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Wulff theorem and best constant in Sobolev inequality. (English) Zbl 0676.46031
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 \[ {\mathcal M}=\{(u,v)\in H^ 1(-1,1)\times H^ 1(-1,1);\quad u(- 1)=u(1), \] \[ v(-1)=v(1),\quad \int^{1}_{-1}f^ 0(u,v)\partial f^ 0/\partial u(u,v)d\theta =\int^{1}_{-1}f^ 0(u,v)\partial f^ 0/\partial v(u,v)d\theta =0\}. \]
Reviewer: B.Dacorogna

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J27 Existence theories for problems in abstract spaces
46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables