scholarly journals On the Dirichlet problem for quasi-linear elliptic differential equations of the second order

1961 ◽  
Vol 13 (1) ◽  
pp. 45-62 ◽  
Author(s):  
Kiyoshi AKo
Keyword(s):  
Dirichlet Problem ◽  
Differential Equations ◽  
Second Order ◽  
Elliptic Differential Equations

On principally linear elliptic differential equations of the second order

1993 ◽  
pp. 349-371
Author(s):  
Masaya Yamaguti ◽  
Louis Nirenberg ◽  
Sigeru Mizohata ◽  
Yasutaka Sibuya
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Elliptic Differential Equations

The problem of dirichlet for quasilinear elliptic differential equations with many independent variables

1969 ◽  
Vol 264 (1153) ◽  
pp. 413-496 ◽  
Keyword(s):  
Dirichlet Problem ◽  
Differential Equations ◽  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Central Position ◽  
Second Primary ◽  
Elliptic Differential Equations ◽  
High Level ◽  
Quasilinear Elliptic

This paper is concerned with the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order, the conclusions being in the form of necessary conditions and sufficient conditions for this problem to be solvable in a given domain with arbitrarily assigned smooth boundary data. A central position in the discussion is played by the concept of global barrier functions and by certain fundamental invariants of the equation. With the help of these invariants we are able to distinguish an important class of ‘ regularly elliptic5 equations which, as far as the Dirichlet problem is concerned, behave comparably to uniformly elliptic equations. For equations which are not regularly elliptic it is necessary to impose significant restrictions on the curvatures of the boundaries of the underlying domains in order for the Dirichlet problem to be generally solvable; the determination of the precise form of these restrictions constitutes a second primary aim of the paper. By maintaining a high level of generality throughout, we are able to treat as special examples the minimal surface equation, the equation for surfaces having prescribed mean curvature, and a number of other non-uniformly elliptic equations of classical interest.

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