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Given the Lagrangian

$$ J(u)= \int_{V} \sqrt{1+|\operatorname{grad}(u)|^{2})} $$

with the constraint $ \int_{V}udx =1 $

(1) Why is the volume constraint there ?

(2) For the case of $\mathbb R^{3}$, I know this must satisfy the equation

$$ (1+u_x^2)u_{yy} - 2u_xu_yu_{xy} + (1+u_y^2)u_{xx}=0 $$

however how could i generalize this equation to arbitrary $\mathbb R^{n} $ ?

From the Euler-Lagrange equation I get (plus Lagrange parameter)

$$ \sum_{i} \partial_{x_{i}} \frac{u_{x_{i}}}{\sqrt{1+|gra(u)|^{2}}}= \lambda $$

I think this equation is correct but I need to order this a bit more. :)

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Jose Garcia
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  • Perhaps you should give some motivation for your question, like where did you find this? Otherwise, I would recommend that you go get a good book on minimal surfaces out of the library: for example, the book "A Course in Minimal Surfaces" by Colding and Minicozzi http://books.google.com/books/about/A_Course_in_Minimal_Surfaces.html?id=6DUmshPXJ1kC is quite nice. – Otis Feb 20 '14 at 05:18
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    The constraint is there because you said so. If you impose a constraint, it is your responsibility to justify the constraint, not the readers'. – Ben McKay Feb 20 '14 at 07:23
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    Minimizing $J$ with the volume constraint gives you not the minimal surface, but a surface of constant mean curvature; it is somewhat related to the isoperimetric problem (what's the smallest boundary that can enclose a fixed volume). – Willie Wong Feb 20 '14 at 09:21
  • i found this problem on Evans partial differential equations chapter 8 http://books.google.es/books/about/Partial_Differential_Equations.html?id=Xnu0o_EJrCQC chapter 8 problem 12 :) – Jose Garcia Feb 20 '14 at 09:43
  • io know i should compare my Euler lagrange equation with the differential equation of the mean curvature am i right ? where could i find more hitns ? – Jose Garcia Feb 20 '14 at 09:46
  • The Euler-Lagrange equation you found is the correct one. The equation $(1+u_x^2)u_{yy} \ldots$ is the equation for $\lambda = 0$ in $\mathbb{R}^3$; you can check that this is the case by explicitly evaluating the case where $i$ runs from $1$ to $2$. – Willie Wong Feb 20 '14 at 12:06
  • of course but i know would need the curvature equation in $ R^{n} $ to see that the vanishing of Euler lagrange equation is the vanishing also from the mean curvature in n-dimension – Jose Garcia Feb 20 '14 at 12:07
  • On the page you pointed me to, it says to look at Example 4 in section 8.1.2. Did you read that example? The general form of the mean curvature of a graphical manifold is given there. – Willie Wong Feb 20 '14 at 12:10
  • i have donwloade the book and in example 8.1.2 they put only the Euler lagrange equation i put in my question they do not explicitly evaluate the mean curvature of the surface – Jose Garcia Feb 20 '14 at 16:34
  • Similar question by OP: http://math.stackexchange.com/q/681933/11127 – Qmechanic Mar 10 '14 at 15:20

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