Solvability theorems for linear equations of Tricomi type
The author is concerned with questions of solvability for mixed type equations. By solvability one means that for any \(f\) lying in a specified space, the existence of a classical or distributional solution \(u\) of an equation \(Pu= f\) in \(\Omega\) is shown where no side conditions on \(u\) are imposed. Hereby, the differential operator \(P\) has the form \[ P= D^2_y+ yA(x, y,D_x)+ B(x, y,D_x,D_y), \] where \((x,y)\in\mathbb{R}^n\times \mathbb{R}^1\), \(\Omega\subset\mathbb{R}^{n+ 1}\) is a domain overlapping the interface \(y= 0\), \(B\) is a first-order differential operator, and \(A\) is a second-order differential operator with principle symbol \(a(x,y,\xi)> 0\) for \(|\xi|\neq 0\). Using the theory of Hörmander, theorems on local and semilocal solvability are stated which rely on a priori estimates. All the results are independent of the lower order term \(B\) as they depend only on the principle symbol.
scientific article; zbMATH DE number 1115452
scientific article