# On the infimium of a functional

Let $$(M^n,g)$$ be a closed Riemannian manifold. Define $$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0 where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\log f + bRf^2\right)\;d\nu,$$ and $$R$$ is the scalar curvature of manifold.

Can one be confident that there is a $$0 with $$\int_Mf_0^2\;d\nu=1$$, such that $$\lambda(g)=\mathcal{F}(g,f_0)$$?

• Looks like a job for the direct method. However, the functional is not convex in $f$ due to the term $f^2\log(f)$, so there is some work to do. Another problem may be the "open" constraint $f>0$. Both problems would not be there if there would be $f\log(f)$ instead of $f^2\log(f)$. Since I do not work on manifolds, I have no idea how to formulate the respective Banach spaces (should be similar to some Sobolev or Orlicz-Sobolev spaces…). – Dirk Jan 10 at 12:58
• Does your manifold have finite volume? Otherwise if you choose $f<\epsilon$ everywhere we have $\int f^2 log(f)< log(\epsilon) \int f^2= log(\epsilon)$ and $F(g,f)$ can get arbitrarily small. – user100927 Jan 14 at 12:57
• Yes, the volume is finite. – user162551 Jan 14 at 21:03