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

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.

[1]  James Serrin,et al.  The Dirichlet Problem for Surfaces of Constant Mean Curvature , 1970 .

[2]  E. Giorgi,et al.  Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche , 1969 .

[3]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[4]  J. Serrin,et al.  The Dirichlet problem for the minimal surface equation in higher dimensions. , 1968 .

[5]  J. Serrin.  The dirichlet problem for quasilinear equations with many independent variables. , 1967, Proceedings of the National Academy of Sciences of the United States of America.

[6]  N. Trudinger.  The Dirichlet problem for nonuniformly elliptic equations , 1967 .

[7]  N. Trudinger.  On the Dirichlet problem for quasilinear uniformly elliptic equations in n variables , 1967 .

[8]  G. Stampacchia,et al.  On some non-linear elliptic differential-functional equations , 1966 .

[9]  S. Chern.  On the curvatures of a piece of hypersurface in euclidean space , 1965 .

[10]  Robert Finn,et al.  Remarks relevant to minimal surfaces, and to surfaces of prescribed mean curvature , 1965 .

[11]  J. Nitsche.  On differential equations of mode 2 , 1965 .

[12]  R. Finn.  New estimates for equations of minimal surface type , 1963 .

[13]  R. Redheffer.  An extension of certain maximum principles , 1962 .

[14]  O. Ladyzhenskaya,et al.  QUASI-LINEAR ELLIPTIC EQUATIONS AND VARIATIONAL PROBLEMS WITH MANY INDEPENDENT VARIABLES , 1961 .

[15]  Helmut H. Schaefer,et al.  Über die Methode der a priori-Schranken , 1955 .

[16]  Erhard Heinz.  Über die Existenz einer Fläche konstanter mittlerer Krümmung bei vorgegebener Berandung , 1954 .

[17]  R. Finn.  On Equations of Minimal Surface Type , 1954 .

[18]  Louis Nirenberg,et al.  On nonlinear elliptic partial differential equations and hölder continuity , 1953 .

[19]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[20]  J. Schauder.  Über lineare elliptische Differentialgleichungen zweiter Ordnung. , 1934 .

[21]  J. Leray,et al.  Topologie et équations fonctionnelles , 1934 .

[22]  B. Su.  Contributions to the Theory of Minimal Surfaces , 1929 .

[23]  E. Hope,et al.  Elementare Bemerkungen über die Lösungen partieller differentialgleichungen zweiter Ordnung vom elliptischen Typus , 1927 .

[24]  S. Bernstein.  Sur les équations du calcul des variations , 1912 .

[25]  Ch. Delaunay,et al.  Sur la surface de révolution dont la courbure moyenne est constante. , 1841 .