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Results tagged with calculus-of-variations
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user 8799
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
5
votes
Is an $H_0^1$ function continuous to the boundary if it is continuous in the interior?
The answer to the follow-up question is negative too. For consider the half-ball $\Omega=\{x\,;\,x_3>0,\,|x|<1\}$. Choose a number $\alpha\in(1,\frac32)$, and a function $\phi\in C^\infty({\mathbb R}^ …
- 50.6k
9
votes
Accepted
Variational formulation for bilaplacian
To begn with, your Boundary-Value Problem (BVP) is under-determined, because it lacks one boundary condition: because the PDE is elliptic and fourth-order, you need two boundary conditions, not only o …
- 50.6k
13
votes
Maxwell equations as Euler-Lagrange equation without electromagnetic potential
Yes indeed, the Maxwell's equations are Euler-Lagrange equations. And this is quite interesting. Let me give here a presentation within Special Relativity, in which the light speed is set to $c=1$. Th …
- 50.6k
14
votes
Minimal surface which divides a convex body into two regions of equal volume
This is a classical problem on which we know the existence, thanks to the Geometric Measure Theory. In space dimension $n$, the solution is a hypersurface which is smooth of constant curvature away fr …
- 50.6k
10
votes
Rigorous justification that overdetermined systems do not have a solution
The principle you mention is not always true ! V. Arnold proved that every continuous function in $N$ real variables is a composition of continuous functions of two variables only. More precisely, the …
- 50.6k