Fine structure of the optimal trace-Sobolev curve $\Phi_\Omega$: attainment, rigidity of the ball comparison, endpoint asymptotics, and the combined inequality (Maggi–Villani open problems)

The family of scale-invariant variational problems $\Phi_\Omega(T)=\inf\{(\int_\Omega|\nabla u|^p\,dx)^{1/p}:\|u\|_{L^{p^\star}(\Omega)}=1,\|u\|_{L^{p^\#}(\partial\Omega)}=T\}$, with critical exponents $p^\star=np/(n-p)$ and $p^\#=p(n-1)/(n-p)$, encodes the best Sobolev inequality on a domain $\Omega$ in the presence of a critical boundary trace term. Maggi and Villani (2005) proved that balls have the worst best Sobolev inequalities, $\Phi_\Omega\geq\Phi_B$ on the $T$-range of the ball curve, deduced the sharp linear and $p$-power trace-Sobolev inequalities (the $p$-power one resolving a 1985 conjecture of Brezis and Lieb), and left open the fine structure of $\Phi_\Omega$: attainment of the infimum; whether equality with the ball curve at a single $T$ forces $\Omega$ to be a ball; whether the sharp small-$T$ asymptotic of $\Phi_\Omega$ is universal over domains or characterizes balls; and whether the ball curve satisfies the combined bound $\Phi_B(T)/S+(T/T_+)^p\geq 1$. Maggi, Neumayer and Tomasetti (2023) settled interval-equality rigidity and, for $n>2p$ with $C^2$ boundary, both attainment and single-point rigidity. The remaining core — the range $n\leq 2p$, boundaries below $C^2$, the sharp endpoint asymptotics, and the combined inequality — is open.

Background

Let $n\geq 2$ and $1<p<n$. The sharp Sobolev inequality on $\mathbb{R}^n$ states $\|\nabla u\|_{L^p(\mathbb{R}^n)}\geq S(n,p)\,\|u\|_{L^{p^\star}(\mathbb{R}^n)}$, where $p^\star=np/(n-p)$ is the critical Sobolev exponent; Aubin and Talenti computed the best constant $S(n,p)$ and showed that equality is attained exactly by the bubbles obtained from $U(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ by translation and dilation. On a half-space $H$ the sharp trace (Escobar) inequality $\|\nabla u\|_{L^p(H)}\geq E(n,p)\,\|u\|_{L^{p^\#}(\partial H)}$ involves the boundary exponent $p^\#=p(n-1)/(n-p)$; it was proved by Escobar for $p=2$ and by Nazaret for all $1<p<n$. Both exponents are critical for scale invariance: under $u(x)\mapsto u(\lambda x)$ all three norms above rescale by the same factor.

Maggi and Villani embedded these facts into a one-parameter family of variational problems that encodes the best Sobolev inequality on a domain in the presence of a trace term. For a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$ set

$$\Phi_\Omega(T)=\inf\left\{\Big(\int_\Omega|\nabla u|^p\,dx\Big)^{1/p}\ :\ \|u\|_{L^{p^\star}(\Omega)}=1,\ \ \|u\|_{L^{p^\#}(\partial\Omega)}=T\right\},\qquad T\geq 0,$$

where the infimum is taken over the Sobolev space $W^{1,p}(\Omega)$, whose elements have a well-defined trace on $\partial\Omega$. The family is scale invariant, and its graph records the best possible value of $\|\nabla u\|_{L^p(\Omega)}$ under a normalization of $\|u\|_{L^{p^\star}(\Omega)}$ and a fixed boundary trace, which is why it is called the best Sobolev inequality on $\Omega$. Three landmarks structure the curve: $\Phi_\Omega(0)=S(n,p)$ for every $\Omega$, because Sobolev bubbles concentrating at an interior point have vanishing boundary trace; $\Phi_\Omega(T)=0$ exactly at $T=\mathrm{ISO}(\Omega)^{1/p^\#}$, where $\mathrm{ISO}(\Omega)=\mathcal{H}^{n-1}(\partial\Omega)/|\Omega|^{(n-1)/n}$ is the isoperimetric ratio, so the isoperimetric inequality is equivalent to the ball curve having the leftmost zero; and on a half-space the Escobar inequality is a linear lower bound on $\Phi_H$.

By an optimal mass-transport argument (a technique that proves sharp functional inequalities by rearranging one function onto another along a convex, energy-nonincreasing map), in the spirit of Cordero-Erausquin, Nazaret and Villani, Maggi and Villani proved the comparison theorem that gives their paper its title: if $B$ is a ball and $T_+=\mathrm{ISO}(B)^{1/p^\#}$, then

$$\Phi_\Omega(T)\ \geq\ \Phi_B(T)\qquad\text{for all }T\in[0,T_+],$$

so balls have the worst best Sobolev inequalities. They further showed that the ball curve is strictly concave and decreasing on $[0,T_+]$, that it is implicitly parametrized by the Aubin–Talenti bubbles centered at the center of $B$ and restricted to $B$, and they deduced two sharp two-term inequalities valid on every locally Lipschitz domain: a linear one,

$$\frac{\|\nabla u\|_{L^p(\Omega)}}{S(n,p)}+\frac{\|u\|_{L^{p^\#}(\partial\Omega)}}{\mathrm{ISO}(B)^{1/p^\#}}\ \geq\ \|u\|_{L^{p^\star}(\Omega)},$$

and a $p$-power one,

$$\frac{\|\nabla u\|^p_{L^p(\Omega)}}{S(n,p)^p}+\frac{\|u\|^p_{L^{p^\#}(\partial\Omega)}}{C(n,p)}\ \geq\ \|u\|^p_{L^{p^\star}(\Omega)},$$

the latter resolving a 1985 conjecture of Brezis and Lieb on Sobolev inequalities with a remainder term. In the notation of the source these sharp constants are denoted $S_n(p)$, $T_n(p)$ and $C_n(p)$. Maggi and Villani also proved a conditional rigidity statement: if $\Omega$ is connected (possibly unbounded), $\Phi_\Omega(T_0)=\Phi_B(T_0)$ for some $T_0\in(0,T_+)$ and the infimum defining $\Phi_\Omega(T_0)$ is attained, then $\Omega$ is a ball; and they constructed non-rigidity examples showing that unboundedness or disconnectedness defeat rigidity. The paper left open the fine structure of $\Phi_\Omega$ — attainment, rigidity without the attainment hypothesis, sharp endpoint asymptotics, and sharp quantitative bounds for the ball curve — which is the subject of this problem. Later work built the surrounding theory: Maggi and Neumayer completely characterized the half-space curve $\Phi_H$ for all $p\in(1,n)$ (extending the competing-symmetries theorem of Carlen and Loss for $p=2$, a method that iterates two competing symmetry groups of a scale-invariant functional to identify its extremals) and proved $\Phi_\Omega(T)\leq\Phi_H(T)$, so half-spaces have the best best Sobolev inequalities; Maggi, Neumayer and Tomasetti then proved the partial existence and rigidity results described below, while an adjacent 2024–2026 program studies stability of the half-space family.

Problem Statement

Let $n\geq 2$, $1<p<n$, $p^\star=np/(n-p)$ and $p^\#=p(n-1)/(n-p)$. For a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, whose boundary is locally the graph of a Lipschitz function so that every $u\in W^{1,p}(\Omega)$ has a well-defined trace on $\partial\Omega$, set

$$\Phi_\Omega(T)=\inf\left\{\Big(\int_\Omega|\nabla u|^p\,dx\Big)^{1/p}\ :\ u\in W^{1,p}(\Omega),\ \|u\|_{L^{p^\star}(\Omega)}=1,\ \|u\|_{L^{p^\#}(\partial\Omega)}=T\right\},\qquad T\geq 0.$$

Let $B$ be a ball, let $T_+=\mathrm{ISO}(B)^{1/p^\#}$ be the zero of the ball curve, where $\mathrm{ISO}(\Omega)=\mathcal{H}^{n-1}(\partial\Omega)/|\Omega|^{(n-1)/n}$ is the isoperimetric ratio, and write $S=S(n,p)$ for the sharp constant of the Sobolev inequality on $\mathbb{R}^n$ (so that $\Phi_\Omega(0)=S$ for every $\Omega$) and $C=C(n,p)$ for the sharp constant of the $p$-power inequality displayed in the background. Determine the fine structure of $\Phi_\Omega$ on bounded Lipschitz connected domains, specifically:

(1) Attainment. For which $\Omega$ and $T>0$ is the infimum defining $\Phi_\Omega(T)$ attained? In particular, if $\Omega$ is bounded, Lipschitz and connected and $n\leq 2p$, is the infimum attained for every $T>0$? (Known: for every bounded $C^1$ domain, all $n\geq 2$ and $p\in(1,n)$, relaxed minimizers — generalized minimizers of a relaxed problem in which part of the mass and trace budget may be carried by at most one boundary concentration point, modeled through the half-space curve — exist and $\Phi_\Omega$ equals the relaxed value; for $C^2$ boundaries and $n>2p$ the relaxed minimizers are classical, so the infimum is attained.)

(2) Rigidity of the ball comparison. If $\Omega$ is bounded, Lipschitz and connected and $\Phi_\Omega(T_0)=\Phi_B(T_0)$ for a single $T_0\in(0,T_+]$, must $\Omega$ be a ball? (The endpoint $T_0=T_+$ is classical isoperimetric rigidity, and $T_0=0$ carries no information since $\Phi_\Omega(0)=S$ for all $\Omega$.) (Known for domains with $C^1$ boundary: yes if equality holds on a whole interval $(0,T^*)$, even just on a neighborhood of $T=0$; and, for $n>2p$ with $C^2$ boundary, yes at a single $T_0$. Open for $n\leq 2p$ with any admissible boundary regularity, and for $n>2p$ when the boundary is not $C^2$.)

(3) Sharp left-endpoint asymptotics: universality, rigidity, or neither. The sharp constant $C$ of the $p$-power inequality was determined by Maggi and Villani from the left-endpoint (small-$T$) behavior of the ball curve, where the deficit $S^p-\Phi_B(T)^p$ is of order $T^p$. Determine whether the sharp small-$T$ asymptotic of $\Phi_\Omega$ is the same for every bounded Lipschitz connected $\Omega$ as for the ball — the behavior that forces the optimal constant of the $p$-power inequality on $\Omega$ to equal $C$, so that the ball's left-endpoint behavior is universal in sharp form — whether having that same sharp left-endpoint asymptotic forces $\Omega$ to be a ball, or neither.

(4) Combined inequality for the ball curve. Prove or disprove

$$\frac{\Phi_B(T)}{S}+\left(\frac{T}{T_+}\right)^p\ \geq\ 1\qquad\text{for all }T\in[0,T_+].$$

This merges the concavity-based linear bound $\Phi_B(T)/S+T/T_+\geq 1$ and the $p$-power type bound into a single stronger quantitative relation for the model curve; it was neither proved nor disproved, with numerical evidence suggesting it holds.

These four questions pursue one objective — the fine structure of the curve that encodes the best Sobolev inequality on a domain — and are linked: question (2) currently reduces, through the conditional rigidity, to question (1); question (3) asks whether the sharp small-$T$ asymptotic is universal over domains, forces the domain to be a ball, or neither; question (4) is the corresponding sharp one-variable statement for the ball curve itself.

A resolution of one part settles only that part, except that, by the link noted above, a resolution of (1) also settles (2).

References

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