Euler-Lagrange equation

for a minimal surface $z=z(x,y)$

The equation

$$\left(1+\left(\frac{\partial z}{\partial x}\right)^2\right)\frac{\partial^2z}{\partial y^2}-2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}\frac{\partial^2z}{\partial x\partial y}+\left(1+\left(\frac{\partial z}{\partial y}\right)^2\right)\frac{\partial^2z}{\partial x^2}=0.$$

It was derived by J.L. Lagrange (1760) and interpreted by J. Meusnier as signifying that the mean curvature of the surface $z=z(x,y)$ is zero. Particular integrals for it were obtained by G. Monge. The Euler–Lagrange equation was systematically investigated by S.N. Bernshtein, who showed that it is a quasi-linear elliptic equation of order $p=2$ and that, consequently, its solutions have a number of properties that distinguish them sharply from those of linear equations. Such properties include, for example, the removability of isolated singularities of a solution without the a priori assumption that the solution is bounded in a neighbourhood of the singular point, the maximum principle, which holds under the same conditions, the impossibility of obtaining a uniform a priori estimate for $z(x,y)$ in an arbitrary compact subdomain of a disc in terms of the value of $z$ at the centre of the disc (that is, the absence of an exact analogue of Harnack's inequality), facts relating to the Dirichlet problem, the non-existence of a non-linear solution defined in the entire plane (the Bernstein theorem), etc.

The Euler–Lagrange equation can be generalized with respect to the dimension: The equation corresponding to a minimal hypersurface $z=z(x_1,\dots,x_n)$ in $\mathbf R^{n+1}$ has the form

$$\sum_{i=1}^n\frac{\partial}{\partial x_i}\left(\frac{\partial z/\partial x_i}{\sqrt{1+|\nabla z|^2}}\right)=0,\quad\nabla z=\left(\frac{\partial z}{\partial x_1},\dots,\frac{\partial z}{\partial x_n}\right).$$

For this equation $(n\geq3)$ the solvability of the Dirichlet problem has been studied, the removability of the singularities of a solution, provided that they are concentrated inside the domain on a set of zero $(n-1)$-dimensional Hausdorff measure, has been proved, and the validity of Bernstein's theorem for $n\leq7$ and the existence of counter-examples for $n\geq8$ has been proved.

Comments

For Bernshtein's paper see [a3].

References

[a1]	E. Giusti, "Minimal surfaces and functions of bounded variation" , Birkhäuser (1984) MR0775682 Zbl 0545.49018
[a2]	J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) pp. §455 MR0448224 Zbl 0319.53003
[a3]	S.N. [S.N. Bernshtein] Bernstein, "Sur les surfaces définies au moyen de leur courbure moyenne ou totale" Ann. Sci. Ecole Norm. Sup. , 27 (1910) pp. 233–256
[a4]	E. Bombieri, E. Degiorgi, E. Giusti, "Minimal cones and the Bernstein problem" Inv. Math. , 7 (1969) pp. 243–268