This is the first time I encounter a problem where I need to minimize a function defined in a function space. Define the function space $A=\{\theta:\mathbb{R}^3 \rightarrow\mathbb{R}|\ \theta\ \text{is continuous}\}$. Let $h:A→\mathbb{R}$ be a function which we want to minimize (optimization). What conditions do we need on $A$ and $h$ if we want to have that the solution $argmin_{f\in A}h(f)$ has bounded partial derivatives? I have no idea how to approach this problem.
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1$\begingroup$ Search for calculus of variations. $\endgroup$– JulesCommented Jun 7, 2016 at 11:14
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1$\begingroup$ Jules gave th right buzzword. In fact, this can be tricky and the regularity of minimizers is usually the third step (after showing existence and/or uniqueness which are both not obvious in many cases). $\endgroup$– DirkCommented Jun 7, 2016 at 11:17
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$\begingroup$ I can put regularity conditions on h. $\endgroup$– NadoriCommented Jun 7, 2016 at 13:59
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