Prescribed Mean Curvature Equation

• Prescribed mean curvature equation is a quasilinear geometric PDE that defines graphs with a fixed mean curvature across Euclidean, Lorentzian, and non-Euclidean settings.
• The analysis employs variational, BV, and current-theoretic approaches to address existence, regularity, and boundary behavior under diverse geometric conditions.
• Advanced methods including inverse problems and high-order numerical schemes reveal intricate bifurcation, stability, and singularity phenomena in the equation.

The prescribed mean curvature equation is the quasilinear geometric PDE governing graphs whose mean curvature is fixed by a prescribed datum. In the standard Euclidean graph setting over a domain $\Omega$, it takes the form

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$

while for spacelike graphs in Minkowski spacetime the Lorentzian sign convention gives

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$

Modern treatments include prescriptions by position, height, gradient, Gauss map, angle function, or the normal component of an ambient vector field, as well as anisotropic, sub-Finsler, periodic, and measure-data formulations (Alvarez et al., 2019, Bartolo et al., 2020, Bueno et al., 2018, Leonardi et al., 2023). The subject sits at the intersection of geometric analysis, nonlinear elliptic theory, and the calculus of variations, and it now includes inverse problems and high-order numerical schemes alongside classical existence, regularity, and comparison theory (Leonardi et al., 2016, Liimatainen et al., 26 Sep 2025, Haug et al., 2024).

1. Graph equations and geometric prescriptions

In a complete $n$-dimensional Riemannian manifold $M$, the Dirichlet problem for vertical graphs in $M\times \mathbb R$ is based on

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$

The same equation is written in non-divergence form as

$$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$

and the operator

$Q u := M u - nH(x,u)W^3$

is used in barrier and gradient estimates (Alvarez et al., 2019). This formulation is the standard nonparametric model for prescribed mean curvature of a graph.

A different but closely related prescription fixes the mean curvature as a function of the Gauss map. For an oriented immersed hypersurface $\Sigma\subset \mathbb R^{n+1}$ with Gauss map $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$0, one asks for

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$1

For a local graph $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$2, this becomes

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$3

This framework contains constant mean curvature hypersurfaces and also the linear case $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$4, which is tied to self-translating solitons with forcing term and to constant weighted mean curvature for the density $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$5 with $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$6 (Bueno et al., 2018).

In $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$7, rotational surfaces are treated through the angle function

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$8

and the prescription is

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$9

For rotational profiles this reduces to an autonomous phase-plane system, and the linear family $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$0 contains both constant mean curvature surfaces and translating solitons (Bueno et al., 2020). A closely related idea appears when the prescribed scalar curvature is the normal projection of an ambient vector field: on bounded Euclidean domains one considers

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$1

and on $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$2 the periodic analogue is

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$3

(Tsukamoto, 2019, Tsukamoto, 2020).

The Lorentzian version changes sign and geometry. In $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$4-dimensional Minkowski spacetime $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$5, with sign convention

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$6

a graph $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$7 is spacelike exactly when $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$8, and the mean curvature equation becomes

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$9

This is the graph equation for spacelike hypersurfaces with prescribed mean curvature in the Lorentzian setting (Bartolo et al., 2020).

A further generalization replaces the Euclidean horizontal norm by a Minkowski norm in the Heisenberg group. For $n$0-graphs in the sub-Finsler Heisenberg group $n$1, with horizontal correction term $n$2, the Euler–Lagrange equation away from the singular set $n$3 is

$n$4

where $n$5 is a strictly convex body and $n$6 is the dual support map (2207.13414). This shows that the phrase “prescribed mean curvature equation” now covers isotropic and anisotropic settings, Euclidean and Lorentzian signatures, and several non-Euclidean ambient geometries.

2. Variational, $n$7, and current-theoretic formulations

A central theme in the theory is that the prescribed mean curvature equation is naturally variational, but the correct functional and solution space depend strongly on the regularity of the domain and of the datum. On weakly regular bounded domains $n$8, one minimizes

$n$9

and the Euler–Lagrange equation for compactly supported variations is

$M$0

In the extremal case $M$1, the paper establishes the equivalence between extremality, uniqueness up to additive constants, maximality of $M$2, weak verticality

$M$3

and an integral version of verticality on smooth interior approximations (Leonardi et al., 2016).

The weak regularity theory depends on a generalized Gauss–Green theorem for bounded continuous vector fields with bounded divergence. For

$M$4

one obtains a weak normal trace $M$5 and the identity

$M$6

Together with a weak Young’s law for $M$7-minimizers of perimeter, this replaces smooth boundary tools in the capillarity and extremal theory (Leonardi et al., 2016).

For signed Radon-measure data $M$8, the nonparametric equation is written as

$M$9

The weak formulation is no longer expressed solely in terms of distributions. Instead, $M\times \mathbb R$0 is a weak solution if there exist a vector field $M\times \mathbb R$1 and a Borel map $M\times \mathbb R$2 such that

$M\times \mathbb R$3

and

$M\times \mathbb R$4

This formulation uses Anzellotti’s pairing theory for divergence-measure fields, a refinement of Anzellotti–Giaquinta approximation, and convex duality. Under non-extremal and admissibility assumptions on $M\times \mathbb R$5, the theory yields existence of $M\times \mathbb R$6 solutions, a $M\times \mathbb R$7-convergence theorem for suitable smooth approximations of the prescribed measure, and a maximum principle for continuous weak solutions. The same work constructs examples of non-uniqueness, showing that continuity is genuinely needed in the maximum principle (Leonardi et al., 2023).

A parallel current-theoretic formulation is used when a minimizer need not literally attain the Dirichlet datum on the boundary. If $M\times \mathbb R$8 minimizes the relaxed functional

$M\times \mathbb R$9

one associates the integral current

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$0

with $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$1 supported in the boundary cylinder $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$2. Under $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$3 regularity of the domain and boundary manifold, the support of this current is a $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$4 manifold-with-boundary whose boundary is exactly the prescribed boundary data; under stronger hypotheses, the paper proves $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$5 regularity and improved trace regularity (Bourni, 2010). This viewpoint isolates the geometric object that remains well behaved even when the scalar trace detaches from the boundary datum.

3. Solvability criteria for Dirichlet and exterior problems

For smooth bounded domains in product manifolds, the sharpest existence results are of Serrin type. If $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$6 has $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$7 boundary, $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$8 with $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u) \quad\text{in }\Omega, \qquad u=\varphi \quad\text{on }\partial\Omega.$$9, and

$$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$0

while

$$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$1

then for every $$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$2 there exists a unique solution

$$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$3

of the Dirichlet problem (Alvarez et al., 2019). In $$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$4, the Ricci hypothesis can be replaced by

$$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$5

and on Hadamard manifolds with $$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$6, $$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$7, solvability follows when the principal curvatures of $$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$8 are greater than $$M u=\left(\sigma^{ij}-\frac{u^i u^j}{W^2}\right)u_{ij}=nH(x,u)W^3, \qquad W=\sqrt{1+|\nabla u|^2},$$9 and the same threshold $Q u := M u - nH(x,u)W^3$0 holds (Alvarez et al., 2019).

The proof mechanism is entirely a priori. It uses a comparison lemma for the mean curvature of parallel hypersurfaces, an a priori height estimate, boundary barriers

$Q u := M u - nH(x,u)W^3$1

built from the distance function to the boundary, and a global gradient estimate of Caffarelli–Nirenberg–Spruck type based on

$Q u := M u - nH(x,u)W^3$2

These estimates are then inserted into a Leray–Schauder continuation scheme (Alvarez et al., 2019). A plausible implication is that the classical Euclidean Serrin theory is best viewed as one instance of a curvature-controlled boundary solvability mechanism.

The exterior Lorentzian problem has a different structure because the domain is unbounded and the graph must remain spacelike. The setting is

$Q u := M u - nH(x,u)W^3$3

where the $Q u := M u - nH(x,u)W^3$4 are finitely many pairwise disjoint bounded Lipschitz domains, $Q u := M u - nH(x,u)W^3$5, and the boundary value problem is

$Q u := M u - nH(x,u)W^3$6

The admissible class is

$Q u := M u - nH(x,u)W^3$7

The main theorem states that there exists a spacelike weak solution if and only if

$Q u := M u - nH(x,u)W^3$8

and, when $Q u := M u - nH(x,u)W^3$9 is not convex, $\Sigma\subset \mathbb R^{n+1}$0 must also satisfy

$\Sigma\subset \mathbb R^{n+1}$1

for every $\Sigma\subset \mathbb R^{n+1}$2 such that the open segment $\Sigma\subset \mathbb R^{n+1}$3 (Bartolo et al., 2020). Sufficiency is obtained by minimizing

$\Sigma\subset \mathbb R^{n+1}$4

and strict spacelikeness is recovered by adapting the Bartnik–Simon light-ray argument (Bartolo et al., 2020).

Small-data solvability results complement these geometric criteria. Near a fixed minimal graph $\Sigma\subset \mathbb R^{n+1}$5 on a bounded $\Sigma\subset \mathbb R^{n+1}$6 domain, if $\Sigma\subset \mathbb R^{n+1}$7, $\Sigma\subset \mathbb R^{n+1}$8, and a dominating datum $\Sigma\subset \mathbb R^{n+1}$9 is sufficiently small, then there exists

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$00

solving the prescribed mean curvature equation and satisfying $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$01. The proof is an elliptic fixed-point argument built on the linearized operator

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$02

and the Leray–Schauder theorem (Tsukamoto, 2019). On the torus $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$03, a periodic graph exists under smallness in $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$04, a strict monotonicity condition in the vertical variable,

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$05

and the compatibility condition

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$06

yielding a solution $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$07 by linearized solvability and a weak fixed-point theorem (Tsukamoto, 2020).

4. Global geometry and ambient-geometric extensions

When the prescription depends on the Gauss map, the theory extends far beyond local graph solvability. For complete $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$08-hypersurfaces in $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$09 satisfying $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$10, the basic geometric properties of constant mean curvature theory persist in modified form: the maximum principle holds, translations preserve the class, and symmetries of $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$11 induce ambient isometries preserving the class (Bueno et al., 2018). In the surface case $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$12, the paper proves compactness for sequences of $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$13-surfaces with uniformly bounded second fundamental form, a curvature estimate

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$14

when the mean curvature is bounded below by $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$15 and the Gauss map omits a spherical disk, and several sufficient conditions for uniform height estimates of $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$16-graphs, including failure of the cylindrical obstruction condition

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$17

(Bueno et al., 2018). Under symmetry assumptions, properly embedded finite-topology surfaces with at most one end are then forced into slabs or cylinders, and for $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$18 positive and invariant under three linearly independent reflections, every properly embedded finite-topology $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$19-surface with at most one end is the Guan–Guan sphere $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$20 up to translation (Bueno et al., 2018).

A half-space theory is available for a class of nonconstant prescriptions in $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$21. If $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$22, meaning $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$23, $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$24 for $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$25, and $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$26, then a one-parameter family of rotational $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$27-annuli $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$28 serves as a barrier family analogous to minimal catenoids. Under asymptotic comparison with

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$29

the paper proves that a connected, properly immersed, nonplanar $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$30-surface cannot be contained in a horizontal half-space (Bueno, 2019). The proof follows the moving-barrier method and depends on detailed analysis of the asymptotic behavior of the $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$31-catenoids.

In $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$32, rotational $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$33-surfaces are encoded by a first-order autonomous system on the phase strip $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$34. This phase-plane description yields entire rotational graphs, called $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$35-bowls, whenever there exists $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$36 such that

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$37

and also one-parameter families of $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$38-catenoids when

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$39

For the linear prescription $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$40, the global geometry depends on the thresholds

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$41

producing entire graphs, asymptotically cylindrical surfaces, and properly immersed annuli with self-intersections in different parameter regimes (Bueno et al., 2020). This shows that prescribed mean curvature geometry in negatively curved ambient spaces is already rich at the rotational level.

The sub-Finsler Heisenberg problem adds anisotropy and horizontal degeneracy. The variational functional is

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$42

and the Dirichlet problem for constant $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$43 has a Lipschitz solution provided $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$44 is strictly convex, $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$45 has $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$46 boundary, a compatibility condition on $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$47 holds, and

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$48

together with an integral coercivity condition (2207.13414). The proof uses a two-step approximation: a Finsler desingularization and a small Euclidean regularization, followed by uniform $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$49, boundary gradient, and interior gradient estimates.

5. Comparison principles, stability, singular behavior, and bifurcation

A fundamental qualitative feature is rigidity under touching. For the prescribed mean curvature equation

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$50

with $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$51 locally Lipschitz, the strong comparison principle for semicontinuous viscosity solutions states that if $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$52 is a viscosity supersolution, $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$53 is a viscosity subsolution, $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$54 in $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$55, and $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$56 at some interior point, then

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$57

In bounded domains this yields the weak comparison principle from boundary ordering. The proof uses only the definition of viscosity solutions, inf and sup convolutions, and classical quasilinear elliptic theory (Ohnuma et al., 2018).

Boundary singularities can nonetheless force subtle local behavior. In planar domains with a corner at $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$58, for bounded $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$59 solutions of

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$60

the paper proves that cusp solutions cannot occur if $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$61 is weakly increasing in $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$62 near $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$63, or if $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$64 along the cusp axis for relevant $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$65 (Echart et al., 2016). It also gives an example with $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$66, showing that the local argument fails in the decreasing case. Under the corresponding corner hypotheses, radial limits then exist and are continuous in all directions of the corner sector (Echart et al., 2016).

For $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$67-surfaces in $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$68, stability is defined not by second variation of a global functional, but by the linearized operator

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$69

Every $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$70-graph is stable, compact closed $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$71-surfaces are never stable, and under the pointwise condition

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$72

the paper proves a universal intrinsic radius estimate and hence nonexistence of complete stable $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$73-surfaces (Bueno et al., 2018). This suggests that stability theory for prescribed mean curvature surfaces is already highly constrained by the geometry of the prescription on the sphere.

Bifurcation phenomena differ sharply from semilinear analogues. In the one-dimensional singular problem

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$74

the solution set depends on both $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$75 and $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$76. There is a saddle-node bifurcation at $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$77 for every $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$78, and a second, splitting bifurcation controlled by a critical half-length

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$79

For $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$80 the diagram has the standard single turning point, while for $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$81 the upper branch splits into two disconnected pieces (Brubaker et al., 2012).

Whole-space Lorentzian problems show a different bifurcation profile. For the radial prescribed mean curvature equation in $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$82 with $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$83,

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$84

there is no radial ground state when

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$85

whereas for

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$86

there are infinitely many radial ground states, and they do not belong to $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$87; the same paper also proves infinitely many radial sign-changing bound states for any $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$88 (Azzollini, 2015). A later global bifurcation result on $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$89 proves an unbounded connected continuum of smooth radial positive solutions bifurcating from the principal eigenvalue of the associated linearized problem (Zographopoulos, 13 Mar 2026). Together these results indicate that prescribed mean curvature operators can generate genuinely nonlinear branch structures not visible in semilinear scaling-invariant models.

6. Constructive, inverse, and numerical approaches

Constructive small-data theory on $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$90 can be carried out by explicit series expansion. For entire graphs in Euclidean or Minkowskian space(time), the unified equation is

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$91

with the upper sign for Euclidean geometry and the lower sign for Minkowski geometry. The data are assumed to satisfy

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$92

and the problem is rewritten as a nonlinear Hodge system in terms of a one-form $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$93 or, equivalently, a field $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$94 satisfying

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$95

If the first linear potential $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$96 satisfies

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$97

the paper constructs an absolutely convergent odd-order series

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$98

in $$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)=n\,H(x,u),$$99, leading to a solution $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$00 (Carley et al., 2010). This is an analytic perturbative construction rather than an abstract existence theorem.

An inverse problem has recently been formulated for the planar source equation

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$01

Given boundary data near a background $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$02, the associated Dirichlet-to-Neumann map is

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$03

In two dimensions, if two source terms $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$04 and $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$05 have the same $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$06 for all small perturbations around the same background, then $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$07 in $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$08 (Liimatainen et al., 26 Sep 2025). The first linearization produces an anisotropic conductivity equation

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$09

whose metric $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$10 depends on the background solution $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$11, and can also be rewritten as a Schrödinger-type equation on the Riemannian surface $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$12. The second linearization yields an integral identity attacked with complex geometric optics solutions, and a Liouville-type uniqueness theorem for conformal mappings then removes the anisotropic gauge (Liimatainen et al., 26 Sep 2025). This is presented as the first treatment of an inverse source problem for a quasilinear elliptic equation of this type.

Numerical work now treats the prescribed mean curvature equation as a family of boundary value problems for minimal, constant mean curvature, and capillary surfaces on planar domains. The model equation is

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$13

with $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$14 a rectangle, disk, or annulus, and $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$15, $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$16, or $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$17 depending on whether the target surface is minimal, constant mean curvature, or capillary (Haug et al., 2024). Dirichlet conditions $$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$18 and nonlinear capillary boundary conditions

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$19

are both handled. The discretization uses Chebyshev collocation on rectangles and Chebyshev–Fourier collocation on disks and annuli; the nonlinearity is treated with Newton iteration

$$\operatorname{div}\!\left(\frac{Du}{\sqrt{1-|Du|^2}}\right)=n\,H(x,u), \qquad |Du|<1.$$20

and the algorithm is adaptive, increasing resolution until the residual tolerance is met (Haug et al., 2024). This computational line makes the equation accessible in geometries and boundary regimes where explicit formulas are unavailable.

Across these developments, the prescribed mean curvature equation appears less as a single PDE than as a geometric class of quasilinear equations whose structure is stable under major changes of ambient geometry, boundary model, and notion of weak solution. The common core is the divergence-form curvature operator of a graph; the diversity lies in how the prescription enters, which compactness mechanism survives, and which geometric constraints replace uniform ellipticity.