In [4], the authors have proposed a generalization of the classical Fast Marching Method of Sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign. The goal of this paper is to propose a modified version of the Generalized Fast Marching Method proposed in [4] for which we state a general comparison principle. We also prove the conver...

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm,  such that $f + g $ attains its strong minimum on $X. $ This result extends some of the  well-known varitional principles as that of Ekeland [On the variational principle,  J. Ma...

This work is devoted to the study of a Liouville comparison principle for entire weak solutions of quasilinear differential inequalities of the form A(u) + |u|q−1u ≤ A(v) + |v|q−1v on Rn, where n ≥ 1, q > 0, and the operator A(w) belongs to a class of the so-called α-monotone operators. Typical examples of such operators are the p-Laplacian and its well-known modifications for 1 < p ≤ 2. The re...

We consider weak solutions of the differential inequality of pLaplacian type −∆pu− f(u) ≤ −∆pv − f(v) such that u ≤ v on a smooth bounded domain in RN and either u or v is a weak solution of the corresponding Dirichlet problem with zero boundary condition. Assuming that u < v on the boundary of the domain we prove that u < v, and assuming that u ≡ v ≡ 0 on the boundary of the domain we prove u ...

This note demonstrates that it is possible to bound the expectation of an arbitrary norm of a random matrix drawn from the Stiefel manifold in terms of the expected norm of a standard Gaussian matrix with the same dimensions. A related comparison holds for any convex function of a random matrix drawn from the Stiefel manifold. For certain norms, a reversed inequality is also valid. Mathematics ...

Abstract In this paper, we investigate the problem of p th-moment stability stochastic functional differential equations with Markovian switching and impulsive control via comparison principle. Employing analysis theory an delay inequality, establish a new principle for control. Using principle, derive sufficient conditions by equations. An example is provided to show effectiveness proposed res...

In this paper we introduce a collection of powerful statistical comparison results. We first present the results that we obtained while developing a general comparison concept. After that we introduce a separate lifting procedure that is a comparison concept on its own. We then show how in certain scenarios the lifting procedure basically represents a substantial upgrade over the general strate...

(if μ is the Lebesgue measure, the index μ will be omitted). The problem is to define the behavior of P{||X||μ ≤ ε} as ε → 0. The study of small deviation problem was initiated by Sytaya [S] and continued by many authors. The history of the problem in 20th century is described in reviews by Lifshits [Lf2] and by Li and Shao [LS]. Latest results can be found in [Lf3]. According to the well-known...

We consider gradient descent equations for energy functionals of the type S(u) = 1 2 〈u(x), A(x)u(x)〉L2 + ∫ Ω V (x, u) dx, where A is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration. We consider the steepest descent equation for S where the gradient is an element of the Sobolev spac...