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Equations with nonnegative characteristic form. I. (English. Russian original) Zbl 1200.35158
This is the first part (Chapters 1 and 2) of a two-part work which is aimed at presenting the foundations of the general theory of second order partial differential equations with nonnegative characteristic form. It is devoted to the anniversary of the famous Russian mathematician O. A. Oleinik and is based on a series of joint papers of O. A. Oleinik and E. V. Radkevich written in 1974--1978. The ideas and methods of the latter have found numerous applications which motivated E. V. Radkevich to review the old results on second order partial differential equations with nonnegative characteristic form and to present them in detail for more general type of equations. An equation of the form $$L(u):=a^{kj}(x)u_{x_{k}x_{j}}+b^{k}(x)u_{x_{k}}+c(x)u=f(x),\quad a^{kj}\xi_k\xi_j\geq 0$$ where repeated indices are summed from 1 to $m$ is called an equation with nonnegative characteristic form on a set $G$ if at each point $x$ in $G$ it holds: $a^{kj}\xi_k\xi_j\geq0$ for any real vector $\xi=\{\xi_1,\dots,\xi_m\}$. Such equations are also called degenerate elliptic equations or elliptic-parabolic equations. Note that first results for the boundary-value problems to the equation (2) were obtained by G. Fichera in 1956. In Chapter 1 of the present paper, the existence of a generalized solution of the boundary value problem to equation (2) is obtained as the limit of the boundary-value problem solution to the equation $\varepsilon\Delta u+L(u)=f$ when $\varepsilon\to+0$. In the second chapter, the smoothness of a solution of the boundary-value problem to the equation (2) is considered. The hypoellipticity of some class of degenerated elliptic operators was considered by L. Hormander in 1967 and afterwards in the works of E. V. Radkevich and O. A. Oleinik for the equation (1). These results in a more general form are provided in Chapters 2,3,4. Another application of equation (1) are the Tricomi problem, the equation of the mixed type and boundary layer asymptotic for the compressible fluid. In connection with the Tricomi problem an interest arises in elliptic and hyperbolic equations which are degenerated at the boundary. So in Chapter 2 there are considered the qualitative properties of solutions to the second order degenerated elliptic equation and the maximum principle. Chapter 5 deals with the boundary-value problem for degenerated hyperbolic equation $u_{tt}=L(u)+f$ and a condition for the existence of a solution to such boundary-value problem in the Sobolev spaces is obtained. Chapter 6 treats the question of the uniqueness (in the class of growing functions) of solutions to a boundary-value problem for evolution equations in unbounded domains. Some a priori estimates determine the behavior of solutions as $|x|\to\infty$ or $x\longrightarrow0$. The uniqueness theorems for the general boundary-value problems to the parabolic systems in unbounded domains are proved. The author has provided a panoramic view on the theory of partial differential and pseudo-differential operators. The work can be, in particular, a source of many special courses.

##### MSC:
 35J70 Degenerate elliptic equations 35-02 Research monographs (partial differential equations) 35J25 Second order elliptic equations, boundary value problems
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