Solvability and regularity of solutions for some classes of elliptic functional-differential equations (Q5948382)

The authors of this paper deliver a review of the theory of boundary value problems for some classes of elliptic functional-differential equations and functional-difference equations which are applicable to some mathematical models in nonlinear optics, elasticity theory, control theory, diffusion processes etc. Here the authors mainly consider boundary value problems for elliptic functional-difference equations and elliptic functional-differential equations with contractions and expansions. Some applications of elliptic functional-difference equations as \(-\Delta R_Qu(x)= f_0 (x)\) in \(Q\) with a boundary condition \(u(x)=0\) on \(\partial Q\) are discussed. It is used a nonlocal difference operator \(Ru(x)=u(x_1,x_2)+ \gamma_1u(x_1+1, x_2) +\gamma_2 u(x_1-1,x_2)\), where \(x=(x_1,x_2)\), \(Q=(0,2) \times(0,1)\), \(\gamma_1, \gamma_2 \in\mathbb{R}\), \(f_0\in L_2(Q)\). Necessary and sufficient conditions for strong ellipticity and spectral properties of strongly elliptic functional-differential operators as well the regularity of the solutions are shown.