A minimum problem of the CoV

Question

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)

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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 $B=\infty$?

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.