Equations Of Principal Type Research Articles

On the dimension of the kernel of the Dirichlet problem for fourth-order equations

We study the dimension of the kernel of the Dirichlet problem in the disk for fourth-order elliptic equations with constant complex coefficients in general position. Special attention is paid to improperly elliptic equations, because the kernel of the Dirichlet problem for properly elliptic equations is finite-dimensional. We single out classes of improperly elliptic equations that have properties similar to those of properly elliptic equations: the kernel of the Dirichlet problem is also finite-dimensional and in some cases even trivial. We consider fourth-order equations in general position, including equations of principal type as well as equations that have multiple characteristics and whose characteristic equation has roots ±i.

An example of a locally unsolvable differential equation of quasiprincipal type with a real-valued weighted principal symbol

In this note we show that the equation $$ - \left\{ {\left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }}} \right)^3 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }}} \right)^2 + 6i\left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }}} \right)\left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }}} \right)x_1 } \right\}u = f$$ is locally unsolvable at the origin of the coordinate system. This equation belongs to the class that generalizes the principal type to the case of weighted derivatives. The example is interesting because the weighted principal symbol is real (in this situation, equations of principal type are solvable) but the unsolvability depends on the behavior of the lower-order terms in a neighborhood of the zeros of the weighted principal symbol.

Hypoelliptic partial differential equations of principal type. Sufficient conditions and necessary conditions

Hypoelliptic partial differential equations of principal type. Sufficient conditions and necessary conditions

Hypoelliptic partial differential equations of principal type with analytic coefficients

Hypoelliptic partial differential equations of principal type with analytic coefficients