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Results tagged with calculus-of-variations
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user 165275
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
4
votes
Accepted
What functions are equal to their symmetric decreasing rearrangement?
It should be straightforward to verify that $\mathcal A$ consists exactly of the lower semi-continuous radial and decreasing functions (which are non-negative and vanish at $\infty$): You already know …
- 1,869
1
vote
How do I apply Brouwer fixed-point theorem in this claim?
Only now I realize the condition that $\zeta$ is nonnegative. (Was it really there in the first formulation of the question?)
With this condition, it is possible to get the required a-priori bound req …
- 1,869
1
vote
How do I apply Brouwer fixed-point theorem in this claim?
What is needed is an a-priori $L_\infty$ bound for the solution $v_k$. If you know such an a-priori bound, you can modify $\zeta$ outside of this bound, and you can assume without of generality that $ …
- 1,869
1
vote
Conditions for continuity of an integral functional
Since $\nu$ does not appear anywhere else in the question, I suppose that $L^1(X)=L^1(\nu)$.
In order that the functional be defined, one should then assume (probably without loss of generality) that …
- 1,869