## On some analog of the Bitsadze–Samarskij problem.(English. Russian original)Zbl 0916.35043

Sib. Math. J. 40, No. 1, 153-157 (1999); translation from Sib. Mat. Zh. 40, No. 1, 177-182 (1999).
Consider the degenerate elliptic equation $y^mu_{xx}+u_{yy}+a(x,y)u_y+b(x,y)u_x+c(x,y)u=0 \tag{1}$ in a subdomain $$\Omega$$ of the upper half-plane. The author studies a boundary value problem for (1) with some special nonlocal Bitsadze-Samarskij conditions given on a part of the boundary of $$\Omega$$ and some mixed condition with fractional derivatives given on the degeneration line (the latter may coincide with the Dirichlet or Holmgren condition in particular cases).
For the problem under study, the authors prove an analog of the maximum principle as well as uniqueness and existence theorems.

### MSC:

 35J70 Degenerate elliptic equations 35B50 Maximum principles in context of PDEs

### Keywords:

uniqueness and existence theorems; nonlocal problem
Full Text:

### References:

 [1] M. M. Smirnov, Equations of Mixed Type [in Russian], Vysshaya Shkola, Moscow (1985). · Zbl 0578.35069 [2] A. V. Bitsadze and A. A. Samarskiî, ”On some elementary generalizations of linear elliptic boundary value problems,” Dokl. Akad. Nauk SSSR,185, No. 4, 739–740 (1969). [3] A. M. Nakhushev, ”On a certain mixed problem for degenerate elliptic equations,” Differentsial’nye Uravneniya,11, No. 1, 192–195 (1975). · Zbl 0233.35061 [4] E. Holmgren, ”Sur un probleme aux limites pour l’equationy m z xx +z yy =0,” Ark. Math. Astronom. Fys. 19B,14, 1–3 (1927). · JFM 53.0455.03 [5] S. Gellerstedt, ”Sur un probleme aux limites pour l’equationy 2s u xx +u yy =0,” Ark. Math. Astronom. Fys. 25A,10, 1–12 (1936). · JFM 62.1301.03 [6] M. M. Smirnov, Degenerate Elliptic and Parabolic Equations [in Russian], Nauka, Moscow (1966). · Zbl 0152.09404 [7] S. G. Mikhlin, ”On the E. Tricomi integral equation,” Dolk. Akad. Nauk SSSR,59, No. 6, 1053–1056 (1948). · Zbl 0040.20301 [8] M. M. Smirnov, ”A generalized Tricomi equation,” Uchen. Zap. Belorus. Univ. Ser. Fiz.-Mat. Nauk (Minsk), No. 12, 3–9 (1951). · Zbl 0042.16801
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