Conjugate Problem for Lagrange Multipliers and Consequences for Partial Differential Equations of Mixed Type

Abstract

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.