The Poincaré problem for an equation of mixed type. (Q1432519)

In this paper the Lavrent'ev-Bitsadze equation \[ (\text{sgn\,}y)u_{xx}+ u_{yy}= 0\tag{1} \] is considered in a mixed domain \(D\) of the complex plane \(z= x+ iy\) bounded for \(y\geq 0\) and \(y\leq 0\) by Lyapunov arcs \(\sigma\) and \(\gamma\), respectively, with common ends at \(z=0\) and \(z=1\). The arcs are assumed to be nontangential to the real segment \([0, 1]\) as \(\gamma\) is nontangential to the family of characteristics \(x\pm y=\text{const}\). In particular, the arc \(\gamma\) lies in the characteristic triangle \[ \{y+ x> 0,\,x-y< 1,\,y\leq 0\}\tag{2} \] with the base \([0, 1]\). A solution \(u\) to (1) is a function from \(C^n(D)\) that is harmonic in \(D^+= D\cap\{y> 1\}\) and can be represented by the d'Alembert formula \(u(x, y)= f_1(x +y)+ f_2(x -y)\) in \(D^-= D\cap\{y< 0\}\) with \(f_i\in C^n(0,1)\). For \(n\geq 1\), the functions \(f_i\) can be expressed in terms of the Cauchy data \(\tau_i(x)= u(x,0)\) and \(v(x)= u_y(x,0)\) from the equalities \(f_1+ f_2 =\tau\) and \(f_1'= f_2'= v\). The Poincaré problem (problem P) consists in finding a solution \(u(x, y)\) to (1) in the class \(C(\overline D)\cap C^1(D\overline D\setminus\{0,1\})\) given the boundary condition \[ (a_1 u_x+ a_2 u_y+ a_0 u)|_{\partial D}= g, \] where the coefficients \(a_j\) are Hölder continuous on \(\sigma\) and \(\gamma\), \(a_1+ ia_2\neq 0\) everywhere on \(\sigma\), and one of the functions \(a_1\pm a_2\) is nonzero everywhere on\(\gamma\). Depending on the plus or minus sign in the last condition, problem P is referred to the type 0 or 1. In such way the formulation (2) covers the Dirichlet problem \(u|_{\partial D}= f\) with \(a_0=0\) and \((a_1,a_2)\) coinciding with a tangent vector; the Neumann problem \({\partial u\over\partial\widetilde n}|_{\partial D}= g\); and a pair of mixed problems \(u|_\sigma= f\), \({\partial u\over\partial\widetilde n}|_\gamma= g\), \({\partial u\over\partial\widetilde n}|_\sigma= g\), \(u|_\gamma= f\). The main goal of the paper is to describe the conditions for the problem P to be Fredholm and to find its index.