A Darboux-type problem in the trihedral angle for a third-order equation of hyperbolic type
This paper is devoted to the third-order partial differential equation of the form \[ u_{x_1x_2x_3}=F,\tag{1} \] where \(F\) is a given function. Equation (1) is hyperbolic in the Euclidean space \(\mathbb{R}^3\), for which the planes \(x_1=\text{const}\), \(x_2=\text{const}\), \(x_3=\text{const}\) are characteristic, while the directions, defined by the unit vectors \(\ell_1= (1,0,0)\), \(\ell_2=(0,1,0)\), and \(\ell_3= (0,0,1)\) of the coordinate axes, are bicharacteristic. The author considers (1) in trihedral angles \(D_0\), which is represented in the form \(D_0=\{x\in \mathbb{R}^3 \mid P^0_i (x)>0\), \(i=1,2,3\}\), where \(P^0_i(x) \equiv \alpha_i^0x_1+ \beta^0_ix_2+\gamma_i^0 x_3=0\), \(i=1,2,3\), be arbitrarily given planes in the space \(\mathbb{R}^3\).
scientific article; zbMATH DE number 1541445