Nonlocal problems with integral conditions for elliptic equations

The article consists of three parts. In the first part, we study nonlocal problems for Poisson equations utt+Δu=f(x,t) in the cylinder Q={(x,t): x=(x1,…,xn)∈Ω⊂Rn, t∈(0,T), 0<T<+∞} (Ω is a bounded domain and Δ is the Laplacian with respect to the variables x1, …, xn) by defining the integral condition ∫0TN(t)u(x,t)dt=0(x∈Ω) and some natural boundary conditions on the lower base and on the lateral surfaces of the cylinder Q. The second part of the article is devoted to the study of the solvability of nonlocal problems with integral conditions for elliptic equations with spectral parameter. In the third part, we study some generalizations of the problems presented in the first two parts – to the case of more general integral conditions, to the case of operator-differential equations, and to the case of quasielliptic equations ∂2mu∂t2m+Au=f(m>1).

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