Error estimates for approximations of nonhomogeneous nonlinear uniformly elliptic equations

Abstract

We obtain an error estimate between viscosity solutions and $\delta$-viscosity solutions of nonhomogeneous fully nonlinear uniformly elliptic equations. The main assumption, besides uniform ellipticity, is that the nonlinearity is Lipschitz-continuous in space with linear growth in the Hessian. We also establish a rate of convergence for monotone and consistent finite difference approximation schemes for such equations.

Appendices

Appendix A

In this section we recall the comparison principle for viscosity solutions ([8, Theorem 3.3]) and several related results.

Proposition 10.1

(Comparison for viscosity solutions) Assume (F1). If $u, v\in C(U)$ are, respectively, a subsolution and supersolution of $F(D^2u, x)=f(x)$ in U with $u\le v$ on $\partial U$, then $u\le v$ in U.

We now provide the (quite basic) proof of Lemma 2.6.

Proof of Lemma 2.6

Since $c>0$, $\bar{u}$ is a subsolution of $F(D^2u, x)=f(x)$, so $\bar{u}\le u$ on V by Theorem 10.1.

We denote $R={{\mathrm{diam}}}V$, so there exists $x_0$ such that $V\subset B_R(x_0)$. For $x\in V$ we define

$$\begin{aligned} w(x)=\bar{u}(x) -\frac{c}{\lambda }\left( \frac{|x-x_0|^2}{2}-\frac{R^2}{2}\right) . \end{aligned}$$

If $x\in \partial V$, then $w(x)\ge \bar{u}(x)$. And, since F is uniformly elliptic, we have

$$\begin{aligned} F(D^2w, x)=F\left( D^2\bar{u} -\frac{c}{\lambda }I,x\right) \le F(D^2\bar{u}, x) - c = f(x). \end{aligned}$$

Therefore, w is a supersolution of $F(D^2u, x)=f(x)$ on V, so according to Theorem 10.1, we find that for all $x\in V$,

$$\begin{aligned} u(x)\le w(x). \end{aligned}$$

We have $w(x)\le \bar{u}(x)+\frac{cR^2}{2\lambda }$ for all $x\in V$, which, together with the previous estimate, completes the proof of the lemma. $\square$

We state [8, Theorem 3.2], modified for our setting. This deep result was instrumental in establishing comparison for viscosity solutions; we use it in the proofs of Proposition 3.1 and Proposition 4.2.

Theorem 10.2

Suppose that $u,v\in C(U)$ are viscosity solutions of $F(D^2u, x)=f(x)$ and $G(D^2v, x)=g(x)$ in U. Suppose that $(x_a, y_a)\in U\times U$ is a local maximum of

$$\begin{aligned} u(x) - v(y) - \frac{a}{2}\Bigg |x-y\Bigg |^2. \end{aligned}$$

Then there exist matrices X and Y that satisfy

$$\begin{aligned} -3a\left( \begin{array}{c@{\quad }c} I &{} 0 \\ 0 &{} I \\ \end{array} \right) \le \left( \begin{array}{c@{\quad }c} X &{} 0 \\ 0 &{} -Y \\ \end{array} \right) \le 3a\left( \begin{array}{c@{\quad }c} I &{} -I \\ -I &{} I \\ \end{array} \right) , \end{aligned}$$

(9.17)

and $F(X, x_a)=f(x_a)$, $G(Y, y_a)=g(y_a)$.

Together with Theorem 10.2, we use the following lemma of [9, Lemma III.1].

Lemma 10.3

There is a constant C(n) such that if (X, Y) are $n\times n$ matrices that satisfy (9.17) for some constant a, then

$$\begin{aligned} ||X||,||Y|| \le C(n)\left\{ a^{1/2}||X-Y||^{1/2}+||X-Y||\right\} . \end{aligned}$$

Now we give the proof of Lemma 3.3.

Proof of Lemma 3.3

For any $x\in V$, we have

$$\begin{aligned} \sup _{ y \in V} \left( v(x)-w(y) - \frac{a}{2}|x-y|^2\right)&= v(x) -\inf _{ y \in V} \left( w(y) + \frac{a}{2}|x-y|^2\right) \\&\le v(x) - w(x) + 2||Dw||_{L^{\infty }(V)}^2a^{-1}, \end{aligned}$$

where the inequality follows from the properties of inf-convolutions. Since $v=w$ on the boundary of V, we find

$$\begin{aligned} \sup _{ y \in B_1, x\in \partial V} \left( v(x)-w(y) - \frac{a}{2}|x-y|^2 \right) \le 2||Dw||_{L^{\infty }(V)}^2a^{-1}. \end{aligned}$$

Similarly, if $y\in \partial V$, then

$$\begin{aligned} \sup _{ x \in V, y\in \partial V} \left( v(x)-w(y) - \frac{a}{2}|x-y|^2 \right) \le 2||D v||_{L^{\infty }(V)}^2a^{-1}. \end{aligned}$$

These two bounds imply the first claim of the lemma. We now proceed to give the proof of the second claim. By the definition of $(x_a,y_a)$ as a point at which the supremum is achieved, we have, for any $(x,y)\in V\times V$,

$$\begin{aligned} u(x_a)-v(y_a)-\frac{a}{2}\Bigg |x_a-y_a\Bigg |^2\ge u(x)-v(y)-\frac{a}{2}|x-y|^2, \end{aligned}$$

so in particular, this inequality holds with $(x,y)=(x_a,y_a)$. This implies

$$\begin{aligned} u(x_a) -\frac{a}{2}\Bigg |x_a-y_a\Bigg |^2\ge u(y_a), \end{aligned}$$

so we find

$$\begin{aligned} \frac{a}{2}\Bigg |x_a-y_a\Bigg |^2\le u(x_a)-u(y_a) \le ||Du||_{L^{\infty }(V)}|x_a-y_a|, \end{aligned}$$

from which we easily conclude $|x_a-y_a|\le 2a^{-1}||Du||_{L^{\infty }(V)}$. We find $|x_a-y_a|\le 2a^{-1}||Dv||_{L^{\infty }(V)}$ in a similar way. $\square$

Appendix B

We summarize the basic properties of inf and sup convolutions that we use in this paper. We refer the reader to [7, Proposition 5.3] and [5, Lemma 5.2] for the proof of items (1)–(3). The proof of item (4) is very similar to that of [7, Proposition 5.5] and we omit it.

Proposition 10.4

Assume $u\in C(U)$.

• (1) In the sense of distributions, $D^2u^{+,\theta }(x)\ge -\theta ^{-1}I$ and $D^2u^{-,\theta }(x)\le \theta ^{-1}I$ for all $x\in U$.

• (2) If $u\in C^{\eta }(U)$, for some $\eta \in (0,1]$, then for all $x\in U$,

  $$\begin{aligned} 0\le (u^{+,\theta }-u)(x)&\le [u]_{C^{0,\eta }(U)}(2\theta )^{\frac{\eta }{2-\eta }} , \text { and}\\ 0\le (u-u^{-,\theta })(x)&\le [u]_{C^{0,\eta }(U)}(2\theta )^{\frac{\eta }{2-\eta }} . \end{aligned}$$

• (3) Define $\nu = 4\theta ^{1/2}||u||_{L^{\infty }(U)}^{1/2}$. If u is a subsolution of (1.1) in U, then $u^{\theta , +}$ is a subsolution of

  $$\begin{aligned} F^\nu (D^2u, x)=f_\nu (x) \text { in } U^{\theta }_\delta ; \end{aligned}$$

  if u is a supersolution of (1.1) in U, then $u^{\theta , -}$ is a supersolution of

  $$\begin{aligned} F_\nu (D^2u, x)=f^\nu (x) \text { in } U^{\theta }_\delta . \end{aligned}$$

  (The perturbed nonlinearities $F^\nu$ and $F_\nu$, as well as $f_\nu$ and $f^\nu$, are defined in Definition 4.1).

• (4) Assume that $v\in C(U)$. Let $\nu = 4\theta ^{1/2}||v||_{L^{\infty }(U)}^{1/2}$. If v is a $\delta$-subsolution of (1.1) in U, then $v^{\theta , +}$ is a $\delta$-subsolution of

  $$\begin{aligned} F^\nu (D^2v, x)=f_\nu (x) \text { in } U^{\theta }_\delta ; \end{aligned}$$

  if v is a $\delta$-supersolution of (1.1) in U, then $v^{\theta , -}$ is a $\delta$-supersolution of

  $$\begin{aligned} F_\nu (D^2v, x)=f^\nu (x) \text { in } U^{\theta }_\delta . \end{aligned}$$

1.1 B.1 Inf and sup convolutions of mesh functions

We summarize some basic properties of inf and sup convolutions of mesh functions.

Proposition 10.5

Assume $v\in C^{0,\eta }(U_h)$.

• (1) If $x^{*}\in U_h$ denotes a point where the supremum (resp. infimum) is achieved in the definition of $v_h^{\theta , +}(x)$ (resp. $v_h^{\theta , -}(x)$), then

  $$\begin{aligned} |x-x^{*}|\le 4||v_h||_{L^{\infty }(U_h)}^{1/2}\theta ^{1/2}+\sqrt{n}h. \end{aligned}$$

• (2) In the sense of distributions, $D^2v^{+,\theta }(x)\ge -\theta ^{-1}I$ and $D^2v^{-,\theta }(x)\le \theta ^{-1}I$ for all $x\in U$.

• (3) For all $x\in U_h$, we have $v^{-,\theta }(x)\le v(x)\le v^{+,\theta }(x)$.

• (4) There exists a constant C that depends on $[v]_{C^{0,\eta }(U)}$ such that if $x\in U$ and y is a neighboring mesh point to x, then,

  $$\begin{aligned}&v^{+,\theta }(y)-C\theta ^{\frac{\eta }{2-\eta }} \le v(x)\le v^{-,\theta }(y)+C\theta ^{\frac{\eta }{2-\eta }} . \end{aligned}$$

Proof of (1) of Proposition 10.5

For $x\in U$, we denote by $x_h$ an element of the mesh that is closest to x. Note $|x-x_h|\le \sqrt{n}h$.

Let $x^{*}\in U_h$ be a point where the supremum is achieved in the definition of $v_h^{\theta , +}(x)$. Then

$$\begin{aligned} v_h(x^{*})-\frac{|x-x^{*}|^2}{2\theta }\ge v_h(x_h) - \frac{|x_h-x|^2}{2\theta }. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{|x-x^{*}|^2}{2\theta } \le v_h(x^{*}) - v_h(x_h) +\frac{nh^2}{2\theta } \le 2||v_h||_{L^{\infty }(U_h)} +\frac{nh^2}{2\theta }, \end{aligned}$$

which easily implies the desired bound. The proof for $v_h^{\theta , -}(x)$ is very similar. $\square$

We refer the reader to [6, Proposition 2.3] for the proof of the rest of Proposition 10.5.