W^{2,1}_p-solvability for the parabolic Poincarè problem,

We study the Poincaré problem for linear uniformly parabolic operator P with discontinuous coefficients. The boundary operator is defined in terms of oblique derivative with respect to a vector field l which points outward the domain or becomes tangential to the boundary on a set of possibly positive measure. A’priori estimates and unique strong solvability are obtained in W^(2,1)_p(Q) for all p\in (1,\infty)

Morrey type spaces over unbounded domain

We present an overview of some classical and modern results in Morrey and generalized Morrey spaces defined on Rn and over unbounded domains

Strong solvability for a class of nonlinear parabolic equations

Existence of strong solutions to Cauchy-Dirichlet problem for nonlinear parabolic equation is established. The nonlinear operator is prescribed by Carathéodory's function which satisfies an ellipticity condition due to S. Campanato. The main results are reached through Aleksandrov-Bakel'man-Pucci type maximum principle and topological fixed point theorem.<br /

An integral estimate for the gradient for a class of nonlinear elliptic equations in the plane,

An a priori estimate is established for the gradient of the solution to Dirichlet's problem for a class of nonlinear differential equations on a convex domain in the plane. The nonlinear operator is assumed to be elliptic in the sense of Campanato. By virtue of the Leray-Schauder fixed point theorem an existence result for the problem under consideration is derived

Gradient estimates for nonlinear elliptic equations in Morrey type spaces

We obtain Calderon-Zygmund type estimates in generalized Morrey spaces for nonlinear equations of p-Laplacian type. Our result is obtained under minimal regularity assumptions both on the operator and on the domain. This result allows us to study asymptotically regular operators. As a byproduct, we obtain also generalized Holder regularity of the solutions under some minimal restrictions of the weight functions

From Dido to Morrey: Variational problems and regularity theory!

Although ancient Greek and Roman sources report that Dido, the founder and first queen of Carthage was the first person who formulated a problem in Calculus of Variations, the classical existence theory is connected mainly with the names of Euler, Lagrange and Ostrogradskij. The notorious Euler-Lagrange equation is a second order Partial Differential Equation, the solvability of which ensures the existence of a minimizer of a given functional. The question of regularity for the solutions of this PDE was firstly posed by David Hilbert in his 19th and 20th problem, presented during a celebrated lecture at the International Congress of Mathematics 1900 in Paris. During the last century, these two problems gave a strong impulse to the development of the regularity theory for problems from CV and PDE. Our goal is to present some classical and new results concerning regularity properties of the solutions to the Dirichlet problem for elliptic equations and systems. We obtain essential boundedness of the solution to a class of nonlinear elliptic systems. In addition, we establish estimates in the Morrey spaces for the solutions of a kind of quasilinear and nonlinear elliptic systems

Precise Morrey regularity of the weak solutions to a kind of quasilinear systems with Morrey 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 Carath'eodory 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

Elliptic systems in generalized Morrey spaces

We obtain local regularity in generalized Morrey spaces for the strong solutions to 2b-order linear elliptic systems with discontinuous coefficients