Willmore Equation for Graphs

• Willmore equation for graphs is the Euler–Lagrange equation derived from the Willmore functional when surfaces are expressed as graphs, leading to a quasilinear fourth-order PDE.
• It reformulates geometric quantities like mean curvature, Gauss curvature, and the second fundamental form in graph variables, facilitating divergence and nondivergence analyses.
• The study encompasses Bernstein-type rigidity, low-regularity boundary value problems, and anisotropic numerical schemes, bridging geometric analysis with practical computational methods.

Searching arXiv for recent and foundational papers on the Willmore equation for graphs. The Willmore equation for graphs is the Euler–Lagrange equation associated with the Willmore functional when a surface in $\mathbb{R}^3$ is represented nonparametrically as the graph of a height function. In the graphical setting, one studies surfaces of the form $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$, with $\Omega\subset\mathbb{R}^2$ or $\Omega=\mathbb{R}^2$, and rewrites the geometric equation in terms of $u$, its gradient, and its Hessian. This produces a quasilinear fourth-order PDE whose structure can be expressed either intrinsically through the induced metric and the Laplace–Beltrami operator or explicitly in divergence form. The subject connects geometric analysis, elliptic and parabolic fourth-order PDE, symmetry reduction, Bernstein-type rigidity, boundary value problems, variational relaxation, and numerical approximation (Luo et al., 2011, Gulyak, 25 Sep 2025, Gulyak, 29 Mar 2026, Chen et al., 2014, Deckelnick et al., 2015, Oberhuber, 2011).

1. Geometric formulation for graphical surfaces

Let $\Sigma\subset\mathbb{R}^3$ be represented as a graph

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$

with $u$ smooth. In the notation used across the cited works, one writes $Du=(u_x,u_y)$ and

$v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$

so that $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$0 (Luo et al., 2011, Gulyak, 25 Sep 2025, Gulyak, 29 Mar 2026, Deckelnick et al., 2015). The induced metric is

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$1

with inverse

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$2

or equivalently

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$3

and the area element is

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$4

depending on the notation of the source (Luo et al., 2011, Gulyak, 25 Sep 2025, Gulyak, 29 Mar 2026, Deckelnick et al., 2015). With the upward orientation, the unit normal is

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$5

and for an entire graph the Gauss map image lies in the upper hemisphere $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$6 (Luo et al., 2011, Gulyak, 25 Sep 2025, Gulyak, 29 Mar 2026, Deckelnick et al., 2015).

The second fundamental form coefficients are

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$7

and the mean curvature is taken with the convention

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$8

so minimal graphs satisfy $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$9 (Luo et al., 2011). In graph variables,

$\Omega\subset\mathbb{R}^2$0

and the Gauss curvature is

$\Omega\subset\mathbb{R}^2$1

(Luo et al., 2011, Gulyak, 25 Sep 2025, Gulyak, 29 Mar 2026, Deckelnick et al., 2015). The squared norm of the second fundamental form satisfies the Gauss equation

$\Omega\subset\mathbb{R}^2$2

in the convention of Luo and Sun (Luo et al., 2011). The trace-free second fundamental form $\Omega\subset\mathbb{R}^2$3 is not used directly in that note, but the standard decomposition recorded there is

$\Omega\subset\mathbb{R}^2$4

hence

$\Omega\subset\mathbb{R}^2$5

(Luo et al., 2011).

For a scalar function $\Omega\subset\mathbb{R}^2$6 on the graph, the Laplace–Beltrami operator is

$\Omega\subset\mathbb{R}^2$7

and in particular Luo and Sun record for $\Omega\subset\mathbb{R}^2$8 the formula

$\Omega\subset\mathbb{R}^2$9

(Luo et al., 2011, Gulyak, 25 Sep 2025).

2. Willmore functional and the graphical Euler–Lagrange equation

For a two-dimensional closed surface $\Omega=\mathbb{R}^2$0 with immersion $\Omega=\mathbb{R}^2$1, the Willmore functional is written in Luo–Sun as

$\Omega=\mathbb{R}^2$2

while other cited graph papers write the graph energy as

$\Omega=\mathbb{R}^2$3

or

$\Omega=\mathbb{R}^2$4

reflecting different normalization conventions (Luo et al., 2011, Gulyak, 29 Mar 2026, Deckelnick et al., 2015). The convention $\Omega=\mathbb{R}^2$5 explains why the factor $\Omega=\mathbb{R}^2$6 appears in some formulations and why the graph formula for $\Omega=\mathbb{R}^2$7 carries no additional $\Omega=\mathbb{R}^2$8 (Luo et al., 2011).

The Willmore equation in $\Omega=\mathbb{R}^2$9 is

$u$0

equivalently

$u$1

and in the convention of (Luo et al., 2011) it is also equivalent to

$u$2

(Luo et al., 2011, Gulyak, 25 Sep 2025, Gulyak, 29 Mar 2026). For graphs, this is a quasilinear fourth-order PDE for $u$3. A standard divergence formulation, derived in Deckelnick–Dziuk and used in several later works, is

$u$4

with

$u$5

(Luo et al., 2011, Gulyak, 25 Sep 2025, Gulyak, 29 Mar 2026, Chen et al., 2014).

This divergence identity is central because it exposes the PDE as a conservation law for a vector field built from $u$6. In the low-regularity boundary-value theory of (Gulyak, 25 Sep 2025), isolating the biharmonic term yields

$u$7

where

$u$8

$u$9

and the pointwise bounds

$\Sigma\subset\mathbb{R}^3$0

hold (Gulyak, 25 Sep 2025). The same paper emphasizes that all monomials are at least cubic and that the highest derivatives enter at most quadratically in $\Sigma\subset\mathbb{R}^3$1 and linearly in $\Sigma\subset\mathbb{R}^3$2.

A complementary nondivergence expansion is used in the flow paper (Gulyak, 29 Mar 2026): $\Sigma\subset\mathbb{R}^3$3 where the principal fourth-order part is

$\Sigma\subset\mathbb{R}^3$4

Its coefficients depend only on $\Sigma\subset\mathbb{R}^3$5 via $\Sigma\subset\mathbb{R}^3$6, and satisfy the ellipticity bound

$\Sigma\subset\mathbb{R}^3$7

Thus $\Sigma\subset\mathbb{R}^3$8 is uniformly elliptic while $\Sigma\subset\mathbb{R}^3$9 is bounded (Gulyak, 29 Mar 2026).

3. Entire graphs and Bernstein-type rigidity

A central rigidity result for the Willmore equation on graphs is the Bernstein-type theorem of Luo and Sun. They consider a smooth entire graph

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$0

satisfying the Willmore equation

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$1

and the integrability assumption

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$2

Their theorem states: “Every smooth, entire graphical solution of (2.1) with finite $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$3 norm of the mean curvature is a plane” (Luo et al., 2011).

The argument reduces the result to the earlier Chen–Lamm theorem that “Every smooth, entire graphical solution of (2.1) with finite $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$4 norm of the second fundamental form is a plane” (Luo et al., 2011). Luo and Sun prove that for any smooth entire graph in $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$5 with square integrable mean curvature,

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$6

Combined with

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$7

this yields

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$8

so $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\},$9 if and only if $u$0, and Chen–Lamm then applies (Luo et al., 2011).

The proof proceeds through two geometric ingredients. First, using the calibration argument of Colding–Minicozzi as cited in (Luo et al., 2011), one obtains quadratic area growth: $u$1 Second, because the Gauss map of a global graph takes values in the contractible upper hemisphere $u$2, the area form $u$3 on $u$4 satisfies $u$5 on $u$6, and

$u$7

With a compactly supported cutoff $u$8, the estimate

$u$9

is derived, and a logarithmic cutoff $Du=(u_x,u_y)$0 satisfying

$Du=(u_x,u_y)$1

implies

$Du=(u_x,u_y)$2

(Luo et al., 2011).

This theorem is explicitly described in (Luo et al., 2011) as a Bernstein-type theorem for Willmore graphs, replacing the minimal surface condition by the Willmore Euler–Lagrange equation and assuming $Du=(u_x,u_y)$3. Minimal graphs are included as the case $Du=(u_x,u_y)$4.

4. Radial symmetry, ODE reduction, and inverted catenoids

For radially symmetric graphs $Du=(u_x,u_y)$5, $Du=(u_x,u_y)$6, Chen and Li derive a sharp reduction of the graphical Willmore equation to an ODE (Chen et al., 2014). Writing

$Du=(u_x,u_y)$7

the mean curvature becomes

$Du=(u_x,u_y)$8

and smoothness at the origin implies

$Du=(u_x,u_y)$9

(Chen et al., 2014).

The divergence-form graph equation becomes a radial divergence condition. If the associated radial scalar is denoted by $v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$0, then

$v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$1

For smooth graphs on a full disk this forces $v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$2, whereas on a punctured disk one obtains

$v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$3

for a constant $v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$4 (Chen et al., 2014). In the smooth case, Chen and Li obtain the radial Willmore ODE

$v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$5

and in the punctured-disk case the inhomogeneous version

$v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$6

(Chen et al., 2014).

Their classification theorem states that if $v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$7 is a smooth solution to the graphic Willmore equation on a disk centered at the origin, with $v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$8 and mean curvature at $v=\sqrt{1+|Du|^2},\qquad Q=\sqrt{1+|Du|^2},$9 equal to $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$00, then either $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$01 and $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$02 is constant, or $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$03 and the graph is a spherical cap contained in the half-sphere of radius

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$04

given explicitly by

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$05

In particular, smooth radially symmetric entire Willmore graphs in $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$06 must be flat (Chen et al., 2014).

For smooth radial solutions on a punctured disk with $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$07, there exist a constant $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$08 and a function $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$09 such that

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$10

and the graph of $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$11 is contained in a translated graphical piece of an inverted catenoid uniquely determined by $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$12 and $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$13 (Chen et al., 2014). The inverted catenoid enters because minimal surfaces are mapped to Willmore surfaces under inversion. Chen and Li parametrize the catenoid by

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$14

apply inversion

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$15

and identify the punctured-disk radial Willmore graph with the outermost graphical piece $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$16 of the inverted catenoid, with

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$17

and $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$18 determined by matching the asymptotics (Chen et al., 2014).

The same paper shows that if

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$19

then a radial solution on the punctured disk extends across the puncture as a $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$20 function for any $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$21 and fits the inverted-catenoid model (Chen et al., 2014).

5. Boundary value problems and low-regularity existence theory

For graphs over bounded domains, the Willmore equation is naturally coupled to clamped boundary data. In the modern boundary-value literature, “clamped” means

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$22

so the boundary position and tangent half-planes are fixed along $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$23 (Gulyak, 29 Mar 2026, Gulyak, 25 Sep 2025). In the notation of (Gulyak, 25 Sep 2025), the quantity

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$24

represents the trace of $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$25 on $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$26 and enters the estimates.

A major recent development is the low-regularity theory of graphical Willmore boundary problems. In (Gulyak, 25 Sep 2025), the Willmore equation is rewritten in divergence form so that it can be treated in weighted second-order Sobolev spaces. The resulting framework weakens the regularity assumptions on both the boundary and the Dirichlet data to the $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$27-class while the solution remains smooth in the interior, and extends existence theory to domains with merely Lipschitz boundaries within a purely weighted Sobolev framework (Gulyak, 25 Sep 2025).

The same paper treats two regimes. For $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$28, $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$29, and $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$30, smallness is imposed on the slope: $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$31 with an a priori bound

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$32

Then there exists a solution

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$33

of the clamped problem for any $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$34 (Gulyak, 25 Sep 2025). For Lipschitz $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$35, the analysis is carried out in weighted Sobolev spaces

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$36

with clamped data in the trace space

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$37

and smallness required on

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$38

Under either a BMO-modulo-VMO smallness assumption on the outward normal field or a restricted $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$39 range depending on the Lipschitz constant, one obtains a variational solution

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$40

(Gulyak, 25 Sep 2025).

A distinct but related advance is the low-regularity flow theory for the Willmore flow of graphs with clamped boundary data (Gulyak, 29 Mar 2026). For a graph over a bounded domain, the $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$41-gradient flow of the Willmore energy is

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$42

with

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$43

in immersion form (Gulyak, 29 Mar 2026). The paper develops time-weighted parabolic Hölder spaces that allow derivatives above parabolic order $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$44 to blow up as $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$45 at controlled rates, thereby avoiding the classical fourth-order compatibility condition at $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$46 (Gulyak, 29 Mar 2026).

Its short-time existence theorem states that if $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$47, $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$48, and $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$49, then:

• if $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$50, there is a unique solution

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$51

• if $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$52 with $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$53, there exists a solution

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$54

for some $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$55 (Gulyak, 29 Mar 2026).

In the small-data Lipschitz regime, the same work proves global existence, uniform gradient bounds, and exponential convergence to a stationary solution. Specifically, if

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$56

then the flow admits a global solution and

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$57

Moreover, there exists a unique stationary solution $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$58 satisfying the elliptic Willmore equation with the prescribed boundary data, and for every $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$59,

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$60

(Gulyak, 29 Mar 2026).

6. Variational relaxation, boundary conditions, and generalized graph classes

The Willmore equation for graphs also appears as the Euler–Lagrange equation in variational problems with prescribed boundary data. In (Deckelnick et al., 2015), one considers a bounded $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$61 domain $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$62 with outward unit normal $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$63, and the graph functional

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$64

where

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$65

and the pure Willmore functional is

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$66

(Deckelnick et al., 2015). For smooth critical graphs, the interior Euler–Lagrange equation is given there as

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$67

This is the form adopted in that paper and differs from the normalization used in (Luo et al., 2011), a convention issue that those sources explicitly note.

Two boundary conditions are distinguished in (Deckelnick et al., 2015). The Dirichlet, or clamped, condition prescribes

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$68

Under these constraints, the first variation yields the interior Willmore equation and no additional natural boundary condition. The total Gaussian curvature is then determined by the boundary data and topology through Gauss–Bonnet: $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$69 where the geodesic curvature $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$70 of the boundary curve on the graph is explicitly determined by $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$71 (Deckelnick et al., 2015).

The Navier, or hinged, condition prescribes only

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$72

and the natural boundary condition is

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$73

with $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$74 the normal curvature of the boundary curve in the graph surface (Deckelnick et al., 2015).

A principal result of (Deckelnick et al., 2015) is that for $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$75-regular graphs, bounds for the Willmore energy imply area and diameter bounds. In particular, if $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$76 with $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$77, then there exists $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$78 depending on $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$79 and $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$80 such that

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$81

(Deckelnick et al., 2015). The paper then studies the $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$82-lower semicontinuous relaxation

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$83

If a sequence has bounded Willmore energy, then after passing to a subsequence it converges in $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$84 to some

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$85

and the absolutely continuous contribution

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$86

provides a lower bound: $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$87 (Deckelnick et al., 2015).

The same paper proves the existence of a minimizer

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$88

for the relaxed energy and shows that finite relaxed Willmore energy implies the attainment of the Dirichlet boundary data in an appropriate trace sense on the non-vertical part of the boundary (Deckelnick et al., 2015). A plausible implication is that the graphical Willmore theory naturally extends beyond the classical $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$89 graph class to limit configurations with vertical parts, although the paper formulates the structural statements carefully in BV language rather than as a generalized PDE theory.

7. Flow, anisotropy, and numerical formulations

The Willmore equation for graphs is also studied through its gradient flow and through anisotropic generalizations. In the anisotropic graph framework of (Oberhuber, 2011), one considers a rectangle $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$90 and a convex, positive, $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$91-homogeneous anisotropy $\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$92. The anisotropic mean curvature is defined by

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$93

and the anisotropic Willmore functional is

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$94

Introducing

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$95

the stationary Euler–Lagrange equation is written in vector-conservative form as

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$96

where

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$97

(Oberhuber, 2011).

For the isotropic density

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$98

this reduces to

$\Sigma=\{(x,y,u(x,y)):(x,y)\in\Omega\}$99

with

$\Omega\subset\mathbb{R}^2$00

(Oberhuber, 2011). This is the same structural identity as the isotropic graphical Willmore equation recorded in (Luo et al., 2011, Gulyak, 25 Sep 2025), and (Gulyak, 29 Mar 2026), written in the notation of the anisotropic paper.

The anisotropic Willmore flow is formulated as

$\Omega\subset\mathbb{R}^2$01

coupled to

$\Omega\subset\mathbb{R}^2$02

(Oberhuber, 2011). For zero Dirichlet boundary data, the paper proves the continuous energy equality

$\Omega\subset\mathbb{R}^2$03

so the energy decays monotonically along the flow (Oberhuber, 2011). The same work defines weak solutions, derives a complementary finite volume discretization, reformulates it in finite-difference form, and proves the discrete energy equality

$\Omega\subset\mathbb{R}^2$04

for zero Dirichlet data (Oberhuber, 2011).

The bounded-domain isotropic flow theory of (Gulyak, 29 Mar 2026) supplies the analytic counterpart to this numerical and anisotropic perspective. Along the flow with time-independent Dirichlet data, it proves

$\Omega\subset\mathbb{R}^2$05

hence

$\Omega\subset\mathbb{R}^2$06

(Gulyak, 29 Mar 2026). That identity is the precise gradient-flow energy law underlying the global convergence result.

Taken together, these works show that the Willmore equation for graphs admits several mutually reinforcing formulations: intrinsic geometric form via $\Omega\subset\mathbb{R}^2$07 and $\Omega\subset\mathbb{R}^2$08, divergence form suited to weak and low-regularity analysis, biharmonic-relative forms suited to elliptic estimates and fixed-point arguments, ODE reductions under radial symmetry, relaxed variational formulations in $\Omega\subset\mathbb{R}^2$09, and conservative parabolic forms suited to flow theory and structure-preserving numerics (Luo et al., 2011, Chen et al., 2014, Deckelnick et al., 2015, Oberhuber, 2011, Gulyak, 25 Sep 2025, Gulyak, 29 Mar 2026).