| Contents |
| We introduce a class of fully non-linear quasilinear nonlocal equations of the form \begin{equation}F\big(x,u,Du,L[u,Du]\big)=0\tag{1}\end{equation}
in whole space where $L$ is a nonlocal quasilinear Levy type operator. We show well-posedness of bounded viscosity solutions under assumptions that for the first time includes i) a gradient-dendence in $L$ and ii) cover the generators of SDEs driven by general pure jump Levy process.
Next we study limit problems where the non-local operators converge to local ones: $$L_\epsilon[u,Du]\to
\mathrm{tr}\big(\sigma(x,Du))\sigma(x,Du)D^2u\big) $$
as $\epsilon\to 0$. We prove that the solution $u_\epsilon$ of the $(1)$ with $L=L_\epsilon$ converges locally uniformly to the solution of the local problem
\begin{equation*}
F\Big(x,u,Du,\mathrm{tr}\big(\sigma(x,Du)\sigma(x,Du)D^2u\big)\Big)=0\qquad\text{in}\qquad \mathbb{R}^N.
\end{equation*}
We identify general conditions for this convergence, and are able to obtain in the local limit essentially any non-singular gradient-depending quasi-linear equation.
Finally, we present non-local $p$-Laplace, infinity-Laplace and mean curvature of graph operators. |
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