Abstract. While the classical Faber–Krahn inequality shows that the ball uniquely minimizes the first Dirichlet eigenvalue of the Laplacian in the continuum, this rigidity may fail in the discrete setting. We establish quantitative fluctuation estimates for the first Dirichlet eigenvalue of the combinatorial Laplacian on subsets of [Formula: see text] when their cardinality diverges. Our approach is based on a controlled discrete-to-continuum extension of the associated variational problem and the quantitative Faber–Krahn inequality.
Abstract. In this paper, we study the well-posedness and exponential stability of the generalized Korteweg–de Vries–Burgers equation incorporating a damping term and a time-dependent delay feedback. The delay function is examined under two distinct frameworks: one in which it is assumed to be merely continuous, and another in which it satisfies additional regularity conditions, while remaining bounded in both cases. Under suitable assumptions on the delay feedback, we establish the global existence of solutions. Moreover, we derive exponential decay estimates in both delay settings. Notably, in the case where the delay function exhibits higher regularity, exponential stability is obtained without requiring any smallness conditions on the time delay size.
Abstract. In this article, we study a Hénon-type equation in [Formula: see text] driven by a nonlinear operator given by the combination of a local and a nonlocal term. This equation was originally proposed to model spherically symmetric stellar clusters. Here, we prove that, under a suitable relation among the parameters, there exists a threshold separating the existence and nonexistence of solutions. Moreover, we establish regularity properties of the solutions.
We consider the Cauchy problem for an integrable cubic nonlinear Schro"\dinger equation in one space dimension. We prove global spatial analyticity of solutions via the inverse scattering transform. Consequently we remove the size restriction on the analytic norm when initial data have radially or odd symmetry.
We study approximate controllability for a state-dependent sweeping process in a Hilbert space by coupling it with a nonautonomous evolution inclusion. The control acts on the evolution inclusion and induces an indirect feedback mechanism for the sweeping dynamics. We first establish well-posedness of the coupled system by combining a fixed point argument with tools from nonsmooth and nonconvex analysis. We then use resolvent-operator techniques to derive sufficient conditions for approximate controllability. Finally, we apply the abstract results to a thermally coupled frictionless contact problem with an external heat source.
