Strong Uniqueness Theorem Research Articles

On the Uniqueness Theorem

The aim of this paper is to investigate whether the Uniqueness Theorem introduced by Feit and Thompson holds in finite simple groups of Lie type of Lie rank one. We also introduce and examine the Strong Uniqueness Theorem. We define the concept of containment minimality and prove that a non-Abelian finite simple group is minimal if and only if it is containment minimal. Finally, we inquire when a minimal non-soluble group has a faithful doubly transitive representation.

The relative entropy method for inhomogeneous systems of balance laws

General hyperbolic systems of balance laws with inhomogeneity in space and time in all constitutive functions are studied in the context of relative entropy. A framework is developed in this setting that contributes to a measure-valued weak vs. strong uniqueness theorem, a stability theorem of viscous solutions and a convergence theorem as the viscosity parameter tends to zero. The main goal of this paper is to develop hypotheses under which the relative entropy framework can still be applied. Examples of systems with inhomogeneity that have different characteristics are presented and the hypotheses are discussed in the setting of each example.

Asymptotic behavior of D-solutions to the steady Navier–Stokes flow in an exterior domain of a half-space

We consider the problem of a small body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible steady Navier–Stokes equations in an exterior domain in a half-space, with appropriate boundary conditions on the wall, the body and at infinity. In this paper, we first prove in a very general setup the existence of weak solutions for the problem with the body. Then, we show that any such solution can be truncated and then extended to provide a weak solution for a simplified problem where the body is replaced by a (small) source term with compact support. This simplified problem was already shown to possess strong solutions. We then prove a weak–strong uniqueness theorem to show the uniqueness of solutions for the simplified problem. Finally, we show that this also implies the uniqueness of solutions for the problem of the moving body which proves that the solutions of both problems have the same asymptotic behavior at infinity.

Tangents to subsolutions: existence and uniqueness, Part I

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman in the classical plurisubharmonic case. Fundamental to this study is a new invariant of the equation, called the characteristic, which governs asymptotic structures. The existence of tangents to subsolutions is established in general, as is the existence of an upper semi-continuous density function. Two theorems establishing the strong uniqueness of tangents (which means every tangent is a Riesz kernel) are proved. They cover all O(n)-invariant convex cone equations and their complex and quaternionic analogues, with the exception of the homogeneous Monge-Ampere equations, where uniqueness fails. They also cover a large class of geometrically defined subequations which includes those coming from calibrations. A discreteness result for the sets where the density is $\geq c > 0$ is also established in any case where strong uniqueness holds. A further result (which is sharp) asserts the Holder continuity of subsolutions when the Riesz characteristic p satisfies $1 \leq p < 2$. Many explicit examples are examined. The second part of this paper is devoted to the geometric cases. A Homogeneity Theorem and a Second Strong Uniqueness Theorem are proved, and the tangents in the Monge-Ampere case are completely classified.

Relative Entropy for Hyperbolic–Parabolic Systems and Application to the Constitutive Theory of Thermoviscoelasticity

We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an $L^p$ framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosity and heat-conduction. Existing differences in applying the relative entropy method between the general hyperbolic-parabolic theory and the examples are underlined.

Strong uniqueness of the Ricci flow

In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on $\mathbb{R}^{3}$, with the canonical Euclidean metric $E$ as initial data, then $g(t)$ is trivial, i.e. $g(t)\equiv E$.

A strong uniqueness theorem for planar vector fields

This work establishes a strong uniqueness property for a class of planar locally integrable vector fields. A result on pointwise convergence to the boundary value is also proved for bounded solutions.

On the nonuniqueness of equivariant connected sums

In both ordinary and equivariant 3-dimensional topology there are strong uniqueness theorems for connected sum decompositions of manifolds, but in ordinary higher dimensional topology such decompositions need not be unique. This paper constructs families of manifolds with smooth group actions that are equivariantly almost diffeomorphic but have infinitely many inequivalent equivariant connected sum representations for which one summand is fixed. The examples imply the need for restrictions in any attempt to define Atiyah-Singer type invariants for odd dimensional manifolds with nonfree smooth group actions. Applications to other questions are also considered.

On Best Simultaneous Approximation

The problem is considered of best approximation of finite number of functions simultaneously. For a very general class of norms, characterization results are derived. The main part of the paper is concerned with proving uniqueness and strong uniqueness theorems. For a particular subclass, which includes the important special case of the Chebyshev norm, a characterization is given of a uniqueness element.

Strong Unique Continuation Results for Differential Inequalities

The main result of this paper is a strong uniqueness theorem for differential inequalities of the form |Δu(x)|⩽|V(x)u(x)|+|W(x)∇u(x)|, whereVandWare radial functions inLn/2loc(Ω) andLnloc(Ω) respectively, andΩis a connected open subset of Rn. Other results involving other spaces of potentialsVandWare proved. Our method relies on sharp Carleman estimates.

A note on a strong uniqueness theorem of strauss

In this paper we present a correction of the proof of a strong uniqueness theorem given by H. Strauss[1] in 1992 on approximation by reciprocals of functions of an n-dimensional space (u1, … un) satisfying coefficient constraints.

A note on a strong uniqueness theorem of strauss

A note on a strong uniqueness theorem of strauss

Best Copositive Approximation

A theory of best copositive approximation including a "zero in the convex hull" characterization, alternation theorem, and strong uniqueness theorem is developed for a general n-dimensional Chebyshev subspace in C[a, b]. In addition, a computational algorithm is discussed.

Best restricted range approximation

A theory of best restricted range approximation is developed for an extended n-dimensional Chebyshev subspace of C[a,b] of order n without restricting the upper and lower restraining functions. This theory includes a “zero in the convex hull” characterization, an alternation theorem, and a strong uniqueness theorem.

Strong Uniqueness Theorems for Second Order Elliptic Differential Equations

Strong Uniqueness Theorems for Second Order Elliptic Differential Equations

Wave propagation and uniqueness in magneto-elastodynamics

By using the method introduced in [1], some sufficient conditions on the acoustic tensor and on the kinetic and magnetic fields are given in order that a perturbation initially confined in a proper subset of an unbounded magnetoelastic solid, propagate with finite speed. Moreover, a strong uniqueness theorem is proved for regular solutions to the initial-boundary-value problem of magnetoelastodynamics.

Viscosity solutions of Hamilton-Jacobi equations

Problems involving Hamilton-Jacobi equations—which we take to be either of the stationary form H ( x , u , D u ) = 0 H(x,u,Du) = 0 or of the evolution form u t + H ( x , t , u , D u ) = 0 {u_{t}} + H(x,t,u,Du) = 0 , where D u Du is the spatial gradient of u u —arise in many contexts. Classical analysis of associated problems under boundary and/or initial conditions by the method of characteristics is limited to local considerations owing to the crossing of characteristics. Global analysis of these problems has been hindered by the lack of an appropriate notion of solution for which one has the desired existence and uniqueness properties. In this work a notion of solution is proposed which allows, for example, solutions to be nowhere differentiable but for which strong uniqueness theorems, stability theorems and general existence theorems, as discussed herein, are all valid.

Uniform Approximation by Chebyshev Spline Functions. II: Free Knots

Abstract : Existence, uniqueness and characterization of best uniform approximations of continuous functions by (Tchebycheffian) spline functions with free knots in (a,b) are investigated. (Author)

Uniqueness and pseudolocality theorems of the mean curvature flow

Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, inspired by the Ricci flow, we prove a pseudolocality theorem of mean curvature flow. As a consequence, we obtain a strong uniqueness theorem, which removes the assumption on the boundedness of the second fundamental form of the solution.