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The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. [ABSTRACT FROM AUTHOR]

In this paper we develop techniques for computing elementwise conservative approximations of the flux on element boundaries for the continuous Galerkin method. The technique is based on computing a correction of the average normal flux on an edge or face. The correction is a jump in a piecewise constant or linear function. We derive a basic algorithm which is based on solving a global system of equations and a parallel algorithm based on solving local problems on stars. The methods work on meshes with different element types and hanging nodes. We prove existence, uniqueness, and optimal order error estimates. Lastly, we illustrate our results by a few numerical examples. [ABSTRACT FROM AUTHOR]

We consider a parabolic equation with a drift term Δu+b∇u-ut=0. Under the condition divb=0, we prove that solutions possess dramatically better regularity than those provided by standard theory. For example, we prove continuity of solutions when not even boundedness is expected. [ABSTRACT FROM AUTHOR]

We prove the boundary Harnack principle for ratios of solutions u/ v of non-divergence second order elliptic equations Lu = a D u + b D u = 0 in a bounded domain Ω ⊂ $ {\mathbb R} $. We assume that b ∈ L(Ω) and Ω is a twisted Hölder domain of order α ∈ (1/2, 1]. Based on this result, we derive the Hölder regularity of u/ v for uniform domains. Bibliography: 27 titles. [ABSTRACT FROM AUTHOR]

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ≪1, K≫1, s>1, we construct smooth initial data u0 with $\|u_{0}\|_{{H}^{s}}<\delta$, so that the corresponding time evolution u satisfies $\|u(T)\|_{{H}^{s}}>K$ at some time T. This growth occurs despite the Hamiltonian’s bound on $\|u(t)\|_{\dot{H}^{1}}$ and despite the conservation of the quantity $\|u(t)\|_{L^{2}}$. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems. [ABSTRACT FROM AUTHOR]

The article presents a study which examines the stability of a new pexider-type functional equation. This functional equation is f1(x+y+z)+f2(x-y)+f3(z-x)f4(x-y-z)-f5(x+y)-f6(x+z) = 0, a mixed of a quadratic and an additive functional equations. It established the generalized Hyers-Ulam-Rassias stability for new pexider-type functional equation on the punctured domain V\ {0} and obtain its general solution from the stability results. It also let V and X be normed space and Banach space, respectively. Results of the applications and stability of the equation are also given.

Constitutive equations that describe the experimentally observed failure waves are proposed to model inelastic strains of brittle materials. The complete system of equations is hyperbolic, each equation of this system has divergent form. The model is based on the assumption that continual failure is the process of transition from an intact state to a “fully damaged” state described by the kinetics of the order parameter. The structure of stationary traveling compressive waves is analyzed using a simplified model. It is shown that in a certain range of amplitudes, the wave splits into an elastic precursor and a failure wave. [ABSTRACT FROM AUTHOR]

The article presents a discussion that focuses on the two methods of sharpening Jordan's inequality. It is mentioned that the first method conveys the possibility to come up with new strengthened Jordan's inequalities from old ones. On the other hand, it is also said that the second method shows that Jordan's inequality may be sharpened by choosing appropriate functions in the monotone form of L'Hopital's rule. A background of Jordan's inequality is also disclosed. Theorems that support the sharpening of Jordan's inequality are also provided.

The article presents a discussion that focuses on the Kantorovich and the Durrmeyer-type modifications of the generalized favard operators. The discussion is also said to be aimed at proving an inverse approximation theorem for functions f. An analogous inequality for the rth weighted modulus of smoothness of the function f is also presented. The theorems, equations, and calculations associated with such mathematical concept are also included in the discussion. Mathematical proofs linked to such equations are also provided.

A mixed optimal model approximation is presented to obtain reduced-order models for truly fast descriptor systems. By a projection from truly fast descriptor systems to discrete-time systems, a mixed optimal model approximation for truly fast descriptor systems is transformed to a mixed optimal model approximation of the corresponding discrete-time systems. The structure of the fast descriptor systems is preserved in the model approximation procedure. The expression of the error and its gradient are given explicitly in terms of the solutions of certain Lyapunov equations. A numerical example is provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of a shallow water equation and its local wellposedness. [ABSTRACT FROM AUTHOR]