Anisotropic Laplacian Equation Overview

• Anisotropic Laplacian equations are partial differential equations featuring direction-dependent diffusion via non-Euclidean norms and tensor coefficients.
• They generalize classical Laplacian and p-Laplacian operators, enabling sharp classification, regularity, and rigidity results, particularly in convex cones.
• Applications span geometric analysis, continuum mechanics, and image processing, with analytical tools like anisotropic isoperimetric and Pohozaev identities enhancing solution insights.

Anisotropic Laplacian equations constitute a broad class of elliptic and parabolic partial differential equations (PDEs) distinguished by direction-dependent diffusion effects modeled via non-Euclidean norms, variable exponents, or tensor coefficients. These operators generalize the classical Laplacian and $p$-Laplacian by allowing the underlying geometry to break rotational symmetry, resulting in PDEs whose regularity, maximum principles, extremal functions, and qualitative behavior diverge from the isotropic setting. Recent advances provide sharp classification, regularity, and symmetry results for finite-mass and critical exponent problems on cones and general domains, and establish robust existence and regularity theory for fully anisotropic equations in both variational and non-variational frameworks.

1. Fundamental Structures and Operator Definitions

Let $H:\mathbb{R}^N\to[0,\infty)$ be a "gauge," i.e., a convex, $C^2$ function (off the origin), positively homogeneous of degree 1: $H(t\xi)=t H(\xi)$ for $t>0$, $\xi\in\mathbb{R}^N$, and $H(\xi)>0$ for $\xi\ne0$ (Dai et al., 6 Jul 2024). The dual norm $H_0$ is defined by $H_0(x)=\sup\{\langle x,\xi\rangle:\,H(\xi)=1\}$, and is also convex and degree-1 homogeneous.

The anisotropic (Finsler) $N$-Laplacian is the quasilinear operator: $$\Delta_N^H u := \operatorname{div}\left(H(\nabla u)^{N-1}\nabla H(\nabla u)\right)$$ When $H(\xi)=|\xi|$ (standard Euclidean norm), this reduces to the classical $N$-Laplacian $$\Delta_N u = \operatorname{div}(|\nabla u|^{N-2}\nabla u)$$.

Uniform ellipticity is guaranteed when the Wulff ball $\{\xi:H(\xi)<1\}$ is uniformly convex. This gives structure bounds: $$c_1\,|\xi|^{N-2}|\eta|^2 \le \langle \partial a(\xi)\eta,\,\eta\rangle \le c_2\,|\xi|^{N-2}|\eta|^2$$ for $a(\xi)=H(\xi)^{N-1}\nabla H(\xi)$ and appropriate $c_1,c_2>0$.

In lower regularity or more degenerate settings, one finds models like coordinate-separable operators: $$-\sum_{i=1}^N \frac{\partial}{\partial x_i} \left(|u_{x_i}|^{p_i-2}u_{x_i} \right)$$ and tensor-structured equations $\nabla\cdot(\mathbf{A}\nabla u)=0$ for SPD matrix $\mathbf{A}$ (Xiao et al., 23 Dec 2025).

2. Classification and Rigidity of Solutions in Cones

For the anisotropic Liouville equation

$-\Delta_N^H u = e^u \qquad \text{in }\,\mathcal{C}$

where $\mathcal{C}\subseteq\mathbb{R}^N$ is an open convex cone (including whole space, half space, orthants), all finite-mass solutions ($\int_\mathcal{C} e^u\,dx<\infty$) are given explicitly: $$u(x) = \ln\left[\frac{\lambda^N}{C_N H_0(x-x_0)^{N-1}+\lambda^N}\right]$$ where $\lambda>0$, $x_0\in\mathcal{C}$ subject to "rigidity" constraints, and $C_N = N(N^2/(N-1))^{N-1}$ (Dai et al., 6 Jul 2024). The classification is sharp, extending known isotropic Euclidean results and critical $p$-Laplacian cone results (Ciraolo et al., 2019).

Rigidity is enforced by anisotropic isoperimetric inequalities: Wulff balls are the unique extremal sets for the perimeter-to-volume ratio in cones, and the explicit profiles above saturate the associated equalities. Pohozaev-type identities adapted to the anisotropic setting quantize the total mass $\int e^u$ and force uniqueness up to scaling and translation.

3. Regularity, Existence, and Extension to Nonlinear/Variable Exponent Cases

Existence and regularity theory for elliptic anisotropic Laplacian problems proceeds via monotonicity, coercivity, and anisotropic Sobolev embeddings. The variable-exponent model

$$-\sum_{i=1}^n \frac{\partial}{\partial x_i}\left(|u_{x_i}|^{p_i(x)-2}u_{x_i}\right)=f(x,u,\nabla u)$$

with $p_i(x)>1$ and Carathéodory nonlinearities, admits weak solutions in $W_0^{1,p(x)}$ under minimal growth and integrability assumptions, handled via Berkovits's topological degree and compactness properties (Ochoa et al., 5 Nov 2024).

Regularity is nontrivial; local Hölder continuity and boundedness of weak solutions hold under precise anisotropic ellipticity and growth regimes, measured by intrinsic distances and anisotropic Caccioppoli/Troisi embeddings (DiBenedetto et al., 2016). For equations with $L^1$ data, continuity is proved under sharp smallness conditions on the anisotropic Wolff-type potential of $f$ (Savchenko et al., 6 May 2025).

In "one Laplacian" settings (anisotropic BV spaces, Sobolev exponents $p_i=1$ in some coordinates), existence of extremals and boundedness is achieved through concentration-compactness and Anzellotti-pairing theory, with the PDE interpreted in measure-theoretic fashion (Demengel et al., 2018).

4. Analytical Tools: Isoperimetric, Poincaré, and Pohozaev Identities

The core analytical tools driving these results include:

• Anisotropic isoperimetric inequality: In any convex cone $C$, for measurable $E$,

$$P_H(E;C)|E\cap C|^{-\frac{N-1}{N}} \ge P_H(B_{H_0}(1);C)|B_{H_0}(1)\cap C|^{-\frac{N-1}{N}}$$

extremized precisely by Wulff balls (Dai et al., 6 Jul 2024).

• Radial Poincaré inequality: For functions vanishing at the "back-contact" of radial segments in cones, norms are controlled by their gradients scaled by the radial width (Dai et al., 6 Jul 2024).
• Pohozaev identity (anisotropic):

$$\int f(x)(x-y)\cdot \nabla u\,dx = \int_{\partial D}\left[ H(\nabla u)^N (x-y)\cdot\nu - N H(\nabla u)^{N-1}\langle \nabla H(\nabla u),\nu\rangle (x-y)\cdot \nabla u \right]\,dS$$

ensuring mass quantization and rigidity in cones; in Dirichlet/star-shaped domains, obstructs nontrivial solutions for certain nonlinearities (Xia et al., 2018).

5. Applications, Generalizations, and Open Problems

Anisotropic Laplacian equations arise in geometric analysis (isoperimetric profiles, Cheeger constants), continuum mechanics (directional-dependent transport), image processing (non-Euclidean total variation), and phase transition phenomena. Recent developments yield explicit Green’s functions and anisotropic barycentric coordinates for spatial deformation, with gradient/Hessian control utilized in variational design (Xiao et al., 23 Dec 2025).

Open directions include:

• Classification of Liouville-type equations with more general nonlinearities or without strict ellipticity,
• Time-dependent and fractional anisotropic flows,
• Regularity near facets for "very singular" operators (2208.14640), and the extension to BV/measure-valued frameworks,
• Critical exponent and Hénon-type anisotropic problems with Morse index/non-stability constraints (Li, 2021).

6. Comparative Perspective: Isotropic vs. Anisotropic Theory

The anisotropic Laplacian theory parallels, but strictly extends, classical $p$-Laplacian analysis. Under geometric hypotheses like the $(H_M)$-assumption $H(\xi) = \sqrt{\langle M\xi,\xi\rangle}$, there is a precise conjugacy between Finsler and Euclidean Laplacians via linear transformation (Li et al., 2022). Most structure and regularity theory, variational identities, and mean-value properties thereby carry over. Outside such reducible cases, the direction-dependence and cone geometry drive phenomena absent in isotropic settings, including anisotropic rigidity of extremals, non-radially symmetric solutions, sharp anisotropic isoperimetry, and novel regularity thresholds.

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Key References

• Dai, Gui, Luo: "Anisotropic Finsler $N$-Laplacian Liouville equation in convex cones" (Dai et al., 6 Jul 2024)
• Ciraolo, Li: "Classification of solutions to the anisotropic $N$-Liouville equation in $\mathbb{R}^N$" (Ciraolo et al., 2023)
• Ochoa, Ramos Valverde, Silva: "Existence of weak solutions for the anisotropic $p(x)$-Laplacian via degree theory" (Ochoa et al., 5 Nov 2024)
• Savchenko, Skrypnik, Yevgenieva: "On the continuity of solutions to the anisotropic $N$-Laplacian with $L^1$ lower order term" (Savchenko et al., 6 May 2025)
• Ciraolo, Figalli, Roncoroni: "Symmetry results for critical anisotropic $p$-Laplacian equations in convex cones" (Ciraolo et al., 2019)