Singular Integral Operators,Morrey Spaces and Fine Regularity of Solutions to PDE's

Boundedness in Morrey spaces is studied for singular integral operators with kernels of mixed homogeneity and their commutators with multiplication by a BMO-function. The results are applied in obtaining fine (Morrey and Hölder) regularity of strong solutions to higher-order elliptic and parabolic equations with VMO coefficients.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which 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 coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

Boundedness of the solutions to non-linear systems with Morrey data

We consider non-linear elliptic systems satisfying componentwise coercivity condition. The non-linear terms have controlled growths with respect to the solution and its gradient, while the behaviour in x is governed by functions in Morrey spaces. We firstly prove essential boundedness of the weak solution and then we obtain Morrey regularity of its gradient.     

Gradient estimates for parabolic equations in generalized weighted Morrey spaces

We consider the Cauchy–Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg flat domain.The coefficients supposed to be only measurable in one of the space variables and small BMO with respect to the others.We obtain Calderón–Zygmund type estimate for the gradient of the solution in generalized weighted Morrey spaces with Muckenhoupt weight.     

Global Regularity in Generalized Morrey Spaces of Solutions to Nondivergence Elliptic Equations with VMO Coefficients

We show continuity in generalized Morrey spaces of sublinear integral operators generated by Calderón-Zygmund operator and their commutators with BMO functions. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operators.     

Parabolic oblique derivative problem with discontinuous coefficients in generalized Morrey spaces

We study the global Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with $VMO$ coefficients.     

Singular integral operators in Morrey spaces and interior regularity of solutions to systems of linear PDE’s

We obtain boundedness in Morrey spaces of singular integral operators with Calderón-Zygmund type kernel of mixed homogeneity. These estimates are used for the study of the interior regularity of the solutions of linear elliptic/parabolic systems. The proved Poincaré-type inequality permits to describe the Hölder, Morrey, and BMO regularity of the lower-order derivatives of the solutions.     

Morrey-type regularity of solutions to parabolic problems with discontinuous data

We consider regular oblique derivative problem in cylinder Q _T ?=????× (0, T), ${\Omega\subset {\mathbb R}^n}$ for uniformly parabolic operator ${{{\mathfrak P}}=D_t- \sum_{i,j=1}^n a^{ij}(x)D_{ij}}$ with VMO principal coefficients. Its unique strong solvability is proved in Manuscr. Math. 203?C220 (2000), when ${{{\mathfrak P}}u\in L^p(Q_T)}$ , ${p\in(1,\infty)}$ . Our aim is to show that the solution belongs to the generalized Sobolev?CMorrey space ${W^{2,1}_{p,\omega}(Q_T)}$ , when ${{{\mathfrak P}}u\in L^{p,\omega} (Q_T)}$ , ${p\in (1, \infty)}$ , ${\omega(x,r):\,{\mathbb R}^{n+1}_+\to {\mathbb R}_+}$ . For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy?CDirichlet problem.     

Gradient estimates in generalized Morrey spaces for parabolic operators

We obtain global regularity in generalized Morrey spaces for the gradient of the weak solutions to divergence form linear parabolic operators with measurable data. Assuming partial BMO smallness of the coefficients and Reifenberg flatness of the boundary of the underlying domain, we develop a Calderón‐Zygmund type theory for such operators. Problems like the considered here arise in the modeling of composite materials and in the mechanics of membranes and films of simple nonhomogeneous materials which form a linear laminated medium.