Fully Nonlinear Alt–Phillips Problem

• The fully nonlinear Alt–Phillips problem is a class of free boundary and transmission issues governed by uniformly elliptic nonlinear operators in viscosity form.
• It employs techniques like scaling, blow-up analysis, and Evans–Krylov estimates to establish optimal interior regularity and characterize free boundary geometry.
• Multiple formulations—including one-phase, degenerate, and variable-exponent cases—demonstrate existence, tangential interface contact, and open challenges in uniqueness and blow-up classification.

Searching arXiv for papers on the fully nonlinear Alt–Phillips problem and related free transmission formulations. The fully nonlinear Alt–Phillips problem denotes a family of free boundary and free transmission problems in which the classical Alt–Phillips law is replaced by a uniformly elliptic fully nonlinear operator, typically in viscosity form. In current usage, the phrase encompasses at least two closely related directions: the one-phase equation

$F(D^2u)=u^{\gamma-1},\qquad u\ge 0,$

studied for $\gamma\in(1,2)$, and Alt–Phillips-type transmission problems in which the governing law changes across $\{u>0\}$ and $\{u<0\}$, sometimes with phase-dependent degeneracy of the form $|Du|^{\theta_i}F(D^2u)=f$. Across these formulations, the central themes are scaling, blow-up analysis, viscosity encoding of the interface, optimal interior regularity, and the geometry of the free boundary or transmission set (Wu et al., 2020, Huaroto et al., 2020).

1. Model classes and analytic scope

The one-phase fully nonlinear Alt–Phillips equation is posed in an open set $\Omega\subset \mathbb{R}^d$ as

$F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$

with free boundary $\Gamma(u)=\partial\{u>0\}$. In the local theory, solutions are continuous viscosity solutions in the sense of Caffarelli–Cabré, and in the positivity set the equation becomes classical under the structural assumptions on $F$. The free boundary carries the overdetermined condition

$$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$

in the natural sense of approaching from $\gamma\in(1,2)$0 (Wu et al., 2020).

A second formulation is the fully nonlinear free transmission problem, where the PDE switches according to the sign of the solution. In its degenerate version, for a bounded domain $\gamma\in(1,2)$1,

$\gamma\in(1,2)$2

with $\gamma\in(1,2)$3 and free boundary $\gamma\in(1,2)$4. In the nondegenerate transmission problem, two fully nonlinear operators $\gamma\in(1,2)$5 and $\gamma\in(1,2)$6 act in the two phases: $\gamma\in(1,2)$7 In both settings, the interface is not prescribed a priori, and the transmission condition is encoded by viscosity inequalities rather than by an explicit flux balance (Huaroto et al., 2020, Pimentel et al., 2020).

A variable-exponent extension replaces the phase constants by a spatially dependent degeneracy law

$\gamma\in(1,2)$8

where

$\gamma\in(1,2)$9

Typical phase sets are $\{u>0\}$0, $\{u>0\}$1, and $\{u>0\}$2. This places Alt–Phillips-type transmission in a broader fully nonlinear degenerate framework (Jesus, 2021).

Formulation	Core PDE	Main outputs
One-phase fully nonlinear Alt–Phillips	$\{u>0\}$3, $\{u>0\}$4	$\{u>0\}$5 regularity; $\{u>0\}$6 regularity of $\{u>0\}$7
Degenerate free transmission	$\{u>0\}$8 in each phase	Existence; optimal $\{u>0\}$9 regularity
Nondegenerate free transmission	$\{u<0\}$0, $\{u<0\}$1 in the phases	Local $\{u<0\}$2; global half-space classification
Variable-exponent transmission	$\{u<0\}$3	Pointwise $\{u<0\}$4 regularity

2. Structural assumptions, scaling, and viscosity formulation

In the one-phase problem, the operator $\{u<0\}$5 is uniformly elliptic, convex, and normalized by $\{u<0\}$6. Two additional structural regimes are singled out in the free-boundary theory: either $\{u<0\}$7 is differentiable at $\{u<0\}$8, so small-scale limits are governed by $\{u<0\}$9, or $|Du|^{\theta_i}F(D^2u)=f$0 is $|Du|^{\theta_i}F(D^2u)=f$1-homogeneous, so the rescaled operators are unchanged. The natural homogeneity is

$|Du|^{\theta_i}F(D^2u)=f$2

and the scaling

$|Du|^{\theta_i}F(D^2u)=f$3

preserves the equation: $|Du|^{\theta_i}F(D^2u)=f$4 This is the basic blow-up mechanism at free boundary points (Wu et al., 2020).

The boundary-contact problem in the half-ball introduces a fixed boundary and imposes

$|Du|^{\theta_i}F(D^2u)=f$5

under structural assumptions

$|Du|^{\theta_i}F(D^2u)=f$6

together with

$|Du|^{\theta_i}F(D^2u)=f$7

The same exponent $|Du|^{\theta_i}F(D^2u)=f$8 controls the boundary blow-up scaling (Wang et al., 2 Sep 2025).

In the transmission setting, the interface law is encoded globally by viscosity inequalities. For the degenerate two-phase problem with $|Du|^{\theta_i}F(D^2u)=f$9, solutions satisfy

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

and

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

This replaces a classical Alt–Phillips flux balance by a pair of viscosity constraints that remain meaningful at $\Omega\subset \mathbb{R}^d$2 (Huaroto et al., 2020).

3. One-phase theory: optimal regularity, blow-ups, and regular free boundary

For $\Omega\subset \mathbb{R}^d$3, the one-phase fully nonlinear Alt–Phillips equation admits a Harnack-type estimate: if $\Omega\subset \mathbb{R}^d$4 solves $\Omega\subset \mathbb{R}^d$5 in $\Omega\subset \mathbb{R}^d$6, then

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

with $\Omega\subset \mathbb{R}^d$8 and $\Omega\subset \mathbb{R}^d$9 universal. The optimal interior regularity theorem states that $F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$0 for some universal $F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$1, and if $F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$2,

$F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$3

Near any free boundary point $F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$4, one has the sharp growth regime

$F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$5

The lower bound comes from a barrier of the form $F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$6, while the upper bound follows from Harnack and scaling (Wu et al., 2020).

The blow-up classification at regular points is governed by the planar profile

$F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$7

where the coefficient satisfies

$F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$8

A point $F(D^2 u) = u^{\gamma-1},\qquad u \ge 0,$9 is called regular if the zero set has positive density,

$\Gamma(u)=\partial\{u>0\}$0

At such points, global blow-ups with conic zero set are one-plane solutions, and an improvement-of-monotonicity argument yields directional monotonicity in a cone of directions. Under the additional hypothesis that $\Gamma(u)=\partial\{u>0\}$1 is differentiable at $\Gamma(u)=\partial\{u>0\}$2 or $\Gamma(u)=\partial\{u>0\}$3-homogeneous, the regular set $\Gamma(u)=\partial\{u>0\}$4 is relatively open and locally a $\Gamma(u)=\partial\{u>0\}$5 hypersurface (Wu et al., 2020).

A salient methodological feature is the absence of Weiss- or Spruck-type monotonicity formulas in the fully nonlinear setting. The theory compensates by combining viscosity compactness, subdifferential linearization of convex operators, Evans–Krylov estimates, and a new improvement-of-monotonicity mechanism for directional derivatives. This marks a structural difference from the classical semilinear Alt–Phillips problem, where variational monotonicity formulas play a central role (Wu et al., 2020).

4. Free boundary geometry at the fixed boundary

For the half-space problem with zero Dirichlet data on the fixed boundary, the principal geometric statement is tangential touch. If $\Gamma(u)=\partial\{u>0\}$6, $\Gamma(u)=\partial\{u>0\}$7, and

$\Gamma(u)=\partial\{u>0\}$8

then there exists a universal modulus of continuity $\Gamma(u)=\partial\{u>0\}$9 such that

$F$0

Equivalently, the free boundary approaches the fixed boundary with vanishing angle. According to the paper, for this range of $F$1 the result is new even when the operator is the Laplacian (Wang et al., 2 Sep 2025).

The proof rests on sharp $F$2-growth and matching nondegeneracy. If $F$3 with

$F$4

then

$F$5

If $F$6 and $F$7, then

$F$8

After rescaling, compactness and the normalization $F$9 force blow-up limits to solve the Laplacian Alt–Phillips problem in $$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$0, where Weiss’s monotonicity formula becomes available (Wang et al., 2 Sep 2025).

The boundary blow-up classification then identifies the unique nontrivial global half-space profile as

$$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$1

This profile has empty free boundary in the open half-space, so any hypothetical non-tangential sequence of free boundary points yields a contradiction after blow-up. The result is local, depends only on the structural hypotheses on $$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$2, and leaves open the broader problem of full free boundary regularity near contact points (Wang et al., 2 Sep 2025).

5. Fully nonlinear transmission: existence, regularity, and phase-dependent degeneracy

In the degenerate two-phase transmission problem, the first basic result is solvability of the Dirichlet problem. If $$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$3 is bounded with a uniform exterior sphere condition, $$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$4 is uniformly elliptic, $$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$5, $$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$6, and $$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$7, then there exists a viscosity solution $$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$8 to

$$u=0\quad\text{and}\quad \nabla u=0 \quad \text{on }\Gamma(u),$$9

The proof uses regularized uniformly elliptic approximations,

$\gamma\in(1,2)$00

comparison, barrier construction, Perron’s method, Schauder fixed point, and the limit $\gamma\in(1,2)$01. Uniqueness is established for the regularized problems, but not claimed for the original free transmission problem (Huaroto et al., 2020).

The optimal interior regularity is $\gamma\in(1,2)$02. If $\gamma\in(1,2)$03 is a viscosity subsolution of the global min inequality and a supersolution of the corresponding max inequality in $\gamma\in(1,2)$04, then for

$\gamma\in(1,2)$05

one has $\gamma\in(1,2)$06, with

$\gamma\in(1,2)$07

where $\gamma\in(1,2)$08. If $\gamma\in(1,2)$09 is convex, Evans–Krylov gives $\gamma\in(1,2)$10, and the optimal exponent becomes

$\gamma\in(1,2)$11

The worse phase, namely the larger degeneracy $\gamma\in(1,2)$12, dictates the regularity threshold (Huaroto et al., 2020).

The variable-exponent theory sharpens this picture pointwise. For

$\gamma\in(1,2)$13

with $\gamma\in(1,2)$14, $\gamma\in(1,2)$15, and a modulus $\gamma\in(1,2)$16 satisfying

$\gamma\in(1,2)$17

the solution is pointwise $\gamma\in(1,2)$18 at each $\gamma\in(1,2)$19, where

$\gamma\in(1,2)$20

Equivalently, there exists an affine function $\gamma\in(1,2)$21 such that for sufficiently small $\gamma\in(1,2)$22,

$\gamma\in(1,2)$23

This formulation captures the pointwise dependence of regularity on the local degeneracy rate (Jesus, 2021).

A complementary transmission theory dispenses with gradient degeneracy but allows different operators in the two phases: $\gamma\in(1,2)$24 For $\gamma\in(1,2)$25-strong solutions and under uniform ellipticity, convexity, positive homogeneity, and a geometric smallness assumption on the negative phase,

$\gamma\in(1,2)$26

one obtains local $\gamma\in(1,2)$27 regularity: $\gamma\in(1,2)$28 Under a thickness lower bound

$\gamma\in(1,2)$29

global solutions are classified as half-space profiles

$\gamma\in(1,2)$30

with $\gamma\in(1,2)$31 and $\gamma\in(1,2)$32 (Pimentel et al., 2020).

6. Relation to classical Alt–Phillips theory, misconceptions, and open directions

The fully nonlinear literature is closely connected to, but distinct from, the classical variational Alt–Phillips problem. In the classical one-phase energy

$\gamma\in(1,2)$33

recent work has established uniqueness of blow-ups for several classes of singular minimizing cones, with sharp logarithmic or polynomial convergence rates governed by epiperimetric inequalities and an integrability/sub-integrability dichotomy. That theory covers the variational or semilinear Alt–Phillips problem, not the fully nonlinear Hessian-dependent setting. The same paper explicitly states that it “does not treat fully nonlinear operators” and presents extension to fully nonlinear free boundary problems as an open direction (Carducci et al., 15 Jun 2026).

A similar distinction applies to generic properties and stability theory. Generic uniqueness and generic free-boundary regularity for the Alt–Phillips functional are proved for minimizers of the variational one-phase problem with quadratic Dirichlet energy and $\gamma\in(1,2)$34 potential, while “fully nonlinear Hessian-dependent Alt–Phillips problems” are described there as lying outside the paper’s framework. Likewise, smoothness and stability results for negative exponents $\gamma\in(1,2)$35, and the construction of stable or minimizing cones near $\gamma\in(1,2)$36, are obtained in quasilinear or semilinear formulations rather than for general fully nonlinear $\gamma\in(1,2)$37 (Fernández-Real et al., 2023, Carducci et al., 14 Jul 2025, Savin et al., 25 Feb 2025).

Two misconceptions are therefore recurrent. The first is to identify all recent Alt–Phillips free boundary results with the fully nonlinear theory. The current record is more fragmented: optimal regularity and $\gamma\in(1,2)$38 regularity of the regular free boundary are available for the one-phase fully nonlinear equation (Wu et al., 2020); tangential touch at the fixed boundary is available for $\gamma\in(1,2)$39 in the half-space problem (Wang et al., 2 Sep 2025); existence and optimal $\gamma\in(1,2)$40 estimates are available for degenerate free transmission problems (Huaroto et al., 2020, Jesus, 2021); and $\gamma\in(1,2)$41 plus global half-space classification are available for strong solutions of the two-operator transmission problem (Pimentel et al., 2020). The second misconception is to expect a complete geometric theory comparable to the classical variational case. Several papers explicitly leave uniqueness for the original transmission problem, density and nondegeneracy of phases, and regularity of the free boundary $\gamma\in(1,2)$42 as open (Huaroto et al., 2020, Jesus, 2021).

This suggests that the fully nonlinear Alt–Phillips problem is presently best understood as a viscosity-based program rather than a finished theory. Its established core consists of scaling laws, compactness, perturbation to homogeneous uniformly elliptic equations, sharp interior regularity, and selected geometric statements at regular or boundary-contact points. Its main unresolved directions are uniqueness and classification of blow-ups in the genuinely fully nonlinear setting, a replacement for variational monotonicity and epiperimetric methods, and higher-order regularity or stratification of the free boundary beyond the regular set (Wu et al., 2020, Wang et al., 2 Sep 2025, Carducci et al., 15 Jun 2026).