Critical anisotropic p-Laplacian classification in convex cones and the sharp weighted anisotropic Sobolev inequality beyond smooth uniformly convex norms and concave homogeneous weights

Ciraolo, Figalli and Roncoroni (Geom. Funct. Anal. 2020) classified the positive finite-energy solutions of the critical anisotropic p-Laplacian equation in a convex cone as anisotropic bubbles and, via optimal transport, proved a sharp weighted anisotropic Sobolev inequality in cones - under two structural hypotheses: the anisotropy $H$ satisfies $H^2\in C^2(\mathbb{R}^n\setminus\{0\})$ with a uniformly convex, $C^{1,1}$ anisotropic unit ball, and the weight $w$ is homogeneous of degree $a\ge0$ with $w^{1/a}$ concave when $a>0$. The problem is whether these hypotheses are essential. Component (1): classify the critical anisotropic equation in cones for anisotropies outside that class - non-uniformly-convex, non-$C^2$, or crystalline (polyhedral unit ball) norms, for which the flux $H^{p-1}\nabla H$ is not classically defined and a weak formulation must first be settled - or exhibit a non-bubble solution in such a class. Component (2): extend the sharp weighted inequality beyond homogeneous concave weights, or show that sharpness or attainment of extremals fails.

Background

Let $n\ge2$, $1<p<n$, and let $p^*=np/(n-p)$ denote the critical Sobolev exponent. The critical $p$-Laplace equation $-\Delta_p u = u^{p^*-1}$ in $\mathbb{R}^n$, where $\Delta_p u = \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian, is the Euler-Lagrange equation of the Sobolev inequality: its positive solutions are exactly the extremals ("bubbles") of that inequality, and classifying them - the general-$p$ analogue of the Caffarelli-Gidas-Spruck classification for the semilinear case $p=2$ - is the rigidity input behind blow-up analysis (the description of sequences of approximate solutions via rescaled limits of bubbles), Struwe decomposition (the expansion of energy-bounded sequences of solutions into finitely many rescaled bubbles), and quantitative stability (inequalities bounding the distance of near-extremals to the bubble family) for critical problems. The classification is hard because the moving-planes method (a symmetry-comparison technique that exploits invariance under reflections, available in $\mathbb{R}^n$ and balls) requires symmetries of both the operator and the domain, and the $p$-Laplacian is nonlinear and degenerate.

An anisotropy is a norm $H$ on $\mathbb{R}^n$ playing the role of the Euclidean norm. The anisotropic $p$-Laplacian is $-\Delta_p^H u = \operatorname{div}(a(\nabla u))$ with $a(\xi) = H^{p-1}(\xi)\nabla H(\xi)$, the Euler-Lagrange operator of the energy $\int H^p(\nabla u)\,dx$. Its unit ball $B_1^H = \{\xi : H(\xi)<1\}$ (called the Wulff shape in crystal-growth models) replaces the Euclidean sphere, and the smooth uniform convexity of $B_1^H$ is the substitute for ellipticity. Ciraolo-Figalli-Roncoroni considered, in a convex cone $\Sigma = \mathbb{R}^k\times C$ where $C$ does not contain a line, the problem $\operatorname{div}(a(\nabla u)) + u^{p^*-1} = 0 \ \text{in } \Sigma, \qquad u>0 \ \text{in } \Sigma, \qquad a(\nabla u)\cdot\nu = 0 \ \text{on } \partial\Sigma, \qquad u\in D^{1,p}(\Sigma),$ where $\nu$ is the outward normal and $D^{1,p}(\Sigma)$ is the finite-energy space $\{u\in L^{p^*}(\Sigma): \nabla u\in L^p(\Sigma)\}$. Since moving planes are unavailable both for anisotropic operators and inside cones, the authors developed an integral-identity approach and proved (Theorem 1.1) that under the hypothesis $H^2\in C^2(\mathbb{R}^n\setminus\{0\})$ together with uniform convexity and $C^{1,1}$ regularity, namely $\lambda\,\mathrm{Id} \le H(\xi)D^2H(\xi) + \nabla H(\xi)\otimes\nabla H(\xi) \le \Lambda\,\mathrm{Id} \quad \forall\,\xi\ne0$ for constants $0<\lambda\le\Lambda$, every solution is an anisotropic bubble $U^H_{\lambda,x_0}$ - the anisotropic analogue of the Aubin-Talenti extremal, an explicit translate and dilation of a fixed profile with the anisotropy replacing the Euclidean distance (formula (1.6) of the paper), the apex $x_0$ being constrained by the lineality factor $\mathbb{R}^k$ of the cone (arbitrary when $\Sigma=\mathbb{R}^n$, at the vertex when the cone is pointed). In Appendix A (Theorem A.1), valid for arbitrary norms $H$ with no smoothness assumption at all, they prove via optimal transport the sharp weighted anisotropic Sobolev inequality in cones $\Big(\int_{\Sigma}|f|^{\beta}w\,dx\Big)^{p/\beta} \le C\,\int_{\Sigma}H^p(\nabla f)\,w\,dx, \qquad \beta = \frac{p(n+a)}{n+a-p},$ with explicit sharp constant and equality exactly at the bubbles - but under the structural hypothesis on the weight: $w\in C^0(\Sigma)$, positive in $\Sigma$, homogeneous of degree $a\ge0$, with $w^{1/a}$ concave when $a>0$.

Both hypotheses are binding, and each fails in a regime of physical and geometric interest. Crystalline norms - polyhedral Wulff shapes, used in models of crystalline curvature flow and anisotropic surface tension - are the extreme nonsmooth case: $H$ is nondifferentiable across the normal fan of the polytope, so the flux $a(\xi)=H^{p-1}(\xi)\nabla H(\xi)$ is not single-valued and the equation must be reformulated as a differential inclusion or in variational subdifferential form; Mosconi-Riey-Squassina observe that for such energies solutions are at best Hölder continuous, so the pointwise calculus behind the integral-identity method breaks down. On the inequality side, the optimal-transport argument uses concavity of $w^{1/a}$ through Brunn-Minkowski-type convexity (the superadditivity of volume under Minkowski addition of convex bodies), a structure that fails for non-concave or non-homogeneous weights.

Problem Statement

Let $n\ge2$, $1<p<n$, $p^*=np/(n-p)$, and let $\Sigma=\mathbb{R}^k\times C\subset\mathbb{R}^n$ be a convex cone with $C$ not containing a line. Consider $\operatorname{div}(a(\nabla u)) + u^{p^*-1} = 0 \ \text{in } \Sigma, \qquad u>0 \ \text{in } \Sigma, \qquad a(\nabla u)\cdot\nu = 0 \ \text{on } \partial\Sigma, \qquad u\in D^{1,p}(\Sigma),$ where $\nu$ is the outward normal, $a(\xi)=H^{p-1}(\xi)\nabla H(\xi)$, $H$ is a norm on $\mathbb{R}^n$, and $D^{1,p}(\Sigma)=\{u\in L^{p^*}(\Sigma):\nabla u\in L^p(\Sigma)\}$ is the finite-energy space. Ciraolo-Figalli-Roncoroni classify all such solutions as anisotropic bubbles $U^H_{\lambda,x_0}$ - the explicit extremal family of the source (formula (1.6) there): a fixed profile evaluated at the anisotropic distance of $x$ from an apex $x_0$, rescaled by a parameter $\lambda>0$, with $x_0$ constrained by the cone's lineality factor $\mathbb{R}^k$ - under the hypothesis that $H^2\in C^2(\mathbb{R}^n\setminus\{0\})$ and $H$ is uniformly convex and $C^{1,1}$, namely $\lambda\,\mathrm{Id}\le H(\xi)D^2H(\xi)+\nabla H(\xi)\otimes\nabla H(\xi)\le\Lambda\,\mathrm{Id}$ for some constants $0<\lambda\le\Lambda$ and all $\xi\ne0$; and they prove the sharp weighted anisotropic Sobolev inequality in cones under the hypothesis that the weight $w$ is homogeneous of degree $a\ge0$ with $w^{1/a}$ concave when $a>0$. The problem is to determine how far these structural hypotheses can be weakened:

(1) Nonsmooth and non-uniformly-convex anisotropy. (a) Formulate the critical anisotropic equation in a mathematically sound way - as a differential inclusion, a variational subdifferential equation, or another justified weak formulation - valid for anisotropies $H$ outside the smooth uniformly convex class, including crystalline norms whose unit ball is a polytope. (b) Within that formulation, either classify all nonnegative finite-energy solutions in convex cones (each being either identically zero or an anisotropic bubble) for a class of anisotropies outside the smooth uniformly convex class (for example: uniformly convex but not $C^2$; smooth but not uniformly convex; or crystalline - any one such class suffices), or exhibit a specific norm in such a class together with a nonzero ($u>0$) finite-energy solution that is not an anisotropic bubble, showing the hypothesis is essential.

(2) Weights beyond the homogeneous concave class. Extend the sharp weighted anisotropic Sobolev inequality on convex cones, $\Big(\int_{\Sigma}|f|^{\beta}w\,dx\Big)^{p/\beta}\le C\,\int_{\Sigma}H^p(\nabla f)\,w\,dx, \qquad \beta=\frac{p(n+a)}{n+a-p},$ to weights $w$ outside the class "homogeneous of degree $a\ge0$ with $w^{1/a}$ concave when $a>0$" (for example: non-homogeneous weights, or homogeneous weights for which $w^{1/a}$ is not concave - any one such class suffices) - identifying the sharp constant and characterizing the extremals; or prove that for such weights sharpness fails, the sharp constant is not attained, or extremals are not of bubble form (a single weight with certified failure of sharpness or attainment settles this branch). For weights outside this class the exponent $\beta$ is not fixed by the data: determine the exact exponent $\beta$, together with the class of weights for which the inequality holds with that exponent and its sharp constant (on the homogeneous subclass $\beta$ reduces to $p(n+a)/(n+a-p)$, with $a$ the degree of homogeneity), or exhibit an explicit weight for which no such extension exists.

The common objective is to decide which structural hypotheses of the source theory are essential for the rigidity of the critical anisotropic Sobolev functional on cones: the sharp inequality defines the variational object, the classification describes its critical points, and the extremals of the inequality are exactly the classified bubbles, so the two deliverables are the two faces of one object. Partial results relaxing one hypothesis while retaining the other are admissible components of an answer but do not by themselves settle the problem as stated.

References

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