Construction of a Carleman function based on the Tikhonov regularization method in an ill-posed problem for the Laplace equation
The paper studies the construction of a Carleman-Lavrant'ev function for a mixed problem for the Laplace equation in a cylinder of rectangular cross-section with Cauchy data on a surface of general form. Consider the domain \(D(F,H)\) bounded by the plain \(z=H\) and the surface \[ S = \{ (x,y,z): \;\;0<x<l_x, \;\;0<y<l_y, \;\;z=F(x,y) \}; \] the function \(F\) satisfies the inequality \(F(x,y)<H\) and continuously differentiable. In the domain \(D(F,H)\) consider the mixed boundary value problem \[ \Delta u(M)=0; \;M \in D(F,H); \] \[ u|_S=f; \;\;\frac{\partial u}{\partial n} |_S =g; \;\;u(0, l_x)=0; \;\;u(l_y, 0) =0. \] The mixed problem with the Cauchy conditions is ill-posed. The solution of the posed problem, including the exact solution, is explicitly constructed by reducing the problem to a Fredholm integral equation of the first kind. A stable approximate solution is constructed by using the Tikhonov regularization method. The approximate solution is used to single out a Carleman-Lavrent'ev function and prove that it is precisely that function also by the definition.
scientific article; zbMATH DE number 6909062