# Guaranteeing isoperimetry constraint for non-extremal functional in PDE.

First of all, hello and thank you for your time.

## Context

I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[\mathbf{y}]=\int\limits_{\Omega\subset\mathbb{R}^n}f(\mathbf{x},\mathbf{y}, \mathbf{\nabla y})d\mathbf{x}$$ $$\frac{\partial y_i}{\partial t}=k\Delta\frac{\delta F}{\delta y_i}$$ Where $$\Delta$$ is the Laplacian and $$\frac{\delta F}{\delta y_i}=\sum\limits_k\frac{\partial}{\partial x_k}\frac{\partial f}{\partial (\partial_ky_i)}-\frac{\partial f}{\partial y_i}$$ is the variational derivative ($$\partial _k$$ is the partial derivative with respect to $$x_k$$). Also: $$\mathbf{x}\in\mathbb{R}^n, \mathbf{y}\in\mathbb{R}^m$$

The system should satisfy the global conservation constraints: $$J_i[\mathbf{y}]=\int\limits_{\Omega\subset\mathbb{R}^n}y_id\mathbf{x}=k_i$$ Where the $$k_i$$ are constants.

I ran the program (without adding the Lagrange multipliers to the integral). And noticed that the $$J_i$$ increased with time, which was obviously not intended.

## Question

I want to add the constraints to the solution. At first I naïvely thought that I could just modify the functional by adding lagrange mutipliers: $$K[\mathbf{y}]=F[\mathbf{y}]-\sum\limits_i\lambda_i J_i[\mathbf{y}]$$

But when checking my reference book I noticed the Theorem said (I modified and omitted parts to take what's most relevant to the current question):

Suppose that $$F$$ has an extremum at $$y\in C^2[x_0, x_1]$$ subject to the boundary conditions [...]. Then there exist two numbers $$\lambda_0, \lambda_1$$ not both zero such that $$\frac{\delta K}{\delta y}=0$$ Where $$K=\lambda_0 F-\lambda_1 J$$

As the theorem says, this works when one wishes to find the extremum so my naïve assumption is probably wrong since the system I described only reaches the extremum of the functional when $$\mathbf{y}$$ gets to the steady state (i.e. it approaches it asymptotically).

Is there a way to satisfy the constraint continuously throughout the time evolution of the system despite $$F$$ not being stationary?

I understand this may be more involved than what an answer in the site may allow so if you know of any good textbook where I could find it I would also be very grateful.