The Dirichlet problem for the Gellerstedt equation in an infinite rectangular domain (Q2703943)

The author considers the Gellerstedt equation NEWLINE\[NEWLINET(u)\equiv \text{sgn }y|y|^m u_{xx}+ u_{yy}= 0,\quad m>0NEWLINE\]NEWLINE in the infinite rectangular domain \(D= \{(x, y): 0< x< 1\), \(-\alpha< y<+\infty\); \(\alpha= ((2+ m)/2)^{2/(2+ m)}\}\). The elliptic part \(D^+\) of the domain \(D\) is a half-strip, and the hyperbolic part \(D^-\) is a rectangle with the property that the characteristics NEWLINE\[NEWLINEAB_0: x-(2/2+ m)(-y)^{(m+ 2)/2}= 0,\quad BA_0: x+ (2/(2+ m))(- y)^{(m+ 2)/2}= 1NEWLINE\]NEWLINE pass through its vertices. A function \(u(x,y)\) is called a solution in \(D\) if the following conditions are satisfied: \(u\in C(\overline D)\cap C^2(D^+\cup D\setminus(AB_0\cup BA_0))\); \(u_y(x,y)= O(1)\) as \(y\to +\infty\), \(0\leq x\leq 1\); \(\tau^+(x)= \lim_{y\to +0}u(x,y)= \tau^-(x)= \lim_{y\to -0}u(x,y)\equiv\tau(x)\), \(0\leq x\leq 1\), \(v^+(x)= \lim_{y\to +0} u_y(x, y)= v^-(x)= \lim_{y\to -0}u_y(x, y)\equiv v(x)\), \(0< x< 1\); the integrals \(\int^1_0 u(x,0) u_y(x,0) dx\), \(\int_{D^+} (y^m u^2_x+ u^2_y) dx dy\), and \(\int_{D^-} (u^2_y- (-y)^m u^2_x) dx dy\) exist; the Green formula can be applied to the integral \(\int_D u Tu dx dy\) in the sense that the contour integral occurring in the Green formula exists as the limit of integrals over curves tending to the contour of the domain \(D\) from inside.NEWLINENEWLINENEWLINEThe author considers in details the Dirichlet problem: In the domain \(D\), find a solution \(u(x,y)\) with the boundary conditions NEWLINE\[NEWLINEu(0,y)= \varphi(y),\quad u(1,y)= \psi(y),\quad -\alpha\leq y<+\infty,\quad u(x,-\alpha)= f(x),\quad 0\leq x\leq 1,NEWLINE\]NEWLINE \(\lim_{y\to+\infty} u(x,y)= 0\) uniformly with respect to \(x\in [0,1]\), where the functions \(\varphi(y)\), \(\psi(y)\), and \(f(x)\) satisfy the conditions \(\varphi(y)\), \(\psi(y)\in C[-\alpha,+\infty)\), \(f(x)\in C[0,1]\), NEWLINE\[NEWLINE\varphi(y)= \begin{cases} \varphi_1(y) &\text{if }0< y<+\infty,\\ \psi_1(y) &\text{if }-\alpha< y< 0,\\ \varphi(0)= \psi_1(0) &\text{if }y= 0,\end{cases}\quad \psi(y)= \begin{cases} \varphi_1(y) &\text{if }0< y<+\infty,\\ \psi_2(y) &\text{if }-\alpha< y< 0,\\ \varphi_2(0)= \psi_2(0) &\text{if }y= 0,\end{cases}NEWLINE\]NEWLINE \(\lim_{y\to +\infty} \varphi_i(y)= 0\), \(i= 1,2\), \(f(0)= \psi_1(-\alpha)\), \(f(1)= \psi_2(-\alpha)\).NEWLINENEWLINENEWLINEHe proves the following Theorem 1. Let \(u(x,y)\) be a solution in \(D\) and, moreover, \(u(0,y)= u(1,y)= 0\), \(-\alpha\leq y<+\infty\), \(u(x,-\alpha)= 0\), \(0\leq x\leq 1\), \(\lim_{y\to+\infty} u(x,y)= 0\) uniformly with respect to \(x\in [0,1]\). Then \(u(x,y)\equiv 0\) in \(D\).NEWLINENEWLINENEWLINEThe obtaining of the solution of the Dirichlet problem leads, after standard transformations, to the integral Fredholm equation of the second kind, whose solvability follows from the uniqueness of these solution.