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.
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Acknowledgments
The author thanks her thesis advisor, Professor Takis Souganidis, for suggesting this problem and for his guidance and encouragement.
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Communicated by L. Caffarelli.
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
If \(x\in \partial V\), then \(w(x)\ge \bar{u}(x)\). And, since F is uniformly elliptic, we have
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 \(x\in V\),
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
Then there exist matrices X and Y that satisfy
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
Now we give the proof of Lemma 3.3.
Proof of Lemma 3.3
For any \(x\in V\), we have
where the inequality follows from the properties of inf-convolutions. Since \(v=w\) on the boundary of V, we find
Similarly, if \(y\in \partial V\), then
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 \((x,y)\in V\times V\),
so in particular, this inequality holds with \((x,y)=(x_a,y_a)\). This implies
so we find
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
Therefore,
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.
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Turanova, O. Error estimates for approximations of nonhomogeneous nonlinear uniformly elliptic equations. Calc. Var. 54, 2939–2983 (2015). https://doi.org/10.1007/s00526-015-0890-6
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DOI: https://doi.org/10.1007/s00526-015-0890-6