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| ON DIRICHLET PROBLEMS FOR SECOND ORDER QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS |
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Citation： |
Yue Jiangliang.ON DIRICHLET PROBLEMS FOR SECOND ORDER QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS[J].Chinese Annals of Mathematics B,1984,5(1):43~58 |
| Page view： 780
Net amount： 941 |
Authors： |
Yue Jiangliang; |
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| Abstract： |
The purpose of this paper is to study the existence of the classical solutions of some Dirichlet problems for quasilinear elliptic equations
$$\[{a_{11}}(x,y,u)\frac{{{\partial ^2}u}}{{\partial {x^2}}} + 2{a_{12}}(x,y,u)\frac{{{\partial ^2}u}}{{\partial x\partial y}} + {a_{22}}(x,y,u)\frac{{{\partial ^2}u}}{{\partial {y^2}}} + f(x,y,u,\frac{{\partial u}}{{\partial x}},\frac{{\partial u}}{{\partial y}}) = 0\]$$
Where $\[{a_{ij}}(x,y,u)(i,j = 1,2)\]$ satisfy
$$\[\lambda (x,y,u){\left| \xi \right|^2} \le \sum\limits_{i,j = 1}^2 {{a_{ij}}(x,y,u)} {\xi _i}{\xi _j} \le \Lambda (x,y,u){\left| \xi \right|^2}\]$$
for all $\[\xi \in {R^2}\]$ and $\[(x,y,u) \in \bar \Omega \times [0, + \infty ),i.e.\lambda (x,y,u),\Lambda (x,y,u)\]$ denote the minimum and maximum eigenvalues of the matrix $\[[{a_{ij}}(x,y,u)]\]$ respectively, moreover
$$\[\lambda (x,y,0) = 0,\Lambda (x,u,0) = 0;\Lambda (x,y,u) \ge \lambda (x,y,u) > 0,(u > 0).\]$$
Some existence theorems under tire “ natural conditions imposed on $\[f(x,y,u,p,q)\]$ are obtained. |
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