Characterization of the convex-symmetrization comparison function for elliptic problems with non-smooth anisotropic gauge functions

The 1997 paper of Alvino, Ferone, Trombetti and Lions introduced convex (Wulff-shape) symmetrization and proved comparison results, in the spirit of the classical Talenti principle, for elliptic equations and variational problems whose principal part dominates $H(\xi)^2$ for an anisotropic gauge $H$ satisfying only convexity, positive 1-homogeneity and uniform comparability with the Euclidean norm — a class that already contains polyhedral (crystalline) norms. The comparison function is the Wulff-radial minimizer $z$ of a symmetrized variational functional; it is characterized as solving an Euler-Lagrange equation only under the conditional statement that $H$ is regular enough, and Remark 4.2 of that paper sketches extensions to lower-order terms and to parabolic equations without proof. The sketched extensions to lower-order terms, to parabolic problems, and to exponents $p\neq 2$ have been substantially established for regular gauges by Piscitelli, Della Pietra and Gavitone, Cianchi, Alberico, di Blasio and Feo, Brock, Diaz, Ferone, Gomez-Castro and Mercaldo, Chen and Yang, and others. What remains open is the strand the source left conditional: for gauge functions that are merely convex but not differentiable off the origin (the model case being polyhedral or crystalline norms), the pointwise Euler-Lagrange equation is meaningless, and the comparison function must be identified and characterized through a weak (variational or subdifferential) formulation of the symmetrized problem.

Background

Symmetrization is a classical tool in the theory of elliptic partial differential equations. Given a function $u$ on a domain $\Omega\subset\mathbb{R}^n$, its decreasing rearrangement is the function $u^*$ on $[0,|\Omega|]$ equimeasurable with $u$, and its Schwarz symmetrization is the radially decreasing function with the same rearrangement. A classical comparison principle (called the Talenti comparison principle in the later anisotropic literature) states that if $u$ solves $-\Delta u=f$ with homogeneous Dirichlet data ($u=0$ on $\partial\Omega$), then the rearrangement of $u$ is pointwise dominated by the solution $v$ of the radially symmetric problem on the ball of the same volume with symmetrized data $f^\#$; the extremal role of balls underlies sharp a priori estimates and Faber-Krahn-type bounds (the statement that the ball minimizes the first Dirichlet eigenvalue among sets of given volume).

The 1997 paper [1] developed the anisotropic counterpart. A gauge is a function $H:\mathbb{R}^n\to[0,\infty)$ that is convex and positively 1-homogeneous, $H(t\xi)=|t|H(\xi)$; think of a non-Euclidean norm such as $H(\xi)=\max_i|\xi_i|$. Its polar gauge is $H^\circ(x)=\sup\{\langle x,\xi\rangle:H(\xi)\le 1\}$, and the convex body $K^\circ=\{x:H^\circ(x)\le 1\}$ is the Wulff shape of $H$ — in surface-energy physics the equilibrium shape of a crystal with interfacial energy density $H$; $\kappa_n$ denotes its volume. The anisotropic perimeter of a set $E$ of finite perimeter is $P_H(E)=\int_{\partial^*E}H(\nu_E)\,d\mathcal{H}^{n-1}$, where $\partial^*E$ is the reduced boundary and $\nu_E$ its measure-theoretic outer normal; the anisotropic isoperimetric inequality $P_H(E)\ge n\kappa_n^{1/n}|E|^{1-1/n}$ holds with equality exactly for homothets of the Wulff shape. The convex symmetrization of $u$ is the rearrangement $u^\star(x)=u^*(\kappa_n(H^\circ(x))^n)$, whose level sets are homothetic to the Wulff shape, and the anisotropic Polya-Szego inequality $\int_\Omega H(Du)^p\,dx\ge\int_{\Omega^\star}H(Du^\star)^p\,dx$ holds for $p\ge 1$ on the Wulff-shaped rearranged domain $\Omega^\star$.

Theorem 4.1 of the 1997 paper [1] compares solutions of $-\mathrm{div}\,a(x,u,Du)=f$, $u=0$ on $\partial\Omega$, under the structure condition $a(x,r,\xi)\cdot\xi\ge H^2(\xi)$, which forces the principal part to dominate the quadratic form $H^2$: the rearrangement $u^*$ is bounded above by $(\omega_n/\kappa_n)^{2/n}v^*$, where $\omega_n$ is the volume of the Euclidean unit ball and $v$ solves the symmetrized problem on $\Omega^\star$. Theorem 4.2 gives the analogous statement for minimizers of variational integrals $F(u)=\int(A(Du)-fu)\,dx$ whose recession function (the positively 1-homogeneous leading-growth profile $\tilde A(\xi)=\lim_{t\to\infty}A(t\xi)/t$) dominates $H^2$. The comparison object is the minimizer $z$ of the symmetrized functional $F^\circ(w)=\int(\tfrac12H(Dw)^2-f^\star w)\,dx$. Both theorems are proved for any gauge $H$ satisfying only the standing hypotheses of Section 2 of the paper — convexity, the homogeneity (2.1), and the comparability (2.2), $\alpha|\xi|\le H(\xi)\le\beta|\xi|$ for positive constants $\alpha,\beta$ — together with Caratheodory coefficients subject to the growth hypothesis (4.2) and data $f\in L^{2n/(n+2)}$ for $n\ge 3$ (respectively $f\in L^p$, $p>1$, for $n=2$); no differentiability of $H$ is used. Existence of a Wulff-radial minimizer $z=z^\star$ of $F^\circ$ is established in the paper via the Polya-Szegő and Hardy-Littlewood inequalities. The paper states, conditionally, that if $H$ is regular enough then $z$ solves the Euler-Lagrange equation

$$-\big(H(Dw)H_{\xi_i}(Dw)\big)_{x_i}=f^\star,$$

the anisotropic analogue of the radial ODE in the classical isotropic comparison, which involves the partial derivatives $H_{\xi_i}$ of the gauge. The same regularity is built into the anisotropic $p$-Laplacian $Q_pu=\mathrm{div}(H(Du)^{p-1}H_\xi(Du))$, the Euler-Lagrange operator of $\int H(Du)^p$, treated in later follow-up work for general $p$.

For the open question below, the relevant gauge classes are: smooth gauges, meaning $H$ of class $C^1$ off the origin, typically with strictly convex level sets and uniformly comparable to the Euclidean norm ($\alpha|\xi|\le H(\xi)\le\beta|\xi|$), which essentially all comparison results subsequent to the 1997 paper assume; and non-smooth gauges, still uniformly comparable to the Euclidean norm, the model case being polyhedral or crystalline norms $H(\xi)=\max_{1\le j\le m}\langle\xi,\eta_j\rangle$ determined by finitely many facet normals $\eta_j$, the standard surface-energy anisotropies in the theory of crystalline curvature evolution, for which the formal operator $H H_\xi$ becomes set-valued and constant along whole facets. For non-smooth $H$ the natural weak formulation of the symmetrized problem is variational: $w\mapsto\int H(Dw)^p$ is still convex, and its subdifferential (the set-valued generalization of the Euler-Lagrange operator) or the underlying minimization problem replaces the pointwise equation above.

Problem Statement

Let $H:\mathbb{R}^n\to[0,\infty)$ be a positively 1-homogeneous convex gauge that is uniformly comparable to the Euclidean norm ($\alpha|\xi|\le H(\xi)\le\beta|\xi|$ for some positive constants $\alpha,\beta$) but not of class $C^1$ off the origin; the model case is a polyhedral or crystalline norm $H(\xi)=\max_{1\le j\le m}\langle\xi,\eta_j\rangle$. Consider the Dirichlet problem $-\mathrm{div}\,a(x,u,Du)=f$ in a bounded domain, $u=0$ on the boundary, under the structure condition $a(x,r,\xi)\cdot\xi\ge H^2(\xi)$, and, in parallel, the variational problem for $F(u)=\int(A(Du)-fu)\,dx$ whose recession function dominates $H^2$, in both cases under the standing hypotheses of Theorems 4.1 and 4.2 of Alvino, Ferone, Trombetti and Lions (1997) (Caratheodory coefficients with growth condition (4.2) and data $f\in L^{2n/(n+2)}$ for $n\ge 3$, respectively $f\in L^p$, $p>1$, for $n=2$). For such gauges the comparison theorems of the source already hold: the minimizer $z$ of the symmetrized functional $F^\circ(w)=\int(\tfrac12H(Dw)^2-f^\star w)\,dx$ on the Wulff-shaped domain $\Omega^\star$ exists, is Wulff-radial (constant on level sets homothetic to the boundary of the Wulff shape), and yields the comparison $u^\star\le z$ with the same sharp constants as in the smooth case. What the source leaves conditional is the characterization of $z$: the pointwise Euler-Lagrange equation $-(H(Dw)H_{\xi_i}(Dw))_{x_i}=f^\star$ (problem (4.12) of the paper) has no meaning for non-smooth $H$. Establish, or refute, the extension of this Euler-Lagrange characterization to non-smooth gauges. Establishing the extension amounts to (i) specifying and justifying the weak (variational or subdifferential) formulation replacing the Euler-Lagrange equation, including its well-posedness, and (ii) proving, in that formulation, that the comparison function $z$ is characterized as its solution — in particular, resolving the reduced one-dimensional problem on the Wulff shape, where the formal operator $H H_\xi$ is set-valued and constant along whole facets — or identifying precisely which parts of the characterization survive; refuting the extension amounts to (iii) exhibiting a specific non-smooth gauge, domain, and data for which no weak formulation of the required kind singles out the comparison function $z$.

References

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