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.     

W 2,p -a priori estimates for the emergent Poincaré Problem

We derive W ^2,p (Ω)-a priori estimates with arbitrary p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.      

Discontinuous superlinear elliptic equations of divergence form

We prove existence and global Hölder regularity of the weak solution to the Dirichlet problem $\left\{ {\begin{array}{lc} {{\rm div} \left( a^{ij} (x,u)D_{j} u \right) = b(x,u,Du) \quad {\rm in}\, \Omega \subset {\mathbb R}^{n}, \, n \ge 2,} \\ {u = 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,{\rm on}\, \partial\Omega \in C^{1}. } \\ \end{array}} \right.$ The coefficients a ^ij (x, u) are supposed to be VMO functions with respect to x while the term b(x, u, Du) allows controlled growth with respect to the gradient Du and satisfies a sort of sign-condition with respect to u. Our results correct and generalize the announcements in Ragusa (Nonlinear Differ Equ Appl 13:605–617, 2007, Erratum in Nonlinear Differ Equ Appl 15:277–277, 2008).     

The Calderón-Zygmund property for quasilinear divergence form equations over Reifenberg flat domains

The results by Palagachev (2009) [3] regarding global Hölder continuity for the weak solutions to quasilinear divergence form elliptic equations are generalized to the case of nonlinear terms with optimal growths with respect to the unknown function and its gradient. Moreover, the principal coefficients are discontinuous with discontinuity measured in terms of small BMO norms and the underlying domain is supposed to have fractal boundary satisfying a condition of Reifenberg flatness. The results are extended to the case of parabolic operators as well.     

Morrey regularity of solutions to quasilinear elliptic equations over Reifenberg flat domains

We derive global gradient estimates in Morrey spaces for the weak solutions to discontinuous quasilinear elliptic equations related to important variational problems arising in models of linearly elastic laminates and composite materials. The principal coefficients of the quasilinear operator are supposed to be merely measurable in one variable and to have small-BMO seminorms in the remaining orthogonal directions, and the nonlinear terms are subject to controlled growth conditions with respect to the unknown function and its gradient. The boundary of the domain considered is Reifenberg flat which includes boundaries with rough fractal structure. As outgrowth of the main result we get global Hölder continuity of the weak solution with exact value of the corresponding exponent.     

A degenerate Neumann problem for quasilinear elliptic integro-differential operators

This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998     

A time transformation technique for Switched Systems Optimal Control Problems with fixed sequence of phases

A switched systems optimal control problem regarding a quadcopter trajectory optimization is considered. Introducing a particular time transformation, the problem is reformulated as an optimal control problem, whose solution can be found by the SQP algorithm. A numerical example concludes the paper. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fine regularity for elliptic systems with discontinuous ingredients

We propose results on interior Morrey, BMO and H?lder regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey space L^p,λ. Received: 20 October 2004