Anisotropic Trudinger's Equation

• Anisotropic Trudinger’s equation is a family of nonlinear PDEs characterized by critical exponential growth governed by anisotropic Sobolev energies and Finsler geometry.
• It unifies elliptic and parabolic frameworks, with models featuring sharp Moser–Trudinger and Leray–Trudinger inequalities alongside direction-dependent, doubly nonlinear diffusion.
• Analyses reveal intrinsic geometric mechanisms—such as Wulff shapes, spectral thresholds, and blow-up behavior—that underpin concentration and regularity properties.

Searching arXiv for papers on anisotropic Trudinger equations and related anisotropic Trudinger–Moser frameworks. Anisotropic Trudinger’s equation denotes a family of nonlinear partial differential equations arising from anisotropic Sobolev energies, Finsler geometry, and borderline exponential integrability. In the elliptic literature, the term commonly refers to Euler–Lagrange equations associated with anisotropic Moser–Trudinger or Leray–Trudinger functionals, where the Euclidean gradient norm is replaced by a positively $1$-homogeneous convex function $F$ and the geometry is governed by the polar norm $F^\circ$ and the Wulff shape (Zhou, 2019). In the parabolic literature, the same expression refers more specifically to anisotropic doubly nonlinear evolutions whose prototype is

$$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$

a direction-dependent generalization of the classical Trudinger equation (Ciani et al., 21 Jul 2025). The subject therefore spans two closely related, but distinct, strands: sharp anisotropic Trudinger–Moser inequalities and the PDEs they induce, and anisotropic doubly nonlinear parabolic equations with Trudinger-type structure.

1. Terminology, anisotropy, and geometric setting

In the anisotropic framework, the Euclidean norm $|\nabla u|$ is replaced by a function $F:\mathbb{R}^n\to[0,\infty)$ that is convex, even, positively $1$-homogeneous, $C^2(\mathbb{R}^n\setminus\{0\})$, positive away from the origin, and equivalent to the Euclidean norm in the sense that there exist constants $0<a\le b<\infty$ such that

$$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$

Its polar function is

$F$0

and the associated Wulff shape is

$F$1

More generally, Wulff balls are sets of the form

$F$2

These are the anisotropic analogues of Euclidean balls, and they are the natural level sets in concentration and symmetrization arguments (Zhou, 2019).

The basic Sobolev space is $F$3, but its energy is measured by

$F$4

This gives the constrained class

$F$5

which plays the same role as the unit sphere for the Euclidean Dirichlet norm in the isotropic Moser–Trudinger theory (Zhou, 2019).

A related anisotropic setting appears in Leray–Trudinger inequalities, where $F$6 contains the origin, $F$7, and logarithmic weights are expressed in terms of

$F$8

The anisotropic Hardy–Leray structure then replaces $F$9 by $F^\circ$0 and $F^\circ$1 by $F^\circ$2 throughout (Blasio et al., 19 Jun 2025).

This geometric framework is central because anisotropy is not a lower-order perturbation. It enters the principal part of the operator, the sharp constants, the concentration geometry, and the form of extremals. A common misconception is that anisotropy merely changes constants; the supplied results instead show that Euclidean balls are systematically replaced by Wulff balls, radial symmetry by $F^\circ$3-radial symmetry, and isotropic diffusion by direction-dependent Finsler diffusion (Zhou, 2019).

2. Elliptic anisotropic Moser–Trudinger theory

A sharp anisotropic Moser–Trudinger inequality involving an $F^\circ$4-term was established for bounded smooth domains $F^\circ$5. The first eigenvalue associated with the $F^\circ$6-Finsler-Laplacian is

$F^\circ$7

With

$F^\circ$8

the improved anisotropic Moser–Trudinger functional is

$F^\circ$9

The threshold statement is sharp: if $$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$0, then $$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$1, whereas if $$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$2, then $$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$3. Moreover, for every $$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$4, the supremum is attained by some $$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$5 (Zhou, 2019).

This is the anisotropic counterpart of the Adimurthi–Druet type improvement of the classical Moser–Trudinger inequality. The lower-order $$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$6-term increases the effective exponential growth, but only up to the spectral threshold $$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$7. In this setting, the role played by the Euclidean sharp constant is assumed by $$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$8, which depends on the volume of the unit Wulff ball rather than the Euclidean unit sphere (Zhou, 2019).

The anisotropic Leray–Trudinger theory extends the same borderline philosophy to Hardy–Leray differences. For

$$u_t-\sum_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2}D_i u\Big)=0,\qquad u\ge 0,$$9

one has an anisotropic exponential integrability result with logarithmic correction: $|\nabla u|$0 for suitable $|\nabla u|$1 controlled by $|\nabla u|$2. In the subclass of anisotropically radial functions, a sharper inequality with an optimal constant is proved (Blasio et al., 19 Jun 2025).

These results show that the borderline exponential regime is stable under anisotropic deformation, but the deformation is geometric rather than merely algebraic: the sharp constants, the critical profiles, and the relevant symmetry class all depend on the Finsler structure.

3. Euler–Lagrange equations and the elliptic “anisotropic Trudinger equation”

The natural elliptic operator associated with the anisotropic energy $|\nabla u|$3 is the $|\nabla u|$4-Finsler-Laplacian

$|\nabla u|$5

written in the sign convention

$|\nabla u|$6

When $|\nabla u|$7, this reduces to the usual $|\nabla u|$8-Laplacian $|\nabla u|$9 (Zhou, 2019).

To approach the sharp functional, one introduces the subcritical parameter $F:\mathbb{R}^n\to[0,\infty)$0 and the approximate functional

$F:\mathbb{R}^n\to[0,\infty)$1

For each $F:\mathbb{R}^n\to[0,\infty)$2, the supremum of $F:\mathbb{R}^n\to[0,\infty)$3 over $F:\mathbb{R}^n\to[0,\infty)$4 is achieved by some $F:\mathbb{R}^n\to[0,\infty)$5, and the maximizer satisfies an Euler–Lagrange equation with a Lagrange multiplier for the constraint $F:\mathbb{R}^n\to[0,\infty)$6. In the formulation given in the supplied exposition, there exist constants $F:\mathbb{R}^n\to[0,\infty)$7 and $F:\mathbb{R}^n\to[0,\infty)$8 such that

$F:\mathbb{R}^n\to[0,\infty)$9

in $1$0, with homogeneous Dirichlet condition $1$1 on $1$2 (Zhou, 2019).

This nonlinear PDE with critical exponential growth is the elliptic object most naturally described as an anisotropic Trudinger equation. The first eigenvalue $1$3 is the threshold because the Euler–Lagrange equation contains a lower-order term of the form $1$4, which competes with the spectral scale determined by the anisotropic $1$5-Laplacian (Zhou, 2019).

The same terminological extension appears in the anisotropic Leray–Trudinger setting. There the phrase “anisotropic Trudinger equation” is used for Euler–Lagrange equations associated with anisotropic $1$6-Dirichlet energies, anisotropic Hardy–Leray potentials, and critical exponential nonlinearities of the form

$1$7

leading to equations whose principal part is again

$1$8

and whose lower-order terms involve anisotropic Hardy–Leray weights (Blasio et al., 19 Jun 2025).

4. Blow-up, concentration, and Wulff geometry

The proof of the sharp elliptic anisotropic Moser–Trudinger inequality uses blow-up analysis. For subcritical maximizers $1$9, either $C^2(\mathbb{R}^n\setminus\{0\})$0 remains bounded, in which case compactness yields an extremal for the sharp functional, or the maximum

$C^2(\mathbb{R}^n\setminus\{0\})$1

tends to $C^2(\mathbb{R}^n\setminus\{0\})$2. In the blow-up case, one selects points $C^2(\mathbb{R}^n\setminus\{0\})$3 with $C^2(\mathbb{R}^n\setminus\{0\})$4, introduces a scaling parameter $C^2(\mathbb{R}^n\setminus\{0\})$5, and studies the rescaled functions

$C^2(\mathbb{R}^n\setminus\{0\})$6

After rescaling, $C^2(\mathbb{R}^n\setminus\{0\})$7 and the limit $C^2(\mathbb{R}^n\setminus\{0\})$8 solves

$C^2(\mathbb{R}^n\setminus\{0\})$9

with finite mass

$0<a\le b<\infty$0

The explicit limiting profile is

$0<a\le b<\infty$1

after suitable normalization (Zhou, 2019).

The geometric mechanism behind this profile is the anisotropic isoperimetric inequality

$0<a\le b<\infty$2

with equality if and only if $0<a\le b<\infty$3 is a Wulff ball, together with the anisotropic co-area formula

$0<a\le b<\infty$4

These formulas force the level sets of the limiting bubble to be Wulff balls rather than Euclidean balls. The concentration geometry is therefore intrinsically Finslerian (Zhou, 2019).

In the Leray–Trudinger setting, anisotropic radiality similarly means $0<a\le b<\infty$5. The one-dimensional reduction occurs in the anisotropic radius $0<a\le b<\infty$6, and sharp constants in the radial class are recovered through anisotropic polar coordinates, the co-area formula, and one-dimensional logarithmic inequalities (Blasio et al., 19 Jun 2025).

A plausible implication is that Wulff geometry plays in anisotropic Trudinger problems the structural role that Euclidean spherical geometry plays in the isotropic theory: it governs concentration, symmetry reduction, and the explicit form of bubbles.

5. Parabolic anisotropic Trudinger’s equation

In the parabolic literature, the prototype anisotropic Trudinger’s equation is

$0<a\le b<\infty$7

posed in $0<a\le b<\infty$8, where $0<a\le b<\infty$9 is bounded and open and the exponents satisfy

$$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$0

More generally, the equation considered is

$$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$1

with anisotropic structure conditions comparable to the model operator (Ciani et al., 21 Jul 2025).

The equation is anisotropic because each spatial direction has its own exponent $$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$2, and doubly nonlinear because the nonlinearity affects both the gradient and the unknown $$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$3. To express the structure conditions conveniently, the change of variable

$$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$4

is introduced. In terms of $$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$5, coercivity and growth are written as

$$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$6

and

$$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$7

for structural constants $$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$8 (Ciani et al., 21 Jul 2025).

Weak solutions are defined by anisotropic Sobolev regularity together with the variational inequality

$$a|\xi|\le F(\xi)\le b|\xi| \qquad \forall \xi\in\mathbb{R}^n.$$9

for subsolutions, with the reversed inequality for supersolutions and equality for solutions. The analysis uses Steklov averages to justify time regularization and derive anisotropic Caccioppoli inequalities (Ciani et al., 21 Jul 2025).

The anisotropic geometry is encoded by cubes

$F$00

time scale $F$01, and, in refined arguments, intrinsic anisotropic cylinders depending on a level $F$02: $F$03 Unlike the elliptic anisotropic Moser–Trudinger theory, which is organized around Wulff geometry and critical exponential nonlinearities, the parabolic anisotropic Trudinger equation is organized around direction-dependent diffusion exponents and anisotropic intrinsic cylinders (Ciani et al., 21 Jul 2025).

6. Regularity theory, related flows, and broader analytical context

For the parabolic anisotropic Trudinger equation, the principal qualitative result is a Harnack inequality without restrictions on the gap $F$04. If $F$05 is a nonnegative local weak solution, then there exist constants $F$06, depending only on the data $F$07, such that whenever

$F$08

one has

$F$09

This shows that the anisotropic Trudinger operator retains the “heat-equation-like” Harnack form, with time scale $F$10 and no intrinsic correction depending on the solution size (Ciani et al., 21 Jul 2025).

For Hölder continuity, an additional small-gap assumption is required: $F$11 for some $F$12 depending only on the data. Under this restriction, nonnegative local weak solutions admit a locally Hölder continuous representative (Ciani et al., 21 Jul 2025). This distinguishes two levels of regularity theory: Harnack and local boundedness are available in the full anisotropic regime, whereas Hölder continuity is proved only in a restricted range of diffusion exponents.

A separate isotropic result establishes higher integrability for gradients of positive solutions to the scalar doubly nonlinear Trudinger equation

$F$13

by constructing refined intrinsic cylinders and proving a reverse Hölder inequality. The analysis is scalar-specific and exploits positivity, truncation, and Harnack estimates to remove the upper restriction on $F$14 that had appeared in the vectorial theory (Saari et al., 2019). This suggests that intrinsic geometry, stopping-time constructions, reverse Hölder inequalities, and Gehring-type self-improvement are likely to remain important in anisotropic doubly nonlinear settings, provided anisotropic analogues of the scalar positivity theory are available.

Large-time asymptotics are well understood for the isotropic homogeneous Trudinger flow

$F$15

on bounded domains with Dirichlet, Robin, Neumann, and fractional boundary conditions. After rescaling by $F$16, solutions converge to extremals of the corresponding Poincaré inequality, and the dual variable $F$17 approaches extremals of a dual Poincaré inequality (Hynd et al., 2017). The supplied exposition states that the structural ingredients behind this theory—convexity, $F$18-homogeneity, Poincaré inequalities, dual Poincaré inequalities, and spectral simplicity—are well suited to anisotropic extensions, for instance for

$F$19

This suggests an anisotropic large-time theory, but in the supplied material it is presented as an expected extension rather than as an established theorem (Hynd et al., 2017).

A broader elliptic analytical context is provided by anisotropic Wulff-type energies

$F$20

whose critical points solve

$F$21

For such anisotropic quasilinear equations, pointwise gradient bounds of Modica type and rigidity results were proved, including one-dimensional symmetry when equality holds in the associated $F$22-function identity (Cozzi et al., 2013). This is not the same equation as the parabolic anisotropic Trudinger equation or the Euler–Lagrange equations of anisotropic Moser–Trudinger functionals, but it belongs to the same anisotropic Trudinger-type landscape in the sense of quasilinear anisotropic operators with $F$23-type or Orlicz-type growth.

Taken together, these results show that “anisotropic Trudinger’s equation” is not a single canonical PDE. It names a cluster of anisotropic borderline problems: elliptic Euler–Lagrange equations with critical exponential growth, anisotropic Hardy–Leray perturbations of such equations, and anisotropic doubly nonlinear parabolic evolutions. The unifying structure is the replacement of Euclidean isotropy by Finsler or direction-dependent geometry, coupled with critical growth, spectral thresholds, and intrinsic regularity mechanisms (Zhou, 2019).