Optimal boundary regularity for the planar generalized p-Poisson equation

Abstract: In this talk, I will discuss about sharp regularity for solutions to the following generalized \(p\) -Poisson equation

$$ - div  \big( \langle  A\nabla u, \nabla u \rangle ^{\frac{p-2}{2}} A\nabla u \big) = - div  \mathbf{h}  +  f  $$

in the plane (i.e. in \(R^2)\) for \(p>2\) in the presence of Dirichlet as well as Neumann boundary conditions and with \(\mathbf{h}\in C^{1-2/q}\),\(f\in L^q, 2<q\leq\infty\). The regularity assumptions on the principal part \(A\) as well as that on the Dirichlet/Neumann conditions are exactly the same as in the linear case and therefore sharp. These results should be thought of as the boundary analogues of the sharp interior regularity result established in the recent interesting paper by D. J. Araujo, E.V. Teixeira, J. M. Urbano in the case of

                    \(-\ div\) \((|\nabla u|^{p-2} \nabla u)\) =\(f\)

for more general variable coefficient operators and with an additional divergence term.