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Nonclassical solutions of fully nonlinear elliptic equations. (English) Zbl 1132.35036
The very interesting paper under review deals with nonclassical solvability of the Dirichlet problem for fully nonlinear elliptic equations of the form \[ \begin{cases} F(D^2u)=0\;& \text{in}\;\Omega,\\ u=\varphi\;& \text{on}\;\partial\Omega, \end{cases} \tag{\(*\)} \] where \(\Omega\subset\mathbb R^n\) is a bounded domain with smooth boundary, \(\varphi\in C^0(\partial\Omega)\) and \(F\) is smooth and uniformly elliptic.
The main result of the paper asserts existence of a nonclassical viscosity solution of the equation \(F(D^2u)=0\) in \(\Omega\subset\mathbb R^{12}.\) Precisely, the authors prove the following
Theorem. The function \[ w(x)={{\text{Re}(\omega_1\omega_2\omega_3)}\over {| x| }}, \] where \(\omega_i\in \mathbb H,\) \(i=1,2,3,\) are Hamiltonian quaternions, \(x=(\omega_1,\omega_2,\omega_3)\in \mathbb H^3=\mathbb R^{12},\) is a viscosity solution in \(\mathbb R^{12}\) of the fully nonlinear equation \(F(D^2u)=0.\)
Since the homogeneous order-2 function \(w\) is smooth in \(\mathbb R^{12}\setminus\{0\}\) and has discontinuous second order derivatives at \(0,\) and immediate consequence is the following
Corollary. Let \(\Omega\subset\mathbb R^{12}\) be the unit ball and \(\varphi=w\) on \(\partial\Omega.\) Then there exists a smooth and uniformly \(F\) such that the Dirichlet problem \((*)\) has no classical solution.

35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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