arXiv subjects

Dian K. Palagachev

Publications and source records attributed to Dian K. Palagachev.

9 recordsLinked to original sources

arXiv2025

Non-homogeneous conormal derivative problem for quasilinear elliptic equations with Morrey data

A non-homogeneous conormal derivative problem is considered for quasilinear divergence form elliptic equations modeled on the $m$-Laplacian operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with respect to the solution and its gradient, while their $x$-behaviour is controlled in terms of suitable Morrey spaces. Global essential boundedness is proved for the weak solutions, generalizing thus the classical $L^p$-result of Ladyzhenskaya and Ural'tseva to the framework of the Morrey scales.

math.AP↗

arXiv2021

Global Hoelder continuity of solutions to quasilinear equations with Morrey data

We deal with general quasilinear divergence-form coercive operators whose prototype is the $m$-Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly non-regular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property. We prove global boundedness and Hoelder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical $L^p$-result of Ladyzhenskaya and Ural'tseva to the settings of the Morrey spaces.

math.AP↗

arXiv2020

Venttsel boundary value problems with discontinuous data

We study linear and quasilinear Venttsel boundary value problems involving elliptic operators with discontinuous coefficients. On the base of the a priori estimates obtained, maximal regularity and strong solvability in Sobolev spaces are proved.

math.AP↗

arXiv2020

Boundedness of Solutions to a Class of Coercive Systems with Morrey Data

We prove global essential boundedness for the weak solutions of divergence form quasilinear systems. The principal part of the differential operator is componentwise coercive and supports controlled growths with respect to the solution and its gradient, while the lower order term exhibits componentwise controlled gradient growth. The x-behaviour of the nonlinearities is governed in terms of Morrey spaces.

math.AP↗

arXiv2017

Global Sobolev regularity for general elliptic equations of $p$-Laplacian type

We derive global gradient estimates for $W^{1,p}_0(Ω)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and Hölder continuous in $u.$ In the case when $p\geq n,$ we allow only continuous nonlinearity in $u.$ Our result highly improves the known regularity results available in the literature. In fact, we are able not only to weaken the regularity requirement on the nonlinearity in $u$ from Lipschitz continuity to Hölder one, but we also find a very lower level of geometric assumptions on the boundary of the domain to ensure global character of the obtained gradient estimates.

math.AP↗

arXiv2013

Global gradient estimates in weighted Lebesgue spaces for parabolic operators

We deal with the regularity problem for linear, second order parabolic equations and systems in divergence form with measurable data over non-smooth domains, related to variational problems arising in the modeling of composite materials and in the mechanics of membranes and films of simple non-homogeneous materials which form a linear laminated medium. 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 parabolic operators in the settings of the weighted Lebesgue spaces. As consequence of the main result, we get regularity in parabolic Morrey scales for the spatial gradient of the weak solutions to the problems considered.

math.AP↗

arXiv2012

Weighted $L^p$-estimates for elliptic equations with measurable coefficients in nonsmooth domains

We obtain a global weighted $L^p$ 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 Hoelder continuity of the solution.

math.AP↗

arXiv2011

$W^{2,p}$-A~priori estimates for the neutral Poincaré problem

A degenerate oblique derivative problem is studied for uniformly elliptic operators with low regular coefficients in the framework of Sobolev's classes $W^{2,p}(Ω)$ for {\em arbitrary} $p>1.$ The boundary operator is prescribed in terms of a directional derivative with respect to the vector field $ł$ that becomes tangential to $\partial Ω$ at the points of some non-empty subset $\E\subset \partial Ω$ and is directed outwards $Ω$ on $\partialΩ\setminus\E.$ Under quite general assumptions of the behaviour of $ł,$ we derive {\it a priori} estimates for the $W^{2,p}(Ω)$-strong solutions for any $p\in(1,\infty).$

math.AP↗

arXiv2004

A'priori estimates and precise regularity for parabolic systems with discontinuous data

We deal with linear parabolic (in sense of Petrovskii) systems of order 2b with discontinuous principal coefficients. A'priori estimates in Sobolev and Sobolev--Morrey spaces are proved for the strong solutions by means of potential analysis and boundedness of certain singular integral operators with kernels of mixed homogeneity. As a byproduct, precise characterization of the Morrey, BMO and Hölder regularity is given for the solutions and their derivatives up to order 2b-1. Some counterexamples of recent publications announcing estimates similar to the obtained here are also given.

math.AP↗