Geodesic
On the solvability of the generalized Neumann problem for a higher-order elliptic equation in an infinite domain
Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 564-575.

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We consider the generalized Neumann problem for a $2l$th-order elliptic equation with constant real higher-order coefficients in an infinite domain containing the exterior of some circle and bounded by a sufficiently smooth contour. It consists in specifying of the $(k_j-1)$th-order normal derivatives where $1 \le k_1 \ldots $ for $k_j = j$ it turns into the Dirichlet problem, and for $k_j = j + 1$ into the Neumann problem. Under certain assumptions about the coefficients of the equation at infinity, a necessary and sufficient condition for the Fredholm property of this problem is obtained and a formula for its index in Hölder spaces is given. 
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B. D. Koshanov; A. P. Soldatov. On the solvability of the generalized Neumann problem for a higher-order elliptic equation in an infinite domain. Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 564-575. http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a10/

[1] Bitsadze A. V., “O nekotorykh svoistvakh poligarmonicheskikh funktsii”, Diff. uravn., 24:5 (1988), 825–831 | Zbl

[2] Kondratev V. A., Oleinik O. A., “O periodicheskikh resheniyakh parabolicheskogo uravneniya vtorogo poryadka vo vneshnikh oblastyakh”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 4 (1985), 38–47 | Zbl

[3] Koshanov B. D., Kulimbek Zh. K., “Povedenie reshenii uravneniya Puassona i bigarmonicheskogo uravneniya”, Mat. zh., 16:1 (2016), 118–134 | Zbl

[4] Koshanov B., Soldatov A. P., “Kraevaya zadacha s normalnymi proizvodnymi dlya ellipticheskogo uravneniya na ploskosti”, Diff. uravn., 52:12 (2016), 1666–1681 | Zbl

[5] Malakhova N. A., Soldatov A. P., “Ob odnoi kraevoi zadache dlya ellipticheskogo uravneniya vysokogo poryadka”, Diff. uravn., 44:8 (2008), 1077–1083 | Zbl

[6] Matevosyan O. A., “O edinstvennosti resheniya pervoi kraevoi zadachi teorii uprugosti dlya neogranichennykh oblastei”, Usp. mat. nauk, 48:6 (1993), 159–160 | Zbl

[7] Nazarov S. A., Plamenevskii B. A., Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei, Nauka, M., 1991

[8] Pale R., Seminar po teoreme Ati—Zingera ob indekse, Mir, M., 1970

[9] Soldatov A. P., “Singulyarnye integralnye operatory i ellipticheskie kraevye zadachi”, Sovrem. mat. Fundam. napravl., 63, no. 1, 2016, 1–179

[10] Soldatov A. P., “Ob odnoi kraevoi zadache dlya ellipticheskogo uravneniya na ploskosti v mnogosvyaznoi oblasti”, Vladikavkaz. mat. zh., 19:3 (2017), 51–58 | Zbl

[11] Soldatov A. P., “O fredgolmovosti i indekse obobschennoi zadachi Neimana”, Diff. uravn., 56:2 (2020), 217–225 | Zbl

[12] Soldatov A. P., Chernova O. V., “Zadacha Rimana—Gilberta dlya ellipticheskikh sistem pervogo poryadka na ploskosti s postoyannymi starshimi koeffitsientami”, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril., 149, 2018, 95–102

[13] Koshanov B. D., Soldatov A. P., “About the generalized Dirichlet—Neumann problem for an elliptic equation of high order”, AIP Conference Proceedings, 1997 (2018), 020013 | DOI

[14] Matevossian O. A., “On solutions of the Neumann problem for the biharmonic equation in unbounded domains”, Math. Notes, 98 (2015), 990–994 | DOI | Zbl

[15] Ssheshter M., “General boundary value problems for elliptic partial differential equations”, Commun. Pure Appl. Math., 12 (1950), 467–480