Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A positive solution of an elliptic equation with nonlinear integral boundary condition of the radiation type

  • 35 Accesses

  • 2 Citations


The problem for an elliptic equation with a nonlinear integral boundary condition describing, in particular, a stationary radiative heat transfer according to the Stefan-Boltzmann law in a system of blackbodies is considered. Theorems about the existence, uniqueness, and stability of the positive generalized solution are established.

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    A. N. Tikhonov, “On functional equations of the Volterra type and their applications to some problems of mathematical physics,” Byull. Mosk. Gos. Univ., Ser. Mat. Mekh.,1, No. 8, 1–25 (1938).

  2. 2.

    S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc., Providence, Rhode Island (1969).

  3. 3.

    O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press (1968).

  4. 4.

    S. L. Sobolev, Introduction to the Theory of Quadrature Formulae [in Russian], Nauka, Moscow (1974).

  5. 5.

    E. Hille and R. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc., Providence, Rhode Island (1957).

  6. 6.

    J. L. Lions, Certain Methods of Solution of Nonlinear Boundary-Value Problems [Russian translation], Mir, Moscow (1972).

  7. 7.

    A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylock (1961).

  8. 8.

    R. Siegel and J. Howell, Thermal Radiation Heat Transfer, McGraw-Hill, New York (1969).

Download references

Author information

Additional information

Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 117–128, July, 1977.

The author thanks N. S. Bakhvalov for suggesting the problem and guiding the work.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Amosov, A.A. A positive solution of an elliptic equation with nonlinear integral boundary condition of the radiation type. Mathematical Notes of the Academy of Sciences of the USSR 22, 555–561 (1977). https://doi.org/10.1007/BF01147699

Download citation


  • Radiation
  • Boundary Condition
  • Heat Transfer
  • Generalize Solution
  • Elliptic Equation