Nonclassical Solutions of Fully Nonlinear Elliptic Equations

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Abstract.We prove the existence of non-smooth solutions to fully non-linear elliptic equations.

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Sharp hessian integrability estimates for nonlinear elliptic equations: an asymptotic approach (2015, background): “ere the hessian will be large, and in general ΘK are very irregular sets. We should also caution the readers to the existence of a C1,1 \ C2 viscosity solution to a fully nonlinear elliptic equation, [21]. Hence, it is not possible to establish continuity of the hessian out from the regularity theor”
Partial regularity of solutions of fully nonlinear, uniformly elliptic equations (2011, unknown): “1) are unavailable for general F , as attested by the recent counterexamples of Nadirashvili and Vlăduţ [8, 9].”
Singular Solution to Special Lagrangian Equations (2009, result): “ a theory of weak (viscosity) solutions for the Dirichlet problem gives the uniqueness of such solutions, see [CIL], moreover these solutions lie in C1,ε by [C],[T1],[T2]. However, the recent results [NV1],[NV2],[NV3] show that at least in 12 and more dimensions the viscosity solution of the Dirichl”
The Bernstein Technique for Integro-Differential Equations (2020, unknown): “Therefore, since we will establish one-sided second derivative bounds, the convexity assumption on F in the following theorem cannot be dropped, in view of the classical counterexamples to W 2,∞ regularity by Nadirashvili and Vlăduţ [27] for nonconvex fully nonlinear equations of second order.11 O”