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We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded…
We give necessary and sufficient conditions for differentiating under the integral sign an integral that depends on a parameter. The conditions require the equality of two iterated integrals and depend on being able to integrate every…
A Lyapunov-Krasovskii functional with prescribed derivative whose construction does not require the stability of the system is introduced. It leads to the presentation of stability/instability theorems. By evaluating the functional at…
In this paper we investigate continuity properties of first and second order shape derivatives of functionals depending on second order elliptic PDE's around nonsmooth domains, essentially either Lipschitz or convex, or satisfying a uniform…
In this paper, we show that some fundamental results for smooth mappings (e.g., the Brouwer degree formula, the implicit function and inverse function theorems, the mean value theorem, Sard's theorem, Hadamard's global invertibility…
To characterize the Neumann problem for nonlinear Fokker-Planck equations, we investigate distribution dependent reflecting SDEs (DDRSDEs) in a domain. We first prove the well-posedness and establish functional inequalities for reflecting…
The presence of second-order smoothness for objective functions of optimization problems can provide valuable information about their stability properties and help us design efficient numerical algorithms for solving these problems. Such…
The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations)…
We investigate the existence of non-trivial holomorphic and meromorphic solutions of Fermat functional equations over an open Riemann surface $S$. When $S$ is hyperbolic, we prove that any $k$-term Fermat functional equation always exists…
We give a new equivalent condition for the Riemann hypothesis consisting in an order condition for certain finite rational combinations of the values of the Riemann zeta-function at even positive integers.
In the framework of stochastic functional differential equations (SFDE's) and the corresponding calculus developed in the recent years by F. Yan and S. Mohammed, we provide a series of representation formulae for a variety of highly…
We give sufficient conditions under which solutions of finite-difference schemes in the space variable for second order possibly degenerate parabolic and elliptic equations admit estimates of spatial derivatives up to any given order…
The objective of this paper is to deepen the understanding of the connection between the continuous and smooth dependence of solutions on initial conditions and the regularity of the history functionals for retarded functional differential…
We study, by means of a topological approach, the forced oscillations of second order functional retarded differential equations subject to periodic perturbations. We consider a delay-type functional dependence involving a gamma probability…
In this note, we present an elementary proof for a well-known second-order sufficient optimality condition in nonlinear semidefinite optimization which does not rely on the enhanced theory of second-order tangents. Our approach builds on an…
We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation…
Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws is discussed. In the generic case, nonlocal conservation…
Two results on the completeness of maximal solutions to first and second order ordinary differential equations (or inclusions) over complete Riemannian manifolds, with possibly time-dependent metrics, are obtained. Applications to…
In this paper, we introduce the second-order subdifferentials for functions which are G\^ateaux differentiable on an open set and whose G\^ateaux derivative mapping is locally Lipschitz. Based on properties of this kind of second-order…
We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a re duction of the associated dynamical…