\leq 0$ near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a $(n-1)$-dimensional $C^{1,\alpha}$ manifold by the results obtained by Caffarelli, Salsa, and Silvestre ({\it Invent. Math.} 2008), and a set of singular points, which we prove to be contained in a union of $k$-dimensional $C^1$-submanifold, $k=0,\ldots, n-1$. Such a complete result on the structure of the free boundary, proved by L. A. Caffarelli ({\it Acta Math.} 1977 and {\it J. Fourier Anal. Appl.} 1998), was known only in the case of the classical Laplacian, and it is new even for the Signorini problem (which corresponds to the particular case of the ${1\over 2}$-fractional Laplacian). A key ingredient behind our results is the validity of a new non-degeneracy condition $\sup_{B_r(x_0)}(u-\varphi)\geq c\,r^2$, valid at all free boundary points $x_0$.

Positive solutions for a nonlinear Schrödinger-Poisson system

In this paper, we study the following nonlinear Schrödinger-Poisson system \begin{document}$\left\{\begin{array}{ll} -\Delta u+u+\epsilon K(x)\Phi(x)u = f(u),& x\in \mathbb{R}^{3} , \\ -\Delta \Phi = K(x)u^{2},\,\,& x\in \mathbb{R}^{3}, \\\end{array}\right.$ \end{document} where $K(x)$ is a positive and continuous potential and $f(u)$ is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Under some suitable conditions, which are given in section 1, we prove that there exists some $\epsilon_{0}>0$ such that for $0<\epsilon<\epsilon_{0}$, the above problem has infinitely many positive solutions by applying localized energy method. Our main result can be viewed as an extension to a recent result Theorem 1.1 of Ao and Wei in [3] and a result of Li, Peng and Wang in [26].

A Parameterization Method for Lagrangian Tori of Exact Symplectic Maps of $R^2r$

We are concerned with analytic exact symplectic maps of ${\mathbb R}^{2r}$ endowed with the standard symplectic form. We study the existence of a real analytic torus of dimension $r$, invariant by the map and carrying quasi-periodic motion with a prefixed Diophantine rotation vector. Therefore, this torus is a Lagrangian manifold. We address the problem by the parameterization method in KAM theory. The main aspect of our approach is that we do not look for the parameterization of the torus as a solution of the corresponding invariance equation. Instead, we consider a set of three equations that, all together, are equivalent to the invariance equation. These equations arise from the geometric and dynamical properties of the map and the torus. Suppose that an approximate solution of these equations is known and that a suitable nondegeneracy (twist) condition is satisfied. Then, this system of equations is solved by a quasi-Newton-like method, provided that the initial error is sufficiently small. By “quasi-Newton-like” we mean that the convergence is almost quadratic but that at each iteration we have to solve a nonlinear equation. Although it is straightforward to build a quasi-Newton method for the selected set of equations, proceeding in this way we improve the convergence condition. The selected definition of error reflects the level at which the error associated with each of these three equations contributes to the total error. The map is not required to be close to integrable or expressed in action-angle variables. Suppose the map is $\varepsilon$-close to an integrable one, and consider the portion of the phase space not filled up by Lagrangian invariant tori of the map. Then, the upper bound for the Lebesgue measure of this set that we may predict from the result is of ${\mathcal O}(\varepsilon^{1/2})$. In light of the classical KAM theory for exact symplectic maps, an upper bound of ${\mathcal O}(\varepsilon^{1/2})$ for this measure is the expected estimate. The result also has some implications for finitely differentiable maps.

Higher decay inequalities for multilinear oscillatory integrals

In this paper we establish sharp estimates (up to logarithmic losses) for the multilinear oscillatory integral operator studied by Phong, Stein, and Sturm and Carbery and Wright on any product $\prod_{j=1}^d L^{p_j}(\mathbb R)$ with each $p_j \geq 2$, expanding the known results for this operator well outside the previous range $\sum_{j=1}^d p_j^{-1} = d-1$. Our theorem assumes second-order nondegeneracy condition of Varchenko type, and as a corollary reproduces Varchenko's theorem and implies Fourier decay estimates for measures of smooth density on degenerate hypersurfaces in $\mathbb R^d$.

ON THE ‐COMPLETENESS AND ‐COMPLETENESS OF MULTIFRACTAL DECOMPOSITION SETS

The purpose of this paper twofold. Firstly, we establish -completeness and -completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the -completeness and -completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if is a self-similar measure satisfying the strong separation condition and is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of defined by are -complete provided they are non-empty.

Sensitivity of chemical reaction networks: A structural approach. 3. Regular multimolecular systems

We present a systematic mathematical analysis of the qualitative steady‐state response to rate perturbations in large classes of reaction networks. This includes multimolecular reactions and allows for catalysis, enzymatic reactions, multiple reaction products, nonmonotone rate functions, and non‐closed autonomous systems. Our structural sensitivity analysis is based on the stoichiometry of the reaction network, only. It does not require numerical data on reaction rates. Instead, we impose mild and generic nondegeneracy conditions of algebraic type. From the structural data, only, we derive which steady‐state concentrations are sensitive to, and hence influenced by, changes of any particular reaction rate—and which are not. We also establish transitivity properties for influences involving rate perturbations. This allows us to derive an influence graph which globally summarizes the influence pattern of any given network. The influence graph allows the computational, but meaningful, automatic identification of functional subunits in general networks, which hierarchically influence each other. We illustrate our results for several variants of the glycolytic citric acid cycle. Biological applications include enzyme knockout experiments and metabolic control.

Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman’s method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).

Lens rigidity in scattering by non-trapping obstacles

We prove that if two non-trapping obstacles in \(\mathbb {R}^n\) satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.