I have the following minimum problem:

$$\tag{1} \min_{u\in W^{1,1}(0,B)} \int_0^B (\sqrt{1+|u^\prime (t)|^2} -a)\ u^{k-1} (t) t^{h-1}\ \text{d} t $$

(where $B>0$, $0 < a < 1$, $h,k\in \mathbb{N}$ and $k>2$) under constraints:

$$\int_0^B u^k (t) t^{h-1}\ \text{d}t =\text{some constant},\ u(B)=0,\ u(0)>0,\ u(t)\geq 0.$$

With a great deal of effort I found the function $w:(0,B)\to [0,\infty[$:

$$ w(t) := \sqrt{ \frac{B^2}{ 1 - a^2 } - t^2} - \frac{a B}{\sqrt{1 - a^2}} $$

as a solution for the Euler-Lagrange equation of the constrained problem, which seems to be:

$$ \begin{equation} \begin{split} \frac{\text{d}}{\text{d} t} \Big[ \frac{u^\prime}{\sqrt{1+|u^\prime|^2}}\ u^{k-1} t^{h-1} \Big] &- (k-1) (\sqrt{1+|u^\prime |^2} - a) u^{k-2} t^{h-1} \\ &+\lambda\ k u^{k-1} t^{h-1} = 0 \end{split} \end{equation}$$

where dependence of $u,u^\prime$ on $t$ is omitted and $\lambda$ is a Lagrange multiplier (in particular, $\lambda = -(h+k-1) \sqrt{1-a^2} / (kB)$ is the multiplier working for $w(t)$).

But... As far as I can see, the integrand $F(t,u,p) := (\sqrt{1+p^2} - a)\ u^{k-1} t^{h-1} +\lambda u^k t^{h-1}$ lacks convexity in $(u,p)$, hence I cannot tell whether or not $w(t)$ is actually a minimizer. And, actually, I don't even know if $W^{1,1}$ is the "best" Sobolev space for this kind of problem.

Any hint or advice?

(My first post here; try to forgive all the flaws, I'm just a newbye ;-D)

Problem (1) comes from a geometric inequality for cylindrical-type sets, which gives a lower bound for the difference of perimeter and a particular weighted measure of these sets (when their measure is fixed). Actually, standard isoperimetric inequality can be used to prove that (1) has a positive infimum $\gamma (a)$; hence I was trying to evaluate it.

I succeeded in doing all the computations in the case $u$ is smooth enough (and I even got the equality case), but then again I was wondering: what if I take a bigger function space?

And, then again, what if I consider the problem for $B=\infty$?

  • $\begingroup$ Does this minimization problem come from some kind of application? Might be helpful to include that. Your title is also not very informative at the moment; I would suggest making it more specific. $\endgroup$
    – j.c.
    Sep 29, 2011 at 1:54

1 Answer 1


Perhaps you have simplified this from a multidimensional radial problem, which explains the constraint $\int_0^B u^k(t) t^{h-1}\;dt = C$, which if we view $u$ as a radial function in $h$ dimensions is equivalent to $||u||_{L^k(B_r)}=C$. Then this is equivalent to minimizing the functional

$E(u):= \frac{1}{|S^{h-1}|}\int_{B_R} \left(\sqrt{1+|\partial_r u(r)|^2} - a\right) u^{k-1}(r) r^{h-1}\;dr ds$

over radial functions in $W^{1,1}(B_r)$, and a priori you will not necessarily stay in this space, since there is a lack of compactness in $W^{1,1}$ (though later being radial may save things). Using the direct method, we can define

$C:= \inf E(u) = \lim_n E(u_n)$,

and so for $n$ large we have

$C + a \int_{B_r} u_n^{k-1}(r) r^{h-1}\;drds \geq \int_{B_r} |\partial_r u_n(r)| u_n^{k-1}(r) t^{h-1}\;drds $

But $\int_0^B u_n^{k-1}(t) t^{h-1}\;dt$ is bounded from the constraint and Jensen's inequality. This will give bounds on $u^k_n$ in $L^1$ and $\nabla (u_n^k)$ as well, so that $u^k_n$ converges, up to a subsequence, to some $u \in BV(B_r)$. Thus existence of a minimizer can be established, and then there are is the question of uniqueness and regularity to address. I hope this gives a good start to your question/encourages more thoughts on the subject.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.