Generalized boundary condition for Euler–Lagrange equation

Question

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?

Answer 1

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.

Answer 2

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?