caa a22 4500 605525552 CHVBK 20210128100759.0 cr unu---uuuuu 210128e20150701xx s 000 0 eng 10.1007/s10958-015-2436-z doi (NATIONALLICENCE)springer-10.1007/s10958-015-2436-z Conjugate Problem for Lagrange Multipliers and Consequences for Partial Differential Equations of Mixed Type [Elektronische Daten] [A. Kraiko, N. Tillyaeva] We formulate and solve the conjugate problem for Lagrange multipliers connected with designing a Laval nozzle optimal contour, including its subsonic part. In approximation of an ideal (inviscid and nonheat-conducting) gas, the sought contour provides a thrust maximum under a number of constraints; in particular, given total nozzle length and gas mass flow. For a contour of a contracting (subsonic) part of the nozzle (which is suspected to be optimal) we take an abrupt contraction. Because of the constraint on the nozzle length, the abrupt contraction can be a region of boundary extremum with positive permissible variations of the longitudinal ("axial”) coordinate of the contour. To clarify whether this is true, we use the method of Lagrange multipliers and formulate the conjugate problem for finding the Lagrange multipliers. As in the case of quasilinear Euler equations governing a flow, the linear equations in the conjugate problem are elliptic (hyperbolic) in the subsonic (supersonic) flow domain. The requirement that the conjugate problem be solvable for any contour highlights some features that can be of interest not only for this special problem, but, possibly for the general theory of mixed type partial differential equations. Springer Science+Business Media New York, 2015 Kraiko A. Baranov Central Institute of Aviation Motor, 2. Aviamotornaya St., 111116, Moscow, Russia aut Tillyaeva N. Baranov Central Institute of Aviation Motor, 2. Aviamotornaya St., 111116, Moscow, Russia aut Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 208/2(2015-07-01), 181-198 1072-3374 208:2<181 2015 208 10958 https://doi.org/10.1007/s10958-015-2436-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-015-2436-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kraiko A. Baranov Central Institute of Aviation Motor, 2. Aviamotornaya St., 111116, Moscow, Russia aut NATIONALLICENCE 700 1- Tillyaeva N. Baranov Central Institute of Aviation Motor, 2. Aviamotornaya St., 111116, Moscow, Russia aut NATIONALLICENCE 773 0- Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 208/2(2015-07-01), 181-198 1072-3374 208:2<181 2015 208 10958