Nonlocal problems with integral conditions for elliptic equations

ABSTRACT

The article consists of three parts. In the first part, we study nonlocal problems for Poisson equations $u_{tt}+\mathrm{\Delta }u=f(x,t)$ in the cylinder $Q=\left\{\right(x,t):x=(x_1,\dots ,x_n)\in \mathrm{\Omega }\subset {\mathbb{R}}^n,t\in (0,T),0<T<+\mathrm{\infty }\}$ (Ω is a bounded domain and Δ is the Laplacian with respect to the variables $x_1$, …, $x_n$) by defining the integral condition ${\int }_0^{\mathrm{T}}N\left(t\right)u(x,t)\mathrm{d}t=0(x\in \mathrm{\Omega })$ 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 $\frac{{\mathrm{\partial }}^{2m}u}{\mathrm{\partial }{t}^{2m}}+Au=f(m>1).$

COMMUNICATED BY:

• A. Soldatov

KEYWORDS:

• Elliptic differential and operator-differential equations
• nonlocal problem
• integral condition
• existence
• uniqueness

AMS SUBJECT CLASSIFICATIONS:

• Primary: 35J40
• Secondary: 35P05

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by Russian Foundation for Basic Research [grant number 18-51-41009].