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Bernstein estimates permit to bound the sup-norm of the gradient of the solution with the sup-norm of the solution. We are particularly interested in systems that arise in the stochastic optimal control problems of hybrid systems, that are systems of weakly coupled nonlinear elliptic equations. Our techniques can also be adapted to handle nonlocal integral operators, such as the factional Laplacian. In both cases, a generalization of Bernstein estimates for first and second derivatives of classical solutions will be presented. In the light of the approach of Caffarelli-Cabré, Bernstein estimates follow rather simply from the maximum principle for the associated linearized operator. This is a joint work with Diogo Gomes.
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