Precise Morrey regularity of the weak solutions to a kind of quasilinear systems with discontinuous data

We consider the Dirichlet problem for a class of quasilinear elliptic systems in domain with irregular boundary. The principal part satisfies componentwise coercivity condition and the nonlinear terms are Carathéodory maps having Morrey regularity in x and verifying controlled growth conditions with respect to the other variables. We have obtained boundedness of the weak solution to the problem that permits to apply an iteration procedure in order to find optimal Morrey regularity of its gradient

Parabolic oblique derivative problem in generalized Morrey spaces

We study the regularity of the solutions of the oblique derivative problem for linear uniformly parabolic equations with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized Morrey space than the strong solution belongs to the corresponding generalized Sobolev-Morrey space

Applications of Differential Calculus to Nonlinear Elliptic Boundary Value Problems with Discontinuous Coefficients

We deal with Dirichlet's problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. First we state suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Then we fix a solution $u_0$ such that the linearized in $u_0$ problem is non-degenerate, and we apply the Implicit Function Theorem: For all small perturbations of the coefficient functions there exists exactly one solution $u \approx u_0,$ and $u$ depends smoothly (in $W^{2,p}$ with $p$ larger than the space dimension) on the data. For that no structure and growth conditions are needed, and the perturbations of the coefficient functions can be general $L^\infty$-functions with respect to the space variable $x$. Moreover we show that the Newton Iteration Procedure can be applied to calculate a sequence of approximate (in $W^{2,p}$ again) solutions for $u_0.