Maximum principle for an equation of mixed type. (English. Russian original) Zbl 0677.35074

Differ. Equations 24, No. 11, 1322-1329 (1988); translation from Differ. Uravn. 24, No. 11, 1967-1976 (1988).
Consider the equation \[ L(u)=K(y)u_{xx}+N(x)u_{yy}+a(x,y)u_ x+b(x,y)u_ y+c(x,y)u=0, \] where \(yK(y)>0\) for \(y\neq 0\), \(N(x)>0\) in the domain D bounded by a Jordan arc \(\sigma\) ending at A(0,0) and B(\(\ell,0)\) in \(y>0\), and two characteristics AC and BC of the equation in \(y<0\), where \(\ell >0\). The present paper is devoted to prove the maximum principle of the Tricomi problem for the equation \(L(u)=0\) with boundary condition \(u=\phi (x,y)\) on AC\(\cup \sigma\). The author intends to prove the principle under rather weak conditions of the coefficients and to discuss the principle for various examples of mixed type equations.
Reviewer: M.Saigo


35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35B50 Maximum principles in context of PDEs