Aleksandrov-Bakelman-Pucci maximum principle for \(L^p\)-viscosity solutions of equations with unbounded terms
The authors prove the Aleksandrov-Bakelman-Pucci maximum principle for \(L^p\)-viscosity sub/supersolutions of fully nonlinear uniformly elliptic equations \[F(x,u,Du,D^2u)=f(x)\] in a bounded domain \(\Omega\subset \mathbb{R}^n\), where \(f\in L_+^n(\Omega)\). The function \(F\) is assumed to satisfy the following ellipticity and gradient dependence condition \begin{align*} P^-(X-Y)-\gamma(x)|p-q|-\omega(|r-s|)&\leq F(x,r,p,X)-F(x,s,q,Y)\\ &\leq P^+(X-Y)+\gamma(x)|p-q|+\omega(|r-s|) \end{align*} and \[ F(x,0,0,0)=0, \ \ \ F(x,\cdot,p,X)\text{ is nondecreasing} \] for a.e. \(x\in \Omega\), for all \(r,s\in \mathbb{R}\), \(p,q\in \mathbb{R}^n\), \(X,Y\in S(n)\), and for some modulus \(\omega\). Here, \(\gamma \in L_+^q(\Omega)\), with \(q\geq n\), and \[ P^-(X):=-\Lambda\mathrm{Tr}(X^+)+\lambda\mathrm{Tr}(X^-), \ \ P^+(X):=\Lambda\mathrm{Tr}(X^-)-\lambda\mathrm{Tr}(X^+) \] are the Pucci extremal operators defined on the set \(S(n)\) of all \(n\times n\) symmetric matrices, where \(X^+\) and \(X^-\) denote, respectively, the positive and the negative part of the matrix \(X\). In particular, the authors prove that, for \(q>n\), any \(L^n\)-viscosity subsolution (resp. supersolution) \(u\in C(\overline{\Omega})\) of \(F(x,u,Du,D^2u)=f(x)\) satisfies \[ \max_{\overline{\Omega}}u\leq \max_{\partial \Omega} u+C(\mathrm{diam}(\Omega))^{2-\frac{n}{p}}\|f\|_{L^n(\Gamma^+(u))} \] \[ (\text{resp. }\max_{\overline{\Omega}}(-u)\leq \max_{\partial \Omega}(-u)+C(\mathrm{diam}(\Omega))^{2-\frac{n}{p}}\|f\|_{L^n(\Gamma^+(-u))}, \] where \(C\) is a constant which depends only on \(\lambda,n,\|\gamma\|_{L^n(\Omega)}\), and \[ \Gamma^+(u):=\{x\in \Omega: \ \ \exists p\in \mathbb{R}^n \text{ such that }u(y)\leq u(x)+\langle p,y-x \rangle, \text{ for all }y\in \Omega\} \] is the upper contact set of \(u\). The authors also prove that the same conclusion holds in the case \(q=n\) for \(L^p\)-viscosity solutions with \(\bar{p}<p<n\), where \(\bar{p}\) depends only on \(\lambda,\Lambda,n,\|\gamma\|_{L^n(\Omega)}\). The above results improve previous results where the boundedness of \(\gamma\) was assumed.
fully nonlinear uniformly elliptic equations
Aleksandrov-Bakelman-Pucci maximum principle
scientific article; zbMATH DE number 7625583
scientific article