# Introduction

Chapter

## Summary

The principal objective of this work is the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process. This means we shall be concerned with the solvability of boundary value problems (primarily the Dirichlet problem) and related general properties of solutions of linear equations
and of quasilinear equations
Here

$$Lu \equiv {a^{ij}}\left( x \right){D_{ij}}u + {b^i}\left( x \right){D_i}u + c\left( x \right)u = f\left( x \right),\quad i,j = 1,2, \ldots ,n,$$

(1.1)

$$Qu\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to
{ \equiv } {a^{ij}}\left( {x,u,Du} \right){D_{^{ij}}}u + b\left( {x,u,Du} \right) = 0.$$

(1.2)

*Du*= (*D*_{1}*u*, … ,*D*_{ n }*u*), where*D*_{ i }*u*= ∂*u*/∂*x*_{ i },*D*_{ ij }*u*= ∂^{2}*u*/∂*x*_{i}∂*x*_{ j }, etc., and the summation convention is understood. The ellipticity of these equations is expressed by the fact that the coefficient matrix [*a*^{ ij }] is (in each case) positive definite in the domain of the respective arguments. We refer to an equation as*uniformly elliptic*if the ratio γ of maximum to minimum eigenvalue of the matrix [*a*^{ ij }] is bounded. We shall be concerned with both non-uniformly and uniformly elliptic equations.## Keywords

Weak Solution Elliptic Equation Dirichlet Problem Quasilinear Equation Interior Estimate
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer-Verlag Berlin Heidelberg 1977