Uniqueness of minimizers in a problem in the Calculus of Variations - Part II

Take any convex set $$A\subset\mathbb{R}^n$$ which contains a neighborhood of the origin and let $$f_A$$ be the associated Minkowski functional $$f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},$$ which turns to be a convex, homogeneous function defined on $$\mathbb R^n$$. Given a Lipschitz regular domain $$\Omega \subset \mathbb R^n$$ and a "nice" function $$\varphi \colon \partial \Omega \to \mathbb R$$ I want to study the problem $$\min \left\{ \int_\Omega f_A(Du) \, dx :\, u \in \text{Lip}(\Omega) \text{ and } u = \varphi \text{ on } \partial \Omega \right\}.$$

Existence of minimizers should not be a severe issue and should follow easily (and classically) from the convexity of $$f_A$$ and from the properties of $$\varphi$$ (like e.g. the bounded slope condition).

What about uniqueness of (Lipschitz) minimizers? It seems to me that this is quite a difficult task, as the function $$f$$ is never strictly convex, being 1-homogeneous. So I do not see a way to discuss uniqueness of minimizers, and actually I even doubt that uniqueness holds. Any help? Thanks.

• When $f_A(x,y) = |x| + |y|$, isn't the example from my answer to part I a counterexample? Jan 17 '19 at 13:12
• @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.
– Y.B.
Jan 17 '19 at 14:12
• @Y.B. Sure you can since $|x|+|y|$ is a norm. Take $A=\{(x,y):|x|+|y|\}\leq 1$. Jan 17 '19 at 15:37
• @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.
– Y.B.
Jan 17 '19 at 16:53

Existence of minimizers should not be a severe issue...

The proof of the existence follows form the Arzela-Ascoli theorem, but the proof is not entirely obvious. This is Proposition 1.1 in:

E. Giusti, Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.

The statement is as follows:

Theorem. Let $$F$$ be a convex function, and let $$\Omega\subset\mathbb{R}^n$$ be bouded and open. Let $$\varphi$$ be Lipschitz continuous on $$\partial\Omega$$ with the Lipschitz constant $$\leq k$$. Then the functional $$I(u)=\int_\Omega F(Du)\, dx$$ attains minimum in the class of $$k$$-Lipschitz functions on $$\Omega$$ that agrees with $$\varphi$$ on $$\partial\Omega$$.

• 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.
– Y.B.
Jan 16 '19 at 19:07
• 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.
– Y.B.
Jan 17 '19 at 23:14
• @Y.B. You could suggest to Mateusz that he writes an answer with a link to his other answer and then you accept it. Jan 17 '19 at 23:37

Consider $$n=1$$, $$A=[-1,1]$$, $$\Omega=(0,1)$$. You want to minimize $$\int_0^1 |u'(x)|\,dx$$ subject to Dirichlet conditions, say $$u(0)=0$$ and $$u(1)=1$$. Then it is quite obvious that every monotone function is a minimizer, so uniqueness does not hold.

• 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.
– Y.B.
Jan 17 '19 at 8:18

I note that the result quoted from Giusti's book a priori limits the Lipschitz constant to a specific value. Without this limitation, a minimizer may not exist. Consider, for instance, the annulus $$1 in two dimensions, and the minimization of $$\int_\Omega |\nabla u|$$, with boundary condition u=r. We need to consider only radial functions, so the problem boils down to minimizing $$\int_1^2 2\pi r |v'(r)|\,dr$$ over all functions $$v(r)$$ with $$v(1)=1$$ and $$v(2)=2$$. If we require $$v$$ to have Lipschitz constant 1, the minimum is $$3\pi$$, achieved for $$v(r)=r$$. If we allow $$v$$ to have any Lipschitz constant, the infimum is $$2\pi$$, approximated by functions localized near $$r=1$$, but not achieved.

• 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).
– Y.B.
Jan 18 '19 at 11:03