Solution analysis for a class of set-inclusive generalized equations: a convex analysis approach

In the present paper, classical tools of convex analysis are used to study the solution set to a certain class of set-inclusive generalized equations. A condition for the solution existence and global error bounds is established, in the case the set-valued term appearing in the generalized equation is concave. A functional characterization of the contingent cone to the solution set is provided via directional derivatives. Specializations of these results are also considered when outer prederivatives can be employed

On the biharmonic Dirichlet problem: The higher dimensional case

Withdrawn by authors.Comment: Withdrawn by author

The Dirichlet problem in a class of generalized weighted spaces

We show continuity in generalized weighted Morrey spaces of sub-linear integral operators generated by some classical integral operators and commutators. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operators with discontinuous data.Comment: 25 page

The LpL^p Dirichlet Problem for Elliptic Systems on Lipschitz Domains

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in $R^n$. For $n\ge 4$ and $2-\epsilon<p<2(n-1)/(n-3)+\epsilon$, we establish the solvability of the Dirichlet problem with boundary value data in $L^p(\partial\Omega)$. In the case of the polyharmonic equation $\Delta^{\ell}u=0$ with $\ell\ge 2$, the range of $p$ is sharp if $4\le n \le 2\ell +1$

On a power-type coupled system of Monge-Amp\`{e}re equations

We study an elliptic system coupled by Monge-Amp\`{e}re equations: \begin{center} \left\{ \begin{array}{ll} det~D^{2}u_{1}={(-u_{2})}^\alpha, & \hbox{in \Omega,} det~D^{2}u_{2}={(-u_{1})}^\beta, & \hbox{in \Omega,} u_{1}<0, u_{2}<0,& \hbox{in \Omega,} u_{1}=u_{2}=0, & \hbox{on \partial \Omega,} \end{array} \right. \end{center} here $\Omega$~is a smooth, bounded and strictly convex domain in~$\mathbb{R}^{N}$,~$N\geq2,~\alpha >0,~\beta >0$. When $\Omega$ is the unit ball in $\mathbb{R}^{N}$, we use index theory of fixed points for completely continuous operators to get existence, uniqueness results and nonexistence of radial convex solutions under some corresponding assumptions on $\alpha,\beta$. When $\alpha>0$, $\beta>0$ and $\alpha\beta=N^2$ we also study a corresponding eigenvalue problem in more general domains

Generalized Morrey regularity for parabolic equations with discontinuity data

We obtain continuity in generalized parabolic Morrey spaces of sublinear integrals generated by the parabolic Calder\'{o}n-Zygmund operators and its commutator with $VMO$ functions. The obtained estimates are used to study global regularity of the solutions of the Cauchy-Dirichlet problem for linear uniformly parabolic equations with discontinuous coefficients.Comment: 16 page

Adams-Spanne type estimates for the commutators of fractional type sublinear operators in generalized Morrey spaces on Heisenberg groups

In this paper we give BMO (bounded mean oscillation) space estimates for commutators of fractional type sublinear operators in generalized Morrey spaces on Heisenberg groups. The boundedness conditions are also formulated in terms of Zygmund type integral inequalities

The boundedness of certain sublinear operators with rough kernel generated by Calder\'on-Zygmund operators and their commutators on generalized weighted Morrey spaces

The aim of this paper is to get the boundedness of certain sublinear operators with rough kernel generated by Calder\'on-Zygmund operators on the generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutator operators formed by BMO functions and certain sublinear operators with rough kernel are also bounded on the generalized weighted Morrey spaces. Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.Comment: arXiv admin note: substantial text overlap with arXiv:1602.07853, arXiv:1603.06739, arXiv:1603.04088, arXiv:1603.03469, arXiv:1602.08096; text overlap with arXiv:1212.6928 by other author

The Skrypnik Degree Theory and Boundary Value Problems

The paper presents theorems on the calculation of the index of a singular point and at the infinity of monotone type mappings. These theorems cover basic cases when the principal linear part of a mapping is degenerate. Applications of these theorems to proving solvability and nontrivial solvability of the Dirichlet problem for ordinary and partial differential equations are considered.Comment: 9 page

Multi-sublinear operators generated by multilinear fractional integral operators and commutators on the product generalized local Morrey spaces

The aim of this paper is to get the boundedness of certain multi-sublinear operators generated by multilinear fractional integral operators on the product generalized local Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutators of multilinear operators generated by local campanato functions and multilinear fractional integral operators are also bounded on the product generalized local Morrey spaces.Comment: arXiv admin note: substantial text overlap with arXiv:1603.04088; text overlap with arXiv:1212.6928 by other author