On the Weak Convergence of Monge-Ampère Measures for Discrete Convex Mesh Functions

Abstract

To a mesh function we associate the natural analogue of the Monge-Ampère measure. The latter is shown to be equivalent to the Monge-Ampère measure of the convex envelope. We prove that the uniform convergence to a bounded convex function of mesh functions implies the uniform convergence on compact subsets of their convex envelopes and hence the weak convergence of the associated Monge-Ampère measures. We also give conditions for mesh functions to have a subsequence which converges uniformly to a convex function. Our result can be used to give alternate proofs of the convergence of some discretizations for the second boundary value problem for the Monge-Ampère equation and was used for a recently proposed discretization of the latter. For mesh functions which are uniformly bounded and satisfy a convexity condition at the discrete level, we show that there is a subsequence which converges uniformly on compact subsets to a convex function. The convex envelopes of the mesh functions of the subsequence also converge uniformly on compact subsets. If in addition they agree with a continuous convex function on the boundary, the limit function is shown to satisfy the boundary condition strongly.

Appendix

In [10] convergence of numerical schemes for discrete approximations to viscosity solutions of the Dirichlet problem for dynamic programming principles for the $p$-Laplacian is given. The proof is based on a numerical analysis approach, the Barles-Souganidis framework which consists in checking stability, consistency and monotonicity of the numerical scheme. It is known that the Barles-Souganidis framework requires the so-called strong uniqueness property for the differential equation, which is a comparison principle for equations with the boundary condition in the viscosity sense. In [10] the strong uniqueness property for the Laplace equation is proved for smooth domains and convergence of the numerical scheme on a bounded Lipschitz domain is obtained through barriers on appropriate shrinking rings. We review below their approach for the standard discretization of the Laplace equation.

Recall that $\varOmega \subset \mathbb{R}^{d}$ is a bounded Lipschitz domain and $g \in C(\partial \varOmega )$. We consider the problem

$$\begin{aligned} \begin{aligned} -\Delta u & = 0 \ \text{ in } \varOmega \\ u & = g \ \text{ on } \partial \varOmega . \end{aligned} \end{aligned}$$

(28)

We define for $h>0$ the following sets:

Outer boundary strip: $\varGamma _{h} = \{ \, x \in \mathbb{R}^{d}\setminus \varOmega , d(x, \partial \varOmega ) \geq h\, \}$ and put $O=\varGamma _{1}$.

Inner boundary strips: $I_{h} = \{ \, x \in \varOmega , d(x,\partial \varOmega ) \leq h \, \}$ and put $I=I_{1}$.

Extended domain: $\widetilde{\varOmega } = \varOmega \cup O$ and extended computational domain $\widetilde{\varOmega }_{h} = \varOmega _{h} \cup O$.

Let $G$ be a continuous extension of $g$ to $\widetilde{\varOmega }$. Note that $\partial \varOmega _{h} \subset O$. Given a mesh function $u_{h}$, we extend it to $\varOmega _{h} \cup O$ by $u_{h}(x)=G(x)$ for all $x \in O \setminus \partial \varOmega _{h}$. Analogous to [10, (2.9)], we consider the discrete problem: find a mesh function $u_{h}$ such that

$$\begin{aligned} \begin{aligned} -\Delta _{h} u_{h} & = 0 \ \text{ on } \varOmega _{h} \\ u_{h} & = G \ \text{ on } O. \end{aligned} \end{aligned}$$

(29)

A mesh function solves (27) if and only if it solves (29).

In [10] a notion of viscosity solution of (28) is first given. The authors therein recall the existence and uniqueness of such a viscosity solution. They then introduce a generalized version of viscosity solution where the boundary condition is assumed in a weaker sense. That notion has become known as boundary condition in the viscosity sense.

An upper semi-continuous function on $\overline{\varOmega }$ is a viscosity subsolution of (28) if whenever $x_{0} \in \overline{\varOmega }$ and $\phi \in C^{2}(\overline{\varOmega })$ satisfy

$$ \phi (x_{0}) = u(x_{0}), (u-\phi )(x)< (u-\phi )(x_{0}) \text{ for } x \neq x_{0}, $$

we have

$$\begin{aligned} -\Delta \phi (x_{0}) & \leq 0 \ \text{ if } x_{0} \in \varOmega \end{aligned}$$

(30)

$$\begin{aligned} u(x_{0}) - g(x_{0}) & \leq 0 \ \text{ if } x_{0} \in \partial \varOmega . \end{aligned}$$

(31)

If the condition (31) at the boundary is replaced by

$$ \min \{ \, -\Delta \phi (x_{0}), u(x_{0}) - g(x_{0}) \, \} \leq 0, x_{0} \in \partial \varOmega , $$

(32)

the function $u$ is said to be a generalized viscosity subsolution of (28).

A lower semi-continuous function on $\overline{\varOmega }$ is a viscosity supersolution of (28) if whenever $x_{0} \in \overline{\varOmega }$ and $\phi \in C^{2}(\overline{\varOmega })$ satisfy

$$ \phi (x_{0}) = u(x_{0}), (u-\phi )(x)> (u-\phi )(x_{0}) \text{ for } x \neq x_{0}, $$

we have

$$\begin{aligned} -\Delta \phi (x_{0}) & \geq 0 \ \text{ if } x_{0} \in \varOmega \end{aligned}$$

(33)

$$\begin{aligned} u(x_{0}) - g(x_{0}) & \geq 0 \ \text{ if } x_{0} \in \partial \varOmega . \end{aligned}$$

(34)

If the condition (34) at the boundary is replaced by

$$ \max \{ \, -\Delta \phi (x_{0}), u(x_{0}) - g(x_{0}) \, \} \geq 0, x_{0} \in \partial \varOmega , $$

(35)

the function $u$ is said to be a generalized viscosity supersolution of (28).

A function $u \in C(\overline{\varOmega })$ is a viscosity solution of (28) if it is both a viscosity subsolution and a viscosity supersolution of (28). Note that for this notion the boundary condition is taken in the usual sense.

Theorem 15

[10, Theorem 3.4] Let $\varOmega \subset \mathbb{R}^{d}$ be a bounded Lipschitz domain and $g\in C(\partial \varOmega )$. If $u$ is a viscosity subsolution of (28) and $v$ a supersolution of (28), then $u\leq v$ on $\overline{\varOmega }$.

The authors in [10] proved the strong uniqueness property for smooth domains.

Proposition 1

[5, 10] Let $\varOmega \subset \mathbb{R}^{d}$ be a $C^{2}$ domain and $g \in C(\partial \varOmega )$. Let $u$ and $v$ be respectively generalized viscosity subsolution and supersolution of (28). Then $u \leq v$ in $\overline{\varOmega }$.

Using standard arguments we review below, the above proposition allows for $\varOmega$ a $C^{2}$ domain to claim the convergence of the solution $u_{h}$ of (29) to the viscosity solution of (28). We define for a $C^{2}$ function $\phi$ on $\varOmega _{E}$

$$\begin{aligned} S(h,x,\phi (x),\phi ) &= \textstyle\begin{cases} - \sum _{i=1}^{d} \Delta _{e_{i}} \phi (x), & x \in \varOmega \\ \phi (x)-G(x), & x \in O, \end{cases}\displaystyle \end{aligned}$$

with the operator $\Delta _{e}$ defined as for (9). We write (28) in the standard form $S(h,x,u_{h}(x),u_{h})=0$, $x \in \widetilde{\varOmega }_{h}$ with a slight abuse of notation. We have the following analogues of [10, (2.11)–(2.12)].

If $u_{h}$ solves (28), we have

$$ \inf _{O} G \leq u_{h}(x) \leq \sup _{O} G, x \in \varOmega _{h}. $$

(36)

Let $u_{h}^{1}$ and $u_{h}^{2}$ solve

$$\begin{aligned} -\Delta _{h} u_{h}^{1} & \leq 0 \text{ in } \varOmega _{h} \text{ and } u_{h}^{1} = G^{1} \text{ on } O \\ -\Delta _{h} u_{h}^{2} & \geq 0 \text{ in } \varOmega _{h} \text{ and } u_{h}^{2} = G^{2} \text{ on } O, \end{aligned}$$

with $G^{1} \leq G^{2}$. Then

$$ u_{h}^{1} \leq u_{h}^{2} \text{ on } \widetilde{\varOmega }_{h}. $$

(37)

Equation (36) says that the scheme is stable. It is a consequence of properties of the matrix of the discrete linear problem (28) [15, Theorem 4.77]. The proof is analogous to [15, Remark 4.37].

Equation (37) is the discrete comparison principle. Again, it is a consequence of [15, Theorem 4.77]. The proof is analogous to [15, Theorem 4.38 b].

Next, we recall the monotonicity of the scheme, i.e. if $u_{h} \leq v_{h}$ on $\widetilde{\varOmega }_{h}$ with $u_{h}(x)=v_{h}(x)$ we have $S(h,x,u_{h}(x),u_{h})\leq S(h,x,v_{h}(x),v_{h})$.

Next, we describe the form of consistency of the scheme needed to prove convergence when the boundary condition is taken in the viscosity sense. For $x \in \overline{\varOmega }$ and $\phi \in C^{2}(\varOmega _{E})$

$$\begin{aligned} \limsup _{h \to 0, y\to x, \zeta \to 0} S(h,y, \phi (y)+\zeta , \phi + \zeta ) = \textstyle\begin{cases} -\Delta \phi (x) & \text{ if } x \in \varOmega \\ \max \{ \, -\Delta \phi (x), \phi (x)-G(x) \, \} & \text{ if } x \in \partial \varOmega \end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned} \liminf _{h \to 0, y\to x, \zeta \to 0} S(h,y, \phi (y)+\zeta , \phi + \zeta ) = \textstyle\begin{cases} -\Delta \phi (x) & \text{ if } x \in \varOmega \\ \min \{ \, -\Delta \phi (x), \phi (x)-G(x) \, \} & \text{ if } x \in \partial \varOmega . \end{cases}\displaystyle \end{aligned}$$

For $x\in \varOmega$, we have the usual consistency property. For $x \in \partial \varOmega$, one can approach $x$ with points $y$ in either $\varOmega$ or $O$.

Theorem 16

Assume that $\varOmega$ is a $C^{2}$ domain. The solution $u_{h}$ of (29) converges uniformly on $\overline{\varOmega }$ to the viscosity solution of (28).

Proof

Since the scheme is stable, consistent and monotone, it follows from Proposition 1 and the framework in [6] that the solution $u_{h}$ of (29) converges uniformly on $\overline{\varOmega }$ to the viscosity solution of (28). □

To handle the case $\varOmega$ Lipschitz, a delicate treatment at the boundary is done in [10] using barriers on appropriate shrinking rings. Denote by $B_{r}(x)$ the open ball of center $x$ and radius $r$. The following regularity condition for Lipschitz domains is the one used in the proof.

There exists $\overline{\delta } >0$ and $\mu \in (0,1)$ such that for every $\delta \in (0,\overline{\delta })$ and $y \in \partial \varOmega$, there exists a ball $B_{\mu \delta }(z)$ strictly contained in $B_{\delta }(y) \setminus \varOmega$. The constant $\mu$ is independent of $y \in \partial \varOmega$.

We have the following analogue of [10, Corollary 4.5]

Lemma 17

Given $\eta >0$, there exist $\delta =\delta (\eta ,G,\overline{\delta })$, $k_{0}=k_{0}(\eta ,\mu ,G)$, $h_{0}=h_{0}(\eta ,\delta ,\mu ,k_{0})$ such that

$$ |u_{h}(x)-G(y)| \leq \frac{\eta }{2}, $$

for all $y \in \partial \varOmega$, $x \in B_{\delta /4^{k_{0}}}(y) \cap \varOmega _{h}$ and $h\leq h_{0}$.

Proof

As in [10], we prove only the one sided inequality $u_{h}(x) \leq G(y) + \eta /2$, the other being similar. Also, we consider only the case $d\neq 2$. The case $d=2$ is treated with similar arguments as indicated on [10, p. 16].

Part I: In this part we collect parts of the proof in [10] which do not deal with dynamic programming. Fix $\delta \in (0,\overline{\delta })$. Let $u_{h}$ solve (29). For $y \in \partial \varOmega$ define

$$ m^{h}(y) = \sup _{B_{5 \delta }(y) \cap \varGamma _{h}} G \text{ and } M^{h} = \sup _{ \varGamma _{h}} G. $$

Define

$$ \theta = \frac{1-\frac{1}{2} \big( \frac{\mu }{2-\mu }\big)^{\xi }-\frac{1}{2} \big( \frac{\mu }{2}\big)^{\xi } }{1-\big( \frac{\mu }{2}\big)^{\xi }} \in (0,1) \text{ with } \xi =d-2, $$

and for $k\geq 0$

$$ M_{k}^{h}(y) = m^{h}(y) + \theta ^{k}( M^{h}-m^{h}(y) ). $$

Define $\delta _{k}=\delta /4^{k-1}$. By the regularity assumption on $\varOmega$, one can find balls $B_{\mu \delta _{k+1}}(z_{k}) \subset B_{\delta _{k+1}}(y) \setminus \varOmega$ for all $k$. In particular $||y-z_{k}|| < \delta _{k+1}$.

For notational convenience, denote $m=m^{h}(y)$, $M=M^{h}$ and $M_{k}=M_{k}^{h}(y)$. We consider the problem

$$\begin{aligned} \Delta U_{k} & = 0 \ \text{in} \ B_{\delta _{k}}(z_{k}) \setminus \overline{B}_{\mu \delta _{k+1}}(z_{k}) \\ U_{k} & = m \ \text{on} \ \partial B_{\mu \delta _{k+1}}(z_{k}) \\ U_{k} & = M_{k} \ \text{on} \ \partial B_{\delta _{k}}(z_{k}). \end{aligned}$$

Define

$$ a= \frac{1-\big( \frac{\mu }{2}\big)^{\xi }}{1-\big( \frac{\mu }{4}\big)^{\xi }} \text{ and } b=1-a $$

and (see [10, Fig. 1])

$$ \varGamma _{1}^{h}= B_{\delta _{k}/2+h}(z_{k}) \cap \varGamma _{h} \text{ and } \varGamma _{2}^{h}= ( B_{\delta _{k}/2+h}(z_{k}) \setminus \overline{B}_{\delta _{k}/2}(z_{k}) ) \cap \varOmega . $$

It is proven in [10, page 17] that

$$ U_{k} \geq a M_{k} + b m \ \text{ in } \varGamma _{2}^{h}, $$

(38)

and that for $x \in B_{(2-\mu ) \delta _{k+1}}(z_{k})$

$$ U_{k}(x) \leq b' m + a'M_{k}, $$

(39)

where

$$ a' = \frac{1-\big( \frac{\mu }{2-\mu }\big)^{\xi }}{1-\big( \frac{\mu }{4}\big)^{\xi }} \text{ and } b' = \frac{\big( \frac{\mu }{2-\mu }\big)^{\xi }-\big( \frac{\mu }{4}\big)^{\xi }}{1-\big( \frac{\mu }{4}\big)^{\xi }}. $$

Part II: We assume in this part that $u_{h} \leq M_{k}$ on $B_{\delta _{k}}(y) \cap \varOmega _{h}$ for all $h< h_{k}$ for a fixed $h_{k}$ and that $M_{k}-m\geq \eta /4$. We prove that there exists $h_{k+1}=h_{k+1}(\eta ,\mu ,\delta ,d,G) \in (0,h_{k})$ such that for all $h< h_{k+1}$, we have

$$ u_{h} \leq M_{k+1}^{h}(y) \ \text{in} \ B_{\delta _{k+1}}(y) \cap \varOmega . $$

For $h \leq \mu \delta _{k+1}/2$ the barrier $U_{k}$ is extended to the ring

$$ R_{k,h} = B_{\delta _{k}+2 h}(z_{k}) \setminus \overline{B}_{\mu \delta _{k+1} - 2 h}(z_{k}). $$

Let $U_{k}^{h}$ be a mesh function which solves

$$\begin{aligned} \Delta _{h} U_{k}^{h} & = 0 \ \text{in} \ R_{k} \\ U_{k}^{h} & = U_{k} \ \text{in} \ R_{k,h}\setminus R_{k}, \end{aligned}$$

where $R_{k}=B_{\delta _{k}}(z_{k}) \setminus \overline{B}_{ \mu \delta _{k+1}}(z_{k})$. Note that $R_{k,h}\setminus R_{k}$ is the outer $h$-neighborhood of $R_{k}$. Since $R_{k}$ is smooth, by Theorem 1, $U_{k}^{h}$ converges uniformly to $U_{k}$ in $R_{k,h}$ as $h \to 0$ (recall that $U_{k}^{h}=U_{k}$ outside $R_{k}$). Therefore, given

$$ \gamma =\gamma (\mu ,d,\eta ) = \frac{1}{4} \frac{\big( \frac{\mu }{\mu -2}\big)^{\xi } - \big( \frac{\mu }{2}\big)^{\xi } }{1-\big( \frac{\mu }{4}\big)^{\xi }} \frac{\eta }{4} >0, $$

there exists $h_{k+1} = h_{k+1} (\gamma )$, $0< h_{k+1} \leq \min \{ \, \mu \delta _{k+1}/2, h_{k} \, \}$ such that

$$ |U_{k}^{h}-U_{k}| \leq \gamma , $$

(40)

for all $h\leq h_{k+1}$ and $x \in R_{k,h}$.

On $\varGamma _{1}^{h}$, we have $u_{h}=G \leq m$ and so, using (40) and $\varGamma _{1}^{h} \subset R_{k,h}$ (which follows from $||y-z_{k}|| < \delta _{k+1}$), we get

$$ a u_{h} + b m \leq m=\inf _{R_{k}} U_{k} \leq U_{k}^{h}+ \gamma . $$

Since $h<\delta _{k+1}/2$ and $||y-z_{k}|| < \delta _{k+1}$ we have $B_{\delta _{k}/2+h}(z_{k}) \subset B_{\delta _{k}}(y)$. If $u_{h} \leq M_{k}$ on $B_{\delta _{k}}(y) \cap \varOmega _{h}$, using (38) and (40) we get on $\varGamma _{2}^{h}$

$$ a u_{h} + b m \leq a M_{k}+ b m \leq U_{k} \leq U_{k}^{h}+ \gamma . $$

In summary, we have

$$ a u_{h} + b m \leq U_{k}^{h}+ \gamma \text{ on } \varGamma _{1}^{h} \cup \varGamma _{2}^{h}. $$

Since the $h$-boundary of $B_{\delta _{k}/2}(y) \cap \varOmega _{h}$ is contained in $\varGamma _{1}^{h} \cup \varGamma _{2}^{h}$ (see [10, Fig. 1]), by the discrete comparison principle (37), we obtain

$$ a u_{h} + b m \leq U_{k}^{h}+ \gamma \text{ on } B_{\delta _{k}/2}(y) \cap \varOmega _{h}. $$

Using again (40) we have under the assumption $u_{h} \leq M_{k}$ on $B_{\delta _{k}}(y) \cap \varOmega _{h}$

$$ a u_{h} + b m \leq U_{k}+ 2 \gamma \text{ on } B_{\delta _{k}/2}(y) \cap \varOmega _{h}. $$

(41)

Next, since $B_{\delta _{k+1}}(y) \subset B_{\delta _{k}/2}(z_{k})$ we get by (41)

$$ a u_{h} + b m \leq U_{k}+ 2 \gamma \text{ on } B_{\delta _{k+1}}(y) \cap \varOmega _{h}. $$

(42)

As $B_{\delta _{k+1}}(y) \subset B_{(2-\mu ) \delta _{k+1}}(z_{k})$, we have by (39) $a u_{h} + b m \leq b' m + a'M_{k}+ 2 \gamma$ on $B_{\delta _{k+1}}(y) \cap \varOmega _{h}$. Since by assumption $M_{k}-m\geq \eta /4$, we have

$$ \gamma \leq \frac{1}{4} \frac{\big( \frac{\mu }{\mu -2}\big)^{\xi } - \big( \frac{\mu }{2}\big)^{\xi } }{1-\big( \frac{\mu }{4}\big)^{\xi }} (M_{k}-m). $$

Using in addition $b'-b+a'=a$, we obtain for $h< h_{k+1}$ in $B_{\delta _{k+1}}(y) \cap \varOmega _{h}$

$$\begin{aligned} &u_{h} \leq \frac{b'-b}{a} m + \frac{a'}{a} M_{k} +\frac{2 \gamma }{a} \leq m+\frac{a'}{a} (M_{k}-m) + \frac{b'(M_{k}-m)}{2 a} \\ &\quad =m+\theta (M_{k}-m) = m+\theta ^{k+1} (M-m). \end{aligned}$$

(43)

Part III: As $G$ is uniformly continuous on the compact set $\varGamma _{1}$, there exists $0<\delta < \overline{\delta }$ such that for all $y \in \partial \varOmega$,

$$ m^{h}(y)-G(y)< \frac{\eta }{4}. $$

(44)

By the stability result (36)

$$ u_{h} \leq \max _{x \in \partial \varOmega _{h}} G \leq \sup _{\varGamma _{h}} G = M^{h}, $$

for all $h$. Thus we can take $h_{0}=1$.

If $M^{h}_{0}(y)-m^{h}(y)=M^{h}-m^{h}(y) < \eta /4$, then by (44), we obtain $u_{h} \leq M^{h} < m^{h}(y)+\eta /4 < G(y)+\eta /2$. Otherwise, there exists $0< h_{1}< h_{0}$ such that for all $h< h_{1}$, $u_{h}\leq M_{1}^{h}(y)$ in $B_{\delta _{1}}(y) \cap \varOmega$. If $M^{h}_{1}-m^{h}(y) < \eta /4$ we proceed as before to obtain the desired inequality.

As $0<\theta <1$, for some integer $s$, we have $\theta ^{s}( M^{h}-m^{h}(y) ) <\eta /4$ which with (44) gives the desired inequality if $u_{h} \leq M^{h}_{s}$ on $B_{\delta _{s}}(y) \cap \varOmega$. If for some $k< s$ we have $M^{h}_{k}-m^{h}(y) < \eta /4$, we take $k_{0}=k$. Otherwise $k_{0}=s$. □

We can now state

Theorem 17

Assume that $\varOmega$ is a Lipschitz domain. The solution $u_{h}$ of (29) converges uniformly on compact subsets of $\varOmega$ to the viscosity solution of (28).

Proof

The proof is as on [10, p. 18]. It follows from Lemma 17 that the half relaxed limits satisfy the boundary condition in the classical sense. One can then use Theorem 15 and the framework in [6]. □