Dian K. Palagachev

Elliptic and Parabolic Equations with Discontinuous Coefficients

Introduction. Boundary Value Problems for Linear Operators with Discontinuous Coefficients. Linear and Quasilinear Operators with VMO Coefficients. Nonlinear Operators with Discontinuous Coefficients. Appendix A1: Functional and Real Analysis Tools. Appendix A2: Maximum Principles. Bibliography. Index. Functional Spaces and Their Respective Norms.

Boundary value problem with an oblique derivative for uniformly elliptic operators with discontinuous coefficients

Abstract Strong solvability is proved in the Sobolev space W 2, p (Ω), 1 < p < ∞, for the regular oblique derivative problem assuming .

Weighted L p -estimates for Elliptic Equations with Measurable Coefficients in Nonsmooth Domains

We obtain a global weighted Lp estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg flat, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the Morrey spaces which implies global Hölder continuity of the solution.

Parabolic systems with measurable coefficients in weighted Orlicz spaces

We consider a parabolic system in divergence form with measurable coefficients in a cylindrical space–time domain with nonsmooth base. The associated nonhomogeneous term is assumed to belong to a suitable weighted Orlicz space. Under possibly optimal assumptions on the coefficients and minimal geometric requirements on the boundary of the underlying domain, we generalize the Calderon–Zygmund theorem for such systems by essentially proving that the spatial gradient of the weak solution gains the same weighted Orlicz integrability as the nonhomogeneous term.

On PID control of dynamic systems with hysteresis using a Prandtl-Ishlinskii model

This papers deals with PI and PID control of second order systems with an input hysteresis described by a modified Prandtl-Ishlinskii model. The problem of the asymptotic tracking of constant references is re-formulated as the stability of a polytopic linear differential inclusion. This offers a simple linear matrix inequality condition that, when satisfied with the chosen PI or PID controller gains, ensures the tracking of constant reference and also allows the designer to establish a performance index. The validation of the approach is performed experimentally on a Magnetic Shape Memory Alloy micrometric positioning system.

W2,p-a priori estimates for the emergent Poincaré Problem

We derive W2,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.

Oblique derivative problem for uniformly elliptic operators with VMO coefficients and applications

Abstract Strong solvability in Sobolev spaces W 2, p (Ω) is proved for the regular oblique derivative problem for second order uniformly elliptic operators with VMO-principal coefficients. The results are applied to the study of degenerate (tangential) oblique derivative problem in the case of neutral vector field on the boundary.

The Poincaré Problem in L p -Sobolev Spaces II: Full Dimension Degeneracy

We study a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients within the framework of the Sobolev spaces W 2,p (Ω) with arbitrary p > 1. The boundary operator is prescribed in terms of a directional derivative with respect to a vector field ℓ that becomes tangent to ∂ Ω at the points of a non-empty set ℰ ⊂ ∂ Ω and is of emergent type on ∂Ω. We extend the results from Palagachev (2005) regarding a priori estimates, strong solvability, uniqueness and Fredholmness of the problem under consideration to the general case of arbitrary set of tangency ℰ which may have positive surface measure.

Elliptic Obstacle Problems with Measurable Coefficients in Non-Smooth Domains

We establish a global weighted W 1, p -regularity for solutions to variational inequalities and obstacle problems for divergence form elliptic systems with measurable coefficients in bounded non-smooth domains.

Quasilinear divergence form elliptic equations in rough domains

Existence and global Hölder continuity are proved for the weak solution to the Dirichlet problem over Reifenberg flat domains Ω. The principal coefficients a ij (x, u) are discontinuous with respect to x with small BMO-norms and b(x, u, Du) grows as |Du| r with r < 1 + 2/n.