Old and new results in regularity theory for diagonal elliptic systems via blowup techniques

We consider quasilinear diagonal elliptic systems in bounded domains subject to Dirichlet, Neumann or mixed boundary conditions. The leading elliptic operator is assumed to have only measurable coefficients, and the nonlinearities (Hamiltonians) are allowed to be of quadratic (critical) growth in the gradient variable of the unknown. These systems appear in many applications, in particular in differential geometry and stochastic differential game theory. We impose on the Hamiltonians structural conditions developed between 1972-2002 and also a new condition (sum coerciveness) introduced in recent years (in the context of the pay off functional in stochastic game theory). We establish existence, Hölder continuity, Liouville properties, W2,q estimates, etc. for solutions, via a unified approach through the blow-up method. The main novelty of the paper is the introduction of a completely new technique, which in particular leads to smoothness of the solution also for dimensions d≥3.