Geodesic
About a problem on conductor heating
Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 3, pp. 299-311.

Voir la notice de l'article provenant de la source Math-Net.Ru

Kuiper's problem on conductor heating in a uniform electric field of intensity $\sqrt\lambda$ with a positive parameter $\lambda$ is considered. The conductor temperature distribution is a solution to the Dirichlet problem with homogeneous initial data in a bounded domain for a quasilinear elliptic equation with a discontinuous nonlinearity and a parameter. The heat-conductivity coefficient depends on the spatial variable and temperature, and the specific electric conductivity has discontinuities with respect to the phase variable. The existence of a continuum of generalized positive solutions that connects $(0,0)$ to $\infty$ is proved by a topological method. A sufficient condition is obtained for such solutions to be semiregular. Compared to the papers of H.J. Kuiper and K.C. Chang, the restrictions on the discontinuous nonlinearity (the specific electric conductivity) are weaken.
Keywords: Kuiper's problem, conductor heating, quasilinear elliptic equation, discontinuous nonlinearity, continuum of positive solutions, semiregular solution, topological method. 
@article{CHFMJ_2021_6_3_a3,
     author = {V. N. Pavlenko and D. K. Potapov},
     title = {About a problem on conductor heating},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {299--311},
     publisher = {mathdoc},
     volume = {6},
     number = {3},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_3_a3/}
}
TY  - JOUR
AU  - V. N. Pavlenko
AU  - D. K. Potapov
TI  - About a problem on conductor heating
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2021
SP  - 299
EP  - 311
VL  - 6
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_3_a3/
LA  - ru
ID  - CHFMJ_2021_6_3_a3
ER  - 
%0 Journal Article
%A V. N. Pavlenko
%A D. K. Potapov
%T About a problem on conductor heating
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2021
%P 299-311
%V 6
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_3_a3/
%G ru
%F CHFMJ_2021_6_3_a3
V. N. Pavlenko; D. K. Potapov. About a problem on conductor heating. Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 3, pp. 299-311. http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_3_a3/

[1] Kuiper H. J., “On positive solutions of nonlinear elliptic eigenvalue problems”, Rendiconti del Circolo Matematico di Palermo. Series 2, 20:2–3 (1971), 113–138 | DOI

[2] Krasnosel'skii M.A., Pokrovskii A.V., Systems with Hysteresis, Springer-Verlag, Berlin, 1989 | Zbl

[3] Potapov D.K., “On one problem of electrophysics with discontinuous nonlinearity”, Differential Equations, 50:3 (2014), 419–422 | DOI | Zbl

[4] Kuiper H. J., “Eigenvalue problems for noncontinuous operators associated with quasilinear elliptic equations”, Archive for Rational Mechanics and Analysis, 53:2 (1974), 178–186 | DOI | Zbl

[5] Shragin I.V., “Conditions for measurability of superpositions”, Soviet Mathematics Doklady, 12 (1971), 465–470 | Zbl

[6] Sobolev S.L., Some Applications of Functional Analysis in Mathematical Physics, American Mathematical Society, 1991 | Zbl

[7] Krasnosel'skii M.A., Pokrovskii A.V., “Regular solutions of equations with discontinuous nonlinearities”, Soviet Mathematics Doklady, 17 (1976), 128–132 | Zbl

[8] Krasnosel'skii M.A., Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964 | Zbl

[9] Chang K. C., “Free boundary problems and the set-valued mappings”, Journal of Differential Equatons, 49:1 (1983), 1–28 | DOI | Zbl

[10] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983 | Zbl

[11] Ladyzhenskaya O. A., Ural'tseva N. N., Linear and quasilinear elliptic equations, Academic Press, New York; London, 1968 | Zbl