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''Moreau envelope''

Let $H$ be a real [[Hilbert space|Hilbert space]] and let $f : H \rightarrow ( - \infty , + \infty ]$ be a lower semi-continuous extended-real-valued function (cf. also [[Continuous function|Continuous function]]) such that for a certain $T &gt; 0$,

\begin{equation*} \operatorname { inf } _ { x \in H } \left( f ( x ) + ( 2 T ) ^ { - 1 } \| x \| ^ { 2 } \right) \end{equation*}

is finite. For $t &gt; 0$, the Moreau envelope function $f _ { t }$ is defined by infimal convolution of $f$ with $( 2 t ) ^ { - 1 } \| . \| ^ { 2 }$, i.e.,

\begin{equation} \tag{a1} f _ { t } ( x ) = \operatorname { inf } _ { y \in H } \left( f ( y ) + \frac { 1 } { 2 t } \| x - y \| ^ { 2 } \right) , \quad x \in H. \end{equation}

This operation amounts geometrically to performing vector addition of the strict epigraphs of $f$ and $( 2 t ) ^ { - 1 } \| . \| ^ { 2 }$. The Moreau envelopes are usually utilized as approximants of $f$, although regularization was not the purpose of the seminal paper [[#References|[a12]]].

If $t \in ( 0 , T )$, $f _ { t }$ is everywhere finite and Lipschitz continuous (cf. also [[Lipschitz condition|Lipschitz condition]]) on bounded sets. Moreover, $f _ { t }$ increases pointwise to $f$ as $t$ decreases to $0$; the convergence is in fact uniform on bounded sets (cf. also [[Uniform convergence|Uniform convergence]]) when $f$ is uniformly continuous on bounded sets (cf. also [[Uniform continuity|Uniform continuity]]). One might expect that under some additional assumptions on $f$ the differentiability properties of the square of the norm in $H$ should to some extent carry over to $f _ { t }$, thus giving rise to a smooth regularization of $f$. This is true in the presence of convexity.

==The convex case.==
Suppose $f + ( 2 T ) ^ { - 1 } \| . \| ^ { 2 }$ is convex (cf. also [[Convex function (of a real variable)|Convex function (of a real variable)]]). Then $f _ { t }$ is differentiable and $d f _ { t }$ is globally Lipschitz continuous of rate $\operatorname { max } \{ 1 / t , 1 / ( T - t ) \}$ when $t \in ( 0 , T )$. Furthermore, $u ( t , x ) = f _ { t } ( x )$ is a classical solution of

\begin{equation*} \frac { \partial u ( t , x ) } { \partial t } + \frac { 1 } { 2 } \| d _ { x } u ( t , x ) \| ^ { 2 } = 0 , \quad ( t , x ) \in ( 0 , T ) \times H. \end{equation*}

Let the subdifferential of $f$ be defined by

\begin{equation*} \partial f ( x ) = \partial _ { c } \left( f + ( 2 T ) ^ { - 1 }  \| \cdot \| ^ { 2 } \right) ( x ) - T ^ { - 1 } x , \quad x \in H, \end{equation*}

where the first term at the right-hand side is the subdifferential in the sense of convex analysis of the convex function $f + ( 2 T ) ^ { - 1 } \| . \| ^ { 2 }$ ($\xi \in \partial _ { c } g ( x )$ means that $g ( x )$ is finite and $g ( y ) \geq g ( x ) + \langle y - x , \xi \rangle$ for all $y$). The infimum (a1) is achieved at a unique point $y$, which is denoted by $R _ { t } ( x )$ and is given by $R _ { t } ( x ) = ( I + t \partial f ) ^ { - 1 } ( x )$. Furthermore,

\begin{equation*} d f _ { t } = t ^ { - 1 } ( I - R _ { t } ) = ( ( \partial f ) ^ { - 1 } + t I ) ^ { - 1 }, \end{equation*}

which justifies the alternative term Moreau–Yosida approximation, since $( ( \partial f ) ^ { - 1 } + t l ) ^ { - 1 }$, $t &gt; 0$, are the Yosida approximants of $\partial f$. Moreover, $d f _ { t } \rightarrow \partial f$ in the sense of Kuratowski–Painlevé convergence of graphs in $H \times H$, while $d f _ { t } ( x )$ converges to the element of $\partial f ( x )$ of minimal norm unless $\partial f ( x )$ is empty. Stationary points and values are preserved; as a matter of fact, if $( t , x ) \in ( 0 , T ) \times H$, then

\begin{equation*} d f _ { t } ( x ) = 0 \Leftrightarrow \partial f ( x ) \ni 0 \Leftrightarrow f _ { t } ( x ) = f ( x ). \end{equation*}

For various applications and further properties, consult [[#References|[a3]]], [[#References|[a4]]], [[#References|[a1]]], [[#References|[a2]]].

==Regularization in the non-convex case.==
If one insists on a smooth regularization, the Moreau envelopes cannot be used for arbitrary functions. This is however not a serious drawback. It is easy to see that $- f _ { t } + ( 2 t ) ^ { - 1 } \| . \| ^ { 2 }$ is always a convex function, so that $f _ { t , s } : = - ( - f _ { t } ) _ { s }$ is smooth when $0 < s < t < T$; in fact, $d f _ { t ,\, s }$ is globally Lipschitz continuous of rate $\operatorname { max } \{ 1 / s , 1 / ( t - s ) \}$. Explicitly,

\begin{equation*} f _ { t , s } ( x ) = \operatorname { sup } _ { z \in H } \operatorname { inf } _ { y \in H } \left( f ( y ) + \frac { 1 } { 2 t } \| z - y \| ^ { 2 } - \frac { 1 } { 2 s } \| x - z \| ^ { 2 } \right) \end{equation*}

for all $x \in H$.

Moreover, $f _ { t - s } \leq f _ { t , s } \leq f$ and hence $f _ { t , s } \rightarrow f$ pointwise as $0 < s < t \rightarrow 0$. The double envelopes $f _ { t , s }$, frequently called the Lasry–Lions approximants of $f$, were introduced and investigated by J.-M. Lasry and P.-L. Lions in [[#References|[a10]]]; see also [[#References|[a4]]], [[#References|[a13]]]. One can prove that the equation $f _ { t , s } = f _ { t - s }$ is true for all $0 < s < t < T$ exactly when $f + ( 2 T ) ^ { - 1 } \| . \| ^ { 2 }$ is a convex function [[#References|[a13]]], in which case therefore the approximation method reduces to the previous one.

==Connections with the Hamilton–Jacobi equation.==
As stated above, $u ( t , x ) = f _ { t } ( x )$ furnishes a classical solution of the initial-value problem

\begin{equation*} \frac { \partial u ( t , x ) } { \partial t } + \frac { 1 } { 2 } \| d _ { x } u ( t , x ) \| ^ { 2 } = 0 , \quad ( t , x ) \in ( 0 , T ) \times H, \end{equation*}

\begin{equation*} \operatorname { lim } _ { t \downarrow 0 } u ( t , x ) = f ( x ) \quad \text { for all } x \in H, \end{equation*}

if $f + ( 2 T ) ^ { - 1 } \| . \| ^ { 2 }$ is a convex function. Let, now, $H = {\bf R} ^ { n }$ and drop the convexity hypothesis on $f$. While being non-differentiable in general, $u ( t , x ) = f _ { t } ( x )$ is nonetheless locally Lipschitz continuous in $( 0 , T ) \times \mathbf{R} ^ { n }$ and is known to be the unique viscosity solution of the above initial-value problem [[#References|[a11]]], [[#References|[a14]]]. (This notion of a generalized solution allows merely continuous functions to be solutions, [[#References|[a6]]]; cf. [[Viscosity solutions|Viscosity solutions]]). In this context, (a1) is referred to as the Lax formula; the Lax formula is intimately related to the Hopf formula for conservation laws, see [[#References|[a9]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a11]]].

==Extensions to Banach spaces.==
The Lasry–Lions regularization scheme has been extended to certain classes of Banach spaces $( X , \| \, .\,  \| )$. For the case where $\| .\|$ and the dual norm $\| . \|_{*}$ are simultaneously locally uniformly rotund, an approach by means of the Legendre–Fenchel transformation has been taken in [[#References|[a13]]]. Another extension appears in [[#References|[a5]]], under the hypothesis that $X$ be super-reflexive (cf. also [[Reflexive space|Reflexive space]]).

For more on the theme of regularization, see (the editorial comments to) [[Regularization|Regularization]] and [[Regularization method|Regularization method]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  I. Ekeland,   J.M. Lasry,   "On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface"  ''Ann. of Math.'' , '''112'''  (1980)  pp. 283–319</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R.T. Rockafellar,   R.J.-B. Wets,   "Variational analysis" , Springer  (1998)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  H. Attouch,   "Variational convergence for functions and operators" , ''Applicable Math.'' , Pitman  (1984)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  H. Attouch,   D. Azé,   "Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry–Lions method"  ''Ann. Inst. H. Poincaré Anal. Non Lin.'' , '''10'''  (1993)  pp. 289–312</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. Cepedello–Boiso,   "On regularization in superreflexive Banach spaces by infimal convolution formulas"  ''Studia Math.'' , '''129'''  (1998)  pp. 265–284</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  M.G. Crandall,   P.-L. Lions,   "Viscosity solutions of Hamilton–Jacobi equations"  ''Trans. Amer. Math. Soc.'' , '''277'''  (1983)  pp. 1–42</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  E. Hopf,   "The partial differential equation $u _ { t } + u u _ { x } = \mu u _ { xx }$"  ''Commun. Pure Appl. Math.'' , '''3'''  (1950)  pp. 201–230</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  E. Hopf,   "Generalized solutions of non-linear equations of first order"  ''J. Math. Mech.'' , '''14'''  (1965)  pp. 951–973</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  P.D. Lax,   "Hyperbolic systems of conservation laws II"  ''Commun. Pure Appl. Math.'' , '''10'''  (1957)  pp. 537–566</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  J.-M. Lasry,   P.-L. Lions,   "A remark on regularization in Hilbert spaces"  ''Israel J. Math.'' , '''55'''  (1986)  pp. 257–266</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  P.-L. Lions,   "Generalized solutions of Hamilton–Jacobi equations" , ''Res. Notes Math.'' , '''69''' , Pitman  (1982)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  J-J. Moreau,   "Proximité et dualité dans un espace hilbertien"  ''Bull. Soc. Math. France'' , '''93'''  (1965)  pp. 273–299</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  T. Strömberg,   "On regularization in Banach spaces"  ''Ark. Mat.'' , '''34'''  (1996)  pp. 383–406</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  T. Strömberg,   "Hopf's formula gives the unique viscosity solution"  ''Math. Scand.''  (submitted)</td></tr></table>