arXiv subjects
Publications and source records attributed to Jean-Paul Daniel.
Exploiting some connections between the system $\nabla v\otimes\nabla v + 2$ sym $\nabla w = A$ and the isometric immersion problem in two dimensions, we provide a simple construction of $C^{1,α}$ convex integration solutions for the former from the corresponding result for the latter.
We prove quenched~$L^p$--type estimates for the gradient of a solution of a quasilinear elliptic equation with random coefficients.
For a parabolic equation associated to a uniformly elliptic operator, we obtain a $W^{3, \varepsilon}$ estimate, which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. The argument combines parabolic $W^{2,\varepsilon}$ estimates with a comparison principle argument. As an application, we show, assuming the operator is $C^1$, that a viscosity solution is $C^{2,α}$ on the complement of a closed set of Hausdorff dimension $\varepsilon$ less than that of the ambient space, where the constant $\varepsilon>0$ depends only on the dimension and the ellipticity.
We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter which extend those proposed by Kohn and Serfaty (2010). These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution of the PDE as the parameter tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet-Neumann boundary conditions.