Lasry–Lions Double Envelope

• Lasry–Lions double envelope is a two-sided regularization mechanism that employs alternating infimal and supremal envelope operations to transform bounded functions into explicit C¹,¹ approximants.
• It enforces semiconcavity and semiconvexity simultaneously through a pinching principle, ensuring convergence to the original function and uniform error control as the regularization parameter diminishes.
• Its framework extends across various fields—including Hamilton–Jacobi theory, free-boundary price formation, and nonconvex optimization—providing smooth surrogates and robust comparison principles.

The Lasry–Lions double envelope denotes a two-sided regularization mechanism built from alternating infimal and supremal envelope operations. In its classical form on a Hilbert space, it packages the Lasry–Lions inf/sup-convolution regularization into a symmetric operator

$R_t := \check T_t \circ T_{2t} \circ \check T_t,$

where

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$

This operator yields explicit $C^{1,1}$ approximants, preserves order through a pinching principle, and provides a direct proof of Ilmanen’s insertion lemma (Bernard, 2010). The expression “Lasry–Lions double envelope” is also used more broadly for structurally analogous two-sided envelope constructions in price-formation free boundary models, Hamilton–Jacobi theory, Riemannian and Lorentzian geometry, viscosity methods for fully nonlinear PDE, and nonconvex optimization (Caffarelli et al., 2011, Chen et al., 2015, Azagra et al., 2014, Metsch, 22 Jun 2026, Crasta et al., 2019, Simões et al., 2021).

1. Classical operator and $C^{1,1}$ regularization

In the Hilbert-space setting of bounded functions $u:H\to\mathbb R$, the basic Lasry–Lions envelopes are the inf-convolution $T_t$ and sup-convolution $\check T_t$. They satisfy

$T_t(-u) = -\check T_t(u),$

and form semigroups: $$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$ The symmetric double envelope emphasized in "Lasry-Lions regularization and a Lemma of Ilmanen" (Bernard, 2010) is

$$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$

The classical Lasry–Lions theorem recorded there states that if $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$0 is bounded, then for $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$1, the function

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$2

is $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$3; if $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$4 is uniformly continuous, then $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$5 uniformly as $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$6 (Bernard, 2010). The paper’s reformulation replaces the asymmetric two-parameter expression $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$7 by the symmetric operator $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$8, whose principal advantage is a transparent comparison principle.

The regularity mechanism is expressed in terms of semiconcavity and semiconvexity. A function $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$9 is $C^{1,1}$0-semiconcave if

$C^{1,1}$1

is concave, and $C^{1,1}$2-semiconvex if $C^{1,1}$3 is $C^{1,1}$4-semiconcave. The relevant characterization is that a bounded function $C^{1,1}$5 is $C^{1,1}$6-semiconcave iff it lies in the image of $C^{1,1}$7, while a continuous function is $C^{1,1}$8 iff it is both semiconcave and semiconvex (Bernard, 2010). A key lemma states that if $C^{1,1}$9 is continuous and both $C^{1,1}$0-semiconcave and $C^{1,1}$1-semiconvex, then $C^{1,1}$2, and the gradient is Lipschitz with constant $C^{1,1}$3 (Bernard, 2010). Accordingly, the double envelope regularizes by forcing both one-sided second-order controls simultaneously.

The fundamental theorem is that for every bounded $C^{1,1}$4 and every $C^{1,1}$5,

$C^{1,1}$6

is $C^{1,1}$7 (Bernard, 2010). This makes the double envelope an explicit approximation-to-identity with second-order regularity.

2. Algebraic structure, approximation, and pinching

The elementary envelopes already encode one-sided curvature. The Hilbert-space theory proves that $C^{1,1}$8 is $C^{1,1}$9-semiconcave and $u:H\to\mathbb R$0 is $u:H\to\mathbb R$1-semiconvex. More sharply, if $u:H\to\mathbb R$2 is $u:H\to\mathbb R$3-semiconcave, then for each $u:H\to\mathbb R$4,

$u:H\to\mathbb R$5

and if $u:H\to\mathbb R$6 is $u:H\to\mathbb R$7-semiconvex, then for each $u:H\to\mathbb R$8,

$u:H\to\mathbb R$9

(Bernard, 2010). These facts are proved by direct convexity arguments on the quadratic penalization.

For uniformly continuous data, both $T_t$0 and $T_t$1 converge uniformly to $T_t$2 as $T_t$3. More generally, if $T_t$4 is $T_t$5-continuous with modulus

$T_t$6

then there exists a nondecreasing error function $T_t$7 such that

$T_t$8

with explicit estimate

$T_t$9

(Bernard, 2010). The envelopes preserve the same modulus of continuity.

A central algebraic fact is the fixed-point criterion

$\check T_t$0

with equality iff $\check T_t$1 is $\check T_t$2-semiconvex. Similarly,

$\check T_t$3

with equality iff $\check T_t$4 is $\check T_t$5-semiconvex (Bernard, 2010). The proof uses the Legendre–Fenchel biconjugate viewpoint: $\check T_t$6 is the convex biconjugate of $\check T_t$7. This identifies the double envelope not merely as a smoothing map but as a convex-analytic projector onto functions having controlled second-order geometry.

The comparison principle most characteristic of the symmetric operator is the pinching property. If $\check T_t$8 and $\check T_t$9 are $T_t(-u) = -\check T_t(u),$0-semiconcave and

$T_t(-u) = -\check T_t(u),$1

then for every $T_t(-u) = -\check T_t(u),$2,

$T_t(-u) = -\check T_t(u),$3

Thus a function squeezed between semiconcave barriers remains squeezed after regularization (Bernard, 2010). An immediate consequence is stationarity on sufficiently regular data: if $T_t(-u) = -\check T_t(u),$4 is already $T_t(-u) = -\check T_t(u),$5, then for $T_t(-u) = -\check T_t(u),$6 small enough,

$T_t(-u) = -\check T_t(u),$7

This pinching statement is the mechanism by which the paper derives Ilmanen’s insertion lemma: if $T_t(-u) = -\check T_t(u),$8 and $T_t(-u) = -\check T_t(u),$9 are bounded functions on $$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$0 with $$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$1, and both $$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$2 and $$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$3 are semiconcave, then there exists a $$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$4 function $$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$5 such that

$$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$6

(Bernard, 2010).

3. Intrinsic and geometric variants

In Hamilton–Jacobi theory, the quadratic kernel is replaced by an action functional. For a Tonelli Lagrangian, the fundamental solution

$$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$7

induces the Lax–Oleinik operators

$$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$8

In the quadratic case, these coincide with the classical Lasry–Lions sup/inf convolutions. "Lasry-Lions, Lax-Oleinik and Generalized characteristics" (Chen et al., 2015) treats these operators as the Hamiltonian realization of the Lasry–Lions regularization. For semiconcave $$T_t\circ T_s = T_{t+s}, \qquad \check T_t\circ \check T_s = \check T_{t+s}.$$9, $$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$0 is $$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$1 for small $$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$2, converges to $$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$3 as $$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$4, and its local maximizers generate generalized characteristics. The derivative of the regularized function selects the unique element $$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$5 minimizing $$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$6 over the superdifferential (Chen et al., 2015).

For discounted Hamilton–Jacobi equations,

$$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$7

"Lasry-Lions approximations for discounted Hamilton-Jacobi equations" (Chen et al., 2017) replaces the Euclidean quadratic kernel by the fundamental solution $$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$8 of the time-dependent Lagrangian $$R_t f := \check T_t\bigl(T_{2t}(\check T_t f)\bigr).$$9, and defines the intrinsic regularization

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$00

If $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$01 is a viscosity solution, then $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$02 is of class $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$03, converges uniformly to $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$04 as $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$05, and its gradients converge to the unique $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$06 satisfying

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$07

(Chen et al., 2017). This preserves the selection principle familiar from the quadratic theory while encoding the Hamiltonian dynamics directly into the kernel.

The Riemannian manifold extension replaces squared norms by squared geodesic distances. For a manifold $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$08, the intrinsic double envelope is

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$09

Under bounded sectional curvature, positive injectivity radius, and positive convexity radius, every bounded uniformly continuous $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$10 can be uniformly approximated by globally $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$11 functions $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$12 for $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$13 (Azagra et al., 2014). The paper formulates $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$14 intrinsically through Lipschitz control of $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$15 under parallel transport, and proves preservation of ordering, infima, minimizers, invariance under isometries, and Lipschitz continuity; convexity is also preserved when $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$16 (Azagra et al., 2014). Counterexamples show that global $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$17 regularization can fail without boundedness of $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$18 or bounded sectional curvature.

A Lorentzian analogue appears in "A Lorentzian Lasry-Lions regularization theorem" (Metsch, 22 Jun 2026). On a globally hyperbolic spacetime, with causal action

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$19

the forward and backward Lax–Oleinik operators are

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$20

If the forward evolution $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$21 is locally semiconcave and has future-directed timelike superdifferentials near a point, then for $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$22 near $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$23 and sufficiently small $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$24, the double evolution $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$25 is $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$26 near that point (Metsch, 22 Jun 2026). This is the Lorentzian counterpart of the classical result $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$27 for $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$28.

4. Free-boundary and price-formation interpretation

A different but related use of the term occurs in the Lasry–Lions price-formation model. In "On a price formation free boundary model by Lasry & Lions" (Caffarelli et al., 2011), the unknown is a sign-changing density $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$29 and a free boundary $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$30, interpreted as the transaction price or equilibrium price. The model is

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$31

with

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$32

The paper constructs a transformed function

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$33

and describes this as a “double envelope” construction: on the left of the interface one stacks translated positive parts, while on the right one stacks translated negative parts (Caffarelli et al., 2011). The transformed function satisfies the heat equation

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$34

in the distributional sense, and the free boundary becomes the zero level set of $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$35. Conversely, if the zero level set of a suitably prepared heat solution $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$36 is a smooth graph $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$37, one reconstructs the original $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$38 from discrete differences of $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$39 (Caffarelli et al., 2011).

This usage differs from the Hilbert-space operator $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$40, but the structural idea is analogous: a nonlinear interface problem is encoded by a two-sided envelope transformation whose output evolves linearly. The paper proves global existence of a unique smooth solution and continuity of the price path $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$41, and derives asymptotics: if the total positive and negative masses are unequal, then

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$42

where $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$43 is determined by

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$44

if the masses are balanced, then $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$45 remains bounded and converges to a constant (Caffarelli et al., 2011).

The same free-boundary structure appears as the large-transaction-rate limit of a Boltzmann-type kinetic model in "On a Boltzmann type price formation model" (Burger et al., 2013). There, the unfolded quantities

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$46

lead to a heat-equation potential $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$47, and in the limit $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$48 the transaction measure concentrates to

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$49

under a compatibility condition (Burger et al., 2013). The limiting densities are recovered by

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$50

which matches the envelope viewpoint of the earlier price-formation analysis (Burger et al., 2013).

5. Envelope methods in nonlinear PDE and viscosity theory

The Lasry–Lions and Alvarez–Lasry–Lions envelope methodology also appears as a substitute for unavailable variational structure. In "The Brunn-Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators" (Crasta et al., 2019), the problem is the Dirichlet eigenvalue equation

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$51

for fully nonlinear homogeneous elliptic $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$52. Because there is no variational characterization of $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$53 in general, the proof uses the transformation $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$54, the operator

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$55

and the weighted infimal convolution

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$56

on Minkowski sums of domains (Crasta et al., 2019).

In that paper, “double-envelope” refers to the Lasry–Lions / Alvarez–Lasry–Lions envelope mechanism in two related senses: the infimal convolution envelope used for the Brunn–Minkowski inequality, and the convex envelope $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$57 used for log-concavity of eigenfunctions (Crasta et al., 2019). The crucial technical input is viscosity stability under the envelope, encoded by second-order subjet transfer inequalities. This construction yields a viscosity supersolution on the Minkowski sum domain, from which the maximum principle gives the eigenvalue inequality

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$58

(Crasta et al., 2019). Under stronger assumptions, the convex envelope argument shows that positive eigenfunctions are log-concave (Crasta et al., 2019).

This usage broadens the notion of Lasry–Lions double envelope from explicit smoothing of a single function to a PDE mechanism that propagates differential inequalities through envelope constructions. A plausible implication is that the essential content of the method is not tied to a particular formula such as $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$59, but to the interaction between one-sided regularization, convexification, and comparison principles.

A common source of terminological confusion is the appearance of Lasry–Lions monotonicity in mean field game theory. "Mean Field Game Master Equations with Anti-monotonicity Conditions" (Mou et al., 2022) concerns the sign of monotonicity inequalities in master equations and explicitly states that it does not use, develop, or rely on the Lasry–Lions double envelope construction. The relation is therefore nominal rather than methodological.

6. Nonconvex optimization and homotopy formulations

In optimization, the double envelope is a smooth surrogate for nonsmooth, nonconvex terms. For a proper lsc function $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$60, the Moreau envelope is

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$61

and the Lasry–Lions double envelope is

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$62

For smoothness, one typically requires $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$63, where $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$64 is the prox-boundedness threshold of $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$65 (Simões et al., 2021).

The envelope satisfies the bridging inequalities

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$66

Accordingly, it interpolates between a Moreau-type lower relaxation and the original nonconvex function (Simões et al., 2021). For $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$67, one has

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$68

with Lipschitz gradient. The paper also gives the stationarity relation

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$69

(Simões et al., 2021). These properties underlie a homotopy strategy for composite minimization

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$70

where $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$71 is $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$72 and $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$73 is proper, lsc, prox-bounded, and possibly nonsmooth and nonconvex. One solves instead

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$74

along a continuation path $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$75, with epi-convergence guaranteeing convergence of minima and cluster points of approximate minimizers (Simões et al., 2021). The paper reports applications to signal decoding and spectral unmixing, using L-BFGS as the inner smooth solver (Simões et al., 2021).

"A Lasry-Lions envelope approach for mathematical programs with complementarity constraints" (Wang et al., 7 Jul 2025) applies the same principle to the indicator $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$76 of the complementarity set

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$77

The Moreau envelope is

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$78

and the Lasry–Lions double envelope is

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$79

For prox-bounded $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$80, $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$81 is smooth with Lipschitz gradient, and

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$82

(Wang et al., 7 Jul 2025). For the complementarity indicator, the resulting surrogate is everywhere smooth, unlike the standard Moreau envelope of $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$83, which is only piecewise smooth and nonsmooth along the diagonal in the positive orthant (Wang et al., 7 Jul 2025).

The smoothed MPCC subproblem is

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$84

or equivalently

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$85

with $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$86 (Wang et al., 7 Jul 2025). The method solves a sequence of smooth subproblems while decreasing $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$87. Under BCCQ, cluster points are C-stationary; under well-behaved complementarity constraints and BCQ$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$88, they are M-stationary (Wang et al., 7 Jul 2025). The paper also provides a worst-case gradient-evaluation complexity of

$$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$89

for computing an $$T_t u(x) := \inf_{y\in H}\left(u(y)+\frac{|y-x|^2}{t}\right), \qquad \check T_t u(x) := \sup_{y\in H}\left(u(y)-\frac{|y-x|^2}{t}\right).$$90-approximate C-stationary point (Wang et al., 7 Jul 2025).

Across these optimization papers, the double envelope is no longer mainly a proof device for semiconcavity theory. It becomes a tunable smooth approximation that preserves minimizers, epi-converges to the original objective, and supports continuation methods on nonconvex landscapes. This suggests a modern reinterpretation of the Lasry–Lions double envelope as a general-purpose bridge between exact nonsmooth models and tractable smooth surrogates.