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I am dealing with a minimization of functional of the form:

$$I[f]=\sum_{\Omega=\Omega_1,\Omega_2} \int_{\Omega}\mathcal{L}(x_1,x_2,f,f_1,f_2,f_{11},f_{12},f_{22}) \, d\textbf{x}$$ where $$\textbf{x}=(x_1,x_2),~~~~~~~~~f_i=\dfrac{\partial f}{\partial x_i},~~~~~~~~~f_{ij}=\dfrac{\partial^2 f}{\partial x_i\,\partial x_j}$$ and two surfaces $\Omega_1$ and $\Omega_2$ are bounded in 3D.

The general Euler-Lagrange equation for the similar case is given in Wikipedia, however, they do not provide the general form for boundary conditions. In my problem, the boundary conditions in the matching point of two surfaces $\Omega_1$ and $\Omega_2$ are unknown, and I should find the boundary conditions in addition to Euler–Lagrange equations to satisfy the stationarity.

I would like to know if anyone can introduce a reference for the complete proof of the Euler–Lagrange equation of the similar type which includes boundary conditions as well?

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  • $\begingroup$ Do you really have no knowledge of the boundary conditions in the matching point? Shouldn't $f$ have a unique value in the point, i.e. the field on $\Omega_1$ and the field on $\Omega_2$ should be equal in the matching point? $\endgroup$ – md2perpe Jul 12 '18 at 19:20
  • $\begingroup$ Finding the boundary condition in matching point is what I am looking for in this problem. What I am missing is the boundary expression for two dimensions case with higher derivatives, similar to the term one can get from integration by parts in the one-dimensional case (en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation#Statement). That's why I need to have the complete proof to see the similar terms which comes from probably similar integration by part. $\endgroup$ – user575755 Jul 12 '18 at 20:49
  • $\begingroup$ You use $d\mathbf{x}$ for the area differential form. What notation do you prefer for the boundary differential form? $\endgroup$ – md2perpe Jul 15 '18 at 18:38
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That is a huge topic. Often in applications, the physical system dictates some boundary conditions (BCs). However OP seems to ask what BCs make the variational problem well-posed from a purely mathematical perspective?

  1. Sufficient essential BCs are obviously $$ \left. f\right|_{\partial \Omega_k}~=~0, \qquad \left. f_i\right|_{\partial \Omega_k}~=~0. $$

  2. Other sufficient BCs include natural BCs.

  3. It is possible to impose different BCs on disconnected boundaries.

  4. When the Lagrangian $\cal L$ is of some special form, the BCs can sometimes be relaxed.

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You have $$ I[f] = \sum_{\Omega=\Omega_1,\Omega_2} \int_\Omega \mathcal L(\mathbf{x}, f(\mathbf{x}), \partial_i f(\mathbf{x}), \partial_j \partial_k f(\mathbf{x})) \, d\mathbf{x}. $$

Doing a variation of $f$ we get $$ \frac{d}{d\lambda} I[f+\lambda\eta] = \frac{d}{d\lambda} \sum_{\Omega=\Omega_1,\Omega_2} \int_\Omega \mathcal L(\mathbf{x}, f+\lambda\eta, \partial_i (f+\lambda\eta), \partial_j \partial_k (f+\lambda\eta)) \, d\mathbf{x} \\ = \frac{d}{d\lambda} \sum_{\Omega=\Omega_1,\Omega_2} \int_\Omega \mathcal L(\mathbf{x}, f+\lambda\eta, \partial_i f + \lambda\,\partial_i\eta, \partial_j\partial_k f +\lambda\,\partial_j\partial_k\eta)) \, d\mathbf{x} \\ = \sum_{\Omega=\Omega_1,\Omega_2} \int_\Omega \frac{d}{d\lambda} \mathcal L(\mathbf{x}, f+\lambda\,\eta, \partial_i f + \lambda\,\partial_i\eta, \partial_j \partial_k f +\lambda\,\partial_j\partial_k\eta) \, d\mathbf{x} \\ = \sum_{\Omega=\Omega_1,\Omega_2} \int_\Omega \left( \frac{\partial \mathcal L}{\partial f} \eta + \frac{\partial \mathcal L}{\partial(\partial_i f)} \partial_i\eta + \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \partial_j \partial_k \eta \right) d\mathbf{x} $$

Rewriting the second term using integration by parts: $$ \int_\Omega \frac{\partial \mathcal L}{\partial(\partial_i f)} \partial_i\eta \, d\mathbf{x} = \int_\Omega \partial_i \left( \frac{\partial \mathcal L}{\partial(\partial_i f)} \eta \right) d\mathbf{x} - \int_\Omega \partial_i \left( \frac{\partial \mathcal L}{\partial(\partial_i f)} \right) \eta \, d\mathbf{x} \\ = \oint_{\partial\Omega} \frac{\partial \mathcal L}{\partial(\partial_i f)} \eta \, n_i d{S} - \int_\Omega \partial_i \left( \frac{\partial \mathcal L}{\partial(\partial_i f)} \right) \eta \, d\mathbf{x} $$

Rewriting the third term using integration by parts: $$ \int_\Omega \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \partial_j \partial_k \eta \, d\mathbf{x} = \int_\Omega \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \partial_k \eta \right) d\mathbf{x} - \int_\Omega \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \right) \partial_k \eta \, d\mathbf{x} \\ = \int_\Omega \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \partial_k \eta \right) d\mathbf{x} - \int_\Omega \partial_k \left( \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \right) \eta \right) \, d\mathbf{x} + \int_\Omega \partial_k \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \right) \eta \, d\mathbf{x} \\ = \oint_{\partial\Omega} \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \partial_k \eta \, n_j d{S} - \oint_{\partial\Omega} \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \right) \eta \, n_k d{S} + \int_\Omega \partial_k \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \right) \eta \, d\mathbf{x} \\ $$

Thus we get $$ \frac{d}{d\lambda} I[f+\lambda\eta] = \sum_{\Omega=\Omega_1,\Omega_2} \int_\Omega \left( \frac{\partial \mathcal L}{\partial f} - \partial_i \left( \frac{\partial \mathcal L}{\partial (\partial_i f)} \right) + \partial_k \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \right) \right) \eta \, d\mathbf{x} \\ + \sum_{\Omega=\Omega_1,\Omega_2} \left( \oint_{\partial\Omega} \frac{\partial \mathcal L}{\partial(\partial_i f)} \eta \, n_i d{S} + \oint_{\partial\Omega} \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \partial_k \eta \right) n_j d{S} - \oint_{\partial\Omega} \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \right) \eta \, n_k d{S} \right) \\ = \sum_{\Omega=\Omega_1,\Omega_2} \int_\Omega \left( \frac{\partial \mathcal L}{\partial f} - \partial_i \left( \frac{\partial \mathcal L}{\partial (\partial_i f)} \right) + \partial_k \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_k f)} \right) \right) \eta \, d\mathbf{x} \\ + \sum_{\Omega=\Omega_1,\Omega_2} \oint_{\partial\Omega} \left( \frac{\partial \mathcal L}{\partial(\partial_i f)} \eta + \frac{\partial \mathcal L}{\partial(\partial_i \partial_k f)} \partial_k \eta - \partial_j \left( \frac{\partial \mathcal L}{\partial(\partial_j \partial_i f)} \right) \eta \right) \, n_i d{S} $$ Can you extract the equation and boundary conditions from this?

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