# Questions tagged [variational-calculus]

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### Weak form of the Navier-Cauchy equation

I am trying to obtain the weak form of the Navier-Cauchy equation, which is $$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$ ...
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### Prove that the set of maximizers are independent of parameter in the objective function

A maximization problem reads as $$J(y) = \sum_{k=1}^{K} \sigma_k(y)^q \mathop{\rightarrow}^{y} max$$ where $q \in [1,\infty]$ is a user-defined parameter and functions $\sigma_k, k=\{1,\dots,K\}$ ...
223 views

### How to solve calculus of variations problems numerically?

For example, how to solve the well-known isoperimetric problem (i.e., to enclose the largest area with a fixed-length curve)? We can simplify things a bit and fix the two ends of the curve at $[a,0]$,...
77 views

### Deriving weak form of a set of scalar equations

I have the equilibrium equation in elasticity for a static case.i.e Div T=0. For certain implementation, I have to get the x and y component equations and then derive the weak form separately. How is ...
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### Solving a nonlinear poisson equation via variational minimization

I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem. $- \nabla ((1 + u^2) \nabla u) = f$ $u = 0 \ \text{on} \ \Omega$ I am using Newton solver, where I have ...
I am confused. I read an introduction to finite element method where it was derived for the poisson equation: -\Delta u + cu = f,\qquad, u = g_0 \text{ on Dirichlet boundaries},\qquad\partial_n u ...
I want to solve approximately the following variational problem: Given a function $c:[-1,1]^2\rightarrow [0,1]$, constants $p_1...p_n\in \mathbb{R}^+$, $\alpha_1...\alpha_n\in \mathbb{R}$, and \$\...