Wulff theorem and best constant in Sobolev inequality
Let \(f: {\mathbb{R}}^ 2\to {\mathbb{R}}\) be a positively homogeneous function of degree one, lower semi-continuous, with \(f(x)>0\) if \(x\neq 0\). For each such function one defines a convex set of \({\mathbb{R}}^ 2\), \(W_ f\), \[ W_ f=\{x^*\in {\mathbb{R}}^ 2:\quad f^*(x^*)\leq 0\} = \{x^*\in {\mathbb{R}}^ 2:\quad f^ 0(x^*)\leq 1\}, \] where \(f^*\), resp. \(f^ 0\), is the Legendre transform, resp. the polar transform, of f. Let \((u,v)\in W^{1,1}_{per}(a,b)\times W^{1,1}_{per}(a,b)\) with \(u^{'2}+v^{'2}\neq 0\) a.e. in \((a,b)\). Let \[ F(u,v) = \int^{b}_{a}f(v'(\theta),-u'(\theta))d\theta,\quad m(u,v) = \int^{b}_{a}(v'(\theta)u(\theta)-u'(\theta)v(\theta))d\theta. \] Then the following inequality holds \[ (*)\quad F^ 2(u,v)-4| W_ f| m(u,v)\geq 0, \] where \(| W_ f|\) is the Lebesgue measure of \(W_ f\). Equality holds if and only if \((u,v)\) is a parametrization of \(\partial W_ f.\) Inequality (*) is a generalized isoperimetric inequality. Indeed, if \(f\) is the Euclidean norm and \((u,v)\) a parametric representation of the boundary \(\partial A\) of a region \(A\), then \(F(u,v)=\ell (\partial A)\) and \(m(u,v)=| A|\), where \(\ell (\partial A)\) is the length of \(\partial A\) and \(| A|\) the area of A. In that case \(W_ f\) is the unit Euclidean disk and \(| W_ f| =\pi\). The proof of (*) is a consequence of a generalized Wirtinger inequality \[ \inf \{\int^{1}_{-1}(f(v',-u'))^ 2 d\theta /\int^{1}_{-1}(f^ 0(u,v))^ 2 d\theta:\quad (u,v)\in {\mathcal M}\}=(| W_ f|)^ 2, \] where \[ {\mathcal M}=\{(u,v)\in H^ 1(-1,1)\times H^ 1(-1,1);\quad u(- 1)=u(1), \] \[ v(-1)=v(1),\quad \int^{1}_{-1}f^ 0(u,v)\partial f^ 0/\partial u(u,v)d\theta =\int^{1}_{-1}f^ 0(u,v)\partial f^ 0/\partial v(u,v)d\theta =0\}. \]
scientific article; zbMATH DE number 4107461