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In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of mixed fractional operators with different orders include medicine, plasma physics, and population dynamics.
We establish an upper bound of the sum of the eigenvalues for the Dirichlet problem of the fractional Laplacian. Our result is obtained by a subtle computation of the Rayleigh quotient for specific functions.
We establish the existence and multiplicity of positive solutions to the problems involving the fractional Laplacian: begin{equation*} left{begin{array}{lll} &(-Delta)^{s}u=lambda u^{p}+f(u),,,u>0 quad &mbox{in},,Omega, &u=0quad &mbox{in},,mathbb{R}^
We study $L^p$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based Besov classes in Dirichlet spaces that was introduced by the authors in previous works.
We study the graphs associated with Vicsek sets in higher dimensional settings. First, we study the eigenvalues of the Laplacians on the approximating graphs of the Vicsek sets, finding a general spectral decimation function. This is an extension of
We prove the long-standing Payne conjecture that the $k^{text{th}}$ eigenvalue in the buckling problem for a clamped plate is not less than the ${k+1}^{text{st}}$ eigenvalue for the membrane of the same shape which is fixed on the boundary. Moreover,