Boundary Value Problems For Differential Operators Research Articles

A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS

A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The operators of the nonlocal calculus are used to define volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application discussed is the posing of abstract nonlocal balance laws and deriving the corresponding nonlocal field equations; this is demonstrated for heat conduction and the peridynamics model for continuum mechanics.

On the completeness of systems of eigenfunctions and associated functions of differential operators of orders 2 − ε and 1 − ε

In the space of vector-functions, we consider a boundary-value problem for differential operators of fractional orders (2 − e) and (1 − e) and prove the completeness of the system of eigenfunctions and associated functions of this problem in the space \(L_1 \left( {\left[ {0,1} \right],\,\mathbb{C}^p } \right)\).