View source for Euler-Lagrange equation

''for a minimal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e0365101.png" />''

The equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e0365102.png" /></td> </tr></table>

It was derived by J.L. Lagrange (1760) and interpreted by J. Meusnier as signifying that the mean curvature of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e0365103.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e0365104.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e0365105.png" /> in an arbitrary compact subdomain of a disc in terms of the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e0365106.png" /> at the centre of the disc (that is, the absence of an exact analogue of Harnack's inequality), facts relating to the [[Dirichlet problem|Dirichlet problem]], the non-existence of a non-linear solution defined in the entire plane (the [[Bernstein theorem|Bernstein theorem]]), etc.

The Euler–Lagrange equation can be generalized with respect to the dimension: The equation corresponding to a minimal hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e0365107.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e0365108.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e0365109.png" /></td> </tr></table>

For this equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e03651010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e03651011.png" />-dimensional Hausdorff measure, has been proved, and the validity of Bernstein's theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e03651012.png" /> and the existence of counter-examples for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036510/e03651013.png" /> has been proved.



====Comments====
For Bernshtein's paper see [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Giusti, "Minimal surfaces and functions of bounded variation" , Birkhäuser (1984) {{MR|0775682}} {{ZBL|0545.49018}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) pp. §455 {{MR|0448224}} {{ZBL|0319.53003}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Bombieri, E. Degiorgi, E. Giusti, "Minimal cones and the Bernstein problem" ''Inv. Math.'' , '''7''' (1969) pp. 243–268</TD></TR></table>