On the bounds of the sum of eigenvalues for a Dirichlet problem involving mixed fractional Laplacians
Huyuan Chen, Mousomi Bhakta, Hichem Hajaiej
Abstract
Our purpose in this paper is to study of the eigenvalues {λi(μ)}i of the Dirichlet problem(−Δ)s1u=λ((−Δ)s2u+μu)inΩ,u=0inRN∖Ω, where 0<s2<s1<1, N>2s1 and (−Δ)s is the fractional Laplacian operator defined in the principle value sense. We first show the existence of a sequence of eigenvalues, which approaches infinity. Secondly we provide a Berezin–Li–Yau type lower bound for the sum of the eigenvalues of the above problem. Furthermore, using a self-contained and novel method, we establish an upper bound for the sum of eigenvalues of the problem under study.
Topics & Concepts
MathematicsEigenvalues and eigenvectorsDirichlet eigenvalueInfinityUpper and lower boundsOperator (biology)Laplace operatorSequence (biology)Pure mathematicsDirichlet distributionCombinatoricsMathematical analysisApplied mathematicsDirichlet's principleBoundary value problemGeneticsGeneQuantum mechanicsRepressorBiologyTranscription factorChemistryBiochemistryPhysicsNonlinear Partial Differential EquationsAdvanced Mathematical Modeling in EngineeringSpectral Theory in Mathematical Physics