Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userShape optimization for an elliptic operator with infinitely many positive and negative eigenvalues
97
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 spectrum. Rotational symmetry allows a decomposition of the operator into Fourier components. Eigenvalues of infinitely many Fourier components are constructed so that they lie within the essential spectrum of other Fourier components and thus within the essential spectrum of the full NP operator. The proof requires the perturbation to be sufficiently small, with controlled curvature, and the conical singularity to be sufficiently flat.
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 express in terms of the cosine of the Friedrichs angle computed in an infinite product space. In addition, we provide estimates and alternative expressions for the above-mentioned number. Furthermore, we relate this number to the dichotomy theorem and to super-polynomially fast convergence. We also discuss polynomial convergence of the simultaneous projection method which takes place for particularly chosen starting points.
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 single connected component. In our construction, we relate operations on the reaction network to the multivariate polynomial occurring as a common factor in the system of differential equations.
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, onumber u&=0~text{in}~mathbb{R}^NsetminusOmega, onumber end{align} where $Omegasubsetmathbb{R}^N,, Ngeq2$ is a smooth, bounded domain, $lambda>0$, $sin (0,1)$, $gamma(x)in(0,1)$ for all $xinbar{Omega}$, $N>sp(x,y)$ for all $(x,y)inbar{Omega}timesbar{Omega}$ and $(-Delta)_{p(cdot)}^{s}$ is the fractional $p(cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{infty}(bar{Omega})$ estimate of the solution(s) by the Moser iteration technique.
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 $OmegasubsetR^N$ is a smooth bounded domain, $1<p^-=min_{xinoverline{Omega}}p(x)leq p(x)leqmax_{xinoverline{Omega}}p(x)=p^+<N$, $1leq r(x)<p^{*}(x)=frac{Np(x)}{N-p(x)}$, $r^-=min_{xin overline{Omega}}r(x)<p^-$, $r^+=max_{xinoverline{Omega}}r(x)>p^+$ and $Q: overline{Omega}toR$ is a nonnegative continuous function. We prove that eqref{eq0.1} has infinitely many small solutions and infinitely many large solutions by using the Clarks theorem and the symmetric mountain pass lemma.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
