3. Fully Nonlinear Equations

Notice that if we wanted to maximize the payoff, we could simply consider \(-g\) instead of g.

Namely, two players with conflicting interests govern the evolution of the particle by choosing from different sets of indices.

We apply Proposition 3.2.17 to the operator \(\mathcal {I}_g(u, x) = \mathcal {I}(u, x) - g\) (and \(\mathcal {I}_f(u, x) = \mathcal {I}(u, x) - f\)), where we are considering the new \(c_{ab}^{\prime }(x) = c_{ab}(x) - f(x)\), that are still equicontinuous in \(\overline {D}\).

This is satisfied, for example, by any test function \(\eta \in C^\infty _c(\mathbb {R}^n)\) such that \(0 \le \eta \le M\) and \(\eta \equiv M\) in D if M is large enough.

More precisely, the term on the left-hand side of the estimate in Theorem 3.3.1 is a global \(L^1_{\omega _s}\) norm of the solution, while in the local case this would be a local \(L^1\) norm in \(B_{1/2}\). Of course, this comes at a price, which is that the constant C blows up as \(s\uparrow 1\).

The \(\inf \sup \) of \(C^\mu \) functions is \(C^\mu \), whenever \(\mu <1\). If \(\mu \ge 1\), then it is at most Lipschitz (\(C^{0,1}\)). This is why this proof does not obtain higher regularity even if \(\theta \gg 1\).

Actually, \(\omega (r) = Cr^\delta + \sigma (r)\).

If \(f \not \equiv 0\), we would have now \(\mathcal {I}_\varepsilon (v, x) - (f*\varphi _\varepsilon )(x)\) as a regularized version of \(\mathcal {I}(v, x) - f(x)\), since \(\sum _{i, j} M_{ij} = 1\).

It is important to notice that \(\mathcal S\) depends only on n, s, \(\alpha \), and \(\Lambda \). This is what gives the dependence of the constant \(C_\delta \).