Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Yes exactly, the point is indeed to put precisely how B is "between" strictly convex and convex. As for A, if I am not wrong, it should hold something like this: A is the "closure" (in a suitable sense) of strictly convex functions (see e.g. this thesis, Thm. 3.5). I would like to get something like this for B... Thanks for your answer.
I do not think this is really a problem in my situation, as I am assuming something on the boundary datum which makes Giusti's argument work: this is indeed the boundary slope condition (see Giusti Def. 1.2, pag. 19). This assumption, as far as I have understood, allow you to consider the minimization problem in the class of equiLipschitz functions (which is compact by Arzelà-Ascoli so direct method works) and then to infer that the minimizer found in this way is actually a minimizer in the whole Lipschitz class (Giusti, Thm. 1.2 pag. 21).
Sure, I apologize for not doing it earlier, I just wanted to see if M. Renardy updates his answer (so that I will then decide which of the two answers accept). In any case, +1 and thanks again for your remarkable help these days.
Yes, I got it now, thanks (I made a mistake in my proof and it seemed to me we needed $A$ to be open). Your argument is definitively more clear and transparent. Thanks a lot!
@PiotrHajlasz Uhuh that's right. I definitely need to revise Functional Analysis :-) Thanks, I am very satisfied now and your (and Mateusz K.'s) answers/comments have been very helpful to me.
I am sorry to bother you again, but I now fail to see why $\sigma_A$ fulfills Property (3). Maybe we need that $A$ is open? I believe in this case probably one can choose $\gamma=0$ and also $b=0$, but I am not sure. Thanks.
@MateuszKwaśnicki Thanks a lot for your comment and for your help. That example is the best counterexample I can think of and I thank you for that. The point is that I am not sure it is possible to realize $|x|+|y|$ as $f_A$ for a convex $A \subset \mathbb R^N$. Can you prove it? Which $A$ could we use?Thanks.
Interesting, so also in this case uniqueness seems to fail. Just a small question: can you produce a similar example also in $\mathbb R^N$, $N\ge 2$? I am a bit puzzled with the minimizers of $\int |Du|$ subject to Dirichlet bc... Thanks for your help.
Exactly, that reference on the book of Giusti is exactly what I had in mind. The point is indeed not existence, but rather uniqueness of minimizers. Do you know any kind of uniqueness results for the kind of problems Giusti deal with (so without strict convexity assumptions)? Thanks for your interest into the question.
Nice idea, too! I had almost forgotten my old Functional Analysis class :-) Is it possibile to characterize the sets $A$ for which $\sigma_A$ is a strictly convex function?Thank you very much!
