Resolvent estimates and spectral properties of a class of degenerate elliptic operators in a bounded domain

T. P. Konstantinova

Matematičeskie zametki SVFU, Tome 26 (2019), pp. 37-50.

Résumé

The paper is devoted to the study of the spectral asymptotics of elliptic operators of arbitrary even order in a bounded domain with power degeneration along the entire boundary. The operators under study are generated by sesquilinear forms that may not satisfy the coercivity condition. The main part of the published papers in this area refers to the case when the coefficients of the studied operators can be represented as a product of a bounded function and the degree of distance to the boundary. In contrast, here we study elliptic operators whose lower coefficients belong to certain $L_p$-spaces with power weights. Earlier, in many papers, where the estimation of the resolvent of non-self-adjoint operators generated by sesquilinear forms was studied, the inequality of the form $\|(A-\lambda E)^{-1}\|\leq M|\lambda|^{-1/2}$ was proved. Here we prove one representation of the resolvent of the operator $A$ that allows us to obtain an inequality of this type with 1 instead of 1/2. On the basis of such inequalities, we can investigate the summability in the Abel–Lidskiy sense of the system of root vector functions of the operator $A$. It is also proved that the operator $A$ has a discrete spectrum, and the asymptotics of the function $N(t)$, the number of eigenvalues of the operator $A$ whose magnitude is at most $t$, taking into account their algebraic multiplicities, is studied.

Détail

Keywords: elliptic operator, bounded domain, power degeneracy, resolvent estimate, asymptotic behavior of spectrum, noncoercive form.