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The paper deals with an eigenvalue problems possessing infinitely many positive and negative eigenvalues. Inequalities for the smallest positive and the largest negative eigenvalues, which have the same properties as the fundamental frequency, are derived. The main question is whether or not the classical isoperimetric inequalities for the fundamental frequency of membranes hold in this case. The arguments are based on the harmonic transplantation for the global results and the shape derivatives (domain variations) for nearly circular domain.
This article constructs a surface whose Neumann-Poincare (NP) integral operator has infinitely many eigenvalues embedded in its essential spectrum. The surface is a sphere perturbed by smoothly attaching a conical singularity, which imparts essential
We investigate the properties of the simultaneous projection method as applied to countably infinitely many closed and linear subspaces of a real Hilbert space. We establish the optimal error bound for linear convergence of this method, which we expr
We show that weakly reversible mass-action systems can have a continuum of positive steady states, coming from the zeroes of a multivariate polynomial. Moreover, the same is true of systems whose underlying reaction network is reversible and has a si
We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities begin{align} (-Delta)_{p(cdot)}^{s} u&=frac{lambda}{|u|^{gamma(x)-1}u}+f(x,u)~text{in}~Omega, o
This paper is concerned with the $p(x)$-Laplacian equation of the form begin{equation}label{eq0.1} left{begin{array}{ll} -Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &mbox{in} Omega, u=0, &mbox{on} partial Omega, end{array}right. end{equation} where $Omegasubs