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Results tagged with calculus-of-variations
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user 44981
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
2
votes
0
answers
64
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A counterexample to regular boundary points for minimizers of variational integrals under su...
Let $\Omega\subset\mathbb{R}^n$ for some $n\geq 3$ be an open bounded set with at least Lipschitz boundary. Let $p\in (1, 2), N>1$ and $f: \overline{\Omega} \times\mathbb{R}^N\times\mathbb{R}^{Nn}\rig …
- 347
5
votes
0
answers
101
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Minimisers and critical points of variational integrals
In the following we consider $\Omega\subset\mathbb{R}^n\ (n\geq2)$ to be open, bounded and with Lipschitz boundary. Consider the following regular variational integral:
\begin{equation*}
I[u]=\int_{\O …
- 347
15
votes
1
answer
1k
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Who first resolved Hilbert's 20th problem?
Hilbert's 20th problem concerns the existence of solutions to the fundamental problem in the calculus of variations. I understand that Hilbert, Lebesgue and Tonelli were pioneers in this area.
In pa …
- 347
1
vote
0
answers
554
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Reverse Holder Inequality and the higher integrability of the gradient of a solution to Eule...
In Giaquinta-Giusti's (1978) paper "Nonlinear Elliptic Systems with Elliptic Growth" (thm 1.1) they consider the following system:
\begin{equation}
\sum_{i, j=1}^{n}\sum_{\alpha, \beta=1}^N\frac{\par …
- 347