Dian K. Palagachev

Politecnico di Bari | Poliba · Department of Mechanics, Mathematics and Management

Получены новые результаты о сильной разрешимости в пространствах Соболева квазилинейной задачи Вентцеля для параболических уравнений с разрывными старшими коэффициентами.

We obtain some new results on strong solvability in the Sobolev spaces of the linear Venttsel initial-boundary value problems to parabolic equations with discontinuous leading coefficients.

We study linear Venttsel initial-boundary value problems for parabolic operators with discontinuous coefficients. On the base of the a priori estimates obtained, strong solvability in Sobolev spaces is proved.

We study the generalized Morrey regularity of the strong solutions to higher-order uniformly elliptic equations with VMO principal coefficients.KeywordsHigher-order elliptic equationsStrong solutionsVMO coefficientsA priori estimateGeneralized Morrey spaces

We deal with general quasilinear divergence-form coercive operators whose prototype is the [Formula: see text]-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 suppo...

We derive the Calderón–Zygmund property in generalized Morrey spaces for the strong solutions to 2b-order linear parabolic systems with discontinuous principal coefficients.

We develop a global Calderón–Zygmund theory for a quasilinear divergence form parabolic operator with discontinuous entries which exhibit nonlinearities both with respect to the weak solution u and its spatial gradient Du in a nonsmooth domain. The nonlinearity behaves as the parabolic p-Laplacian in Du, its discontinuity with respect to the indepe...

We prove global essential boundedness of the weak solutions u∈W01,p(Ω;RN) to the quasilinear system div,(A(x,u,Du))=b(x,u,Du). The principal part A(x,u,Du) of the differential operator is componentwise coercive and supports controlled growths with respect to u and Du, while the lower order term b(x,u,Du) exhibits componentwise controlled gradient g...

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.

We derive global gradient estimates for $W^{1,p}_0(\Omega)$-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 H\"older...

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...

We deal with the Dirichlet problem for general quasilinear elliptic equations over Reifenberg flat domains. The principal part of the operator supports natural gradient growth and its x-discontinuity is of small-BMO type, while the lower order terms satisfy controlled growth conditions with x-behaviour modelled by Morrey spaces. We obtain a Calderó...

We prove boundedness of the weak solutions to the Cauchy–Dirichlet problem for quasilinear parabolic equations whose prototype is the parabolic m-Laplacian. The nonlinear terms satisfy sub-controlled growth conditions with respect to the unknown function and its spatial gradient, while the behaviour in the independent variables is modelled in Lebes...

We prove global regularity in weighted Lebesgue spaces for the viscosity solutions to the Dirichlet problem for fully nonlinear elliptic equations. As a consequence, regularity in Morrey spaces of the Hessian is derived as well.

We study the obstacle problem for an elliptic equation with discontinuous nonlinearity over a nonsmooth domain, assuming that the irregular obstacle and the nonhomogeneous term belong to suitable weighted Sobolev and Lebesgue spaces, respectively, with weights taken in the Muckenhoupt classes. We establish a Calderón–Zygmund type result by proving...

We deal with general quasilinear divergence-form coercive operators whose prototype is the $m$-Laplacean operator. The nonlinear terms are given by Carath\'eodory 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...

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 underlyin...

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.

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 supp...

Advances in remote sensing technology are now providing tools to support geospatial mapping of the soil properties for the application to the management of agriculture and the environment. In this paper results of visible and near IR spectral reflectance are presented and discussed. A supportable evaluation of organic matter in the soil is the abse...

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...

We prove global essential boundedness of weak solutions to quasilinear coercive divergence form equations with data belonging to Morrey spaces. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth assumptions. As an application of the main result, we get global Hoelder continuity of the solutions to semilinear ellip...

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 med...

We are concerned with optimal regularity theory in weighted Sobolev spaces for discontinuous non-linear parabolic problems in divergence form over a non-smooth, bounded domain. Assuming smallness in BMO of the principal part of the non-linear operator and flatness in Reifenberg sense of the boundary, we establish a global weighted W1,p estimate for...

We obtain global essential boundedness and Hölder continuity of the weak solutions to quasilinear elliptic equations in divergence form with data lying in Morrey spaces.

We consider a parabolic system in divergence form with measurable coefficients in a nonsmooth bounded domain to obtain a global gradient estimate for the weak solution in the setting of Orlicz space which is a natural generalization of Lp space. The coefficients are assumed to be merely measurable in one spatial variable and have small bounded mean...

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...

We derive weak solvability and higher integrability of the spatial gradient of solutions to Cauchy--Dirichlet problem for divergence form quasilinear parabolic equations {u t −div(a ij (x,t,u)D j u+a i (x,t,u))=b(x,t,u,Du) u=0 in Q, on ∂ p Q, where Q is a cylinder in R n ×(0,T) with Reifenberg flat base Ω. The principal coefficients a ij (x,t,u) o...

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...

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.

We present some recent results regarding the W 2,p -theory of a degenerate oblique derivative problem for second order uniformly elliptic operators. The boundary operator is prescribed in terms of directional derivative with respect to a vector field l which is tangent to ¶W\partial \Omega at the points of a nonempty set e Ì ¶W\varepsilon \subse...

We derive global Holder regularity for the W(0)(1,2)(Omega)-weak solutions to the quasilinear, uniformly elliptic equation div(a(ij)(x, u)D(j)u + a(i)(x, u)) + a(x, u, Du) = 0 over a C(1)-smooth domain Omega subset of R(n), n >= 2. The nonlinear terms are all of Caratheodory type with coefficients a(ij)(x, u) belonging to the class VMO of functions...

We prove existence and global Hölder regularity of the weak solution to the Dirichlet problem { lc div ( aij (x,u)Dj u ) = b(x,u,Du) inW Ì \mathbb Rn, n ³ 2, u = 0 on¶W Î C1. \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...

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...

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 se...

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 are dealing with the degenerate oblique derivative problem for uniformly elliptic operators with low regular coefficients in the framework of Sobolev's classes W2,p(Ω) for any p > 1. The boundary operator is prescribed in terms of a directional derivative with respect to a vector field ℓ that becomes tangential to ∂Ω at the points of a nonempty...

We propose results on interior Morrey, BMO and Hölder 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.

We deal with Dirichlet's problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. First we state suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Then we fix a solution $u_0$ such that the linearized in $u_0$ problem is non-degenerate,...

A degenerate oblique derivative problem is studied for uniformly elliptic operators with low regular coefficients in the framework of Sobolev's classes $W^{2,p}(\Omega)$ for {\em arbitrary} $p>1.$ The boundary operator is prescribed in terms of a directional derivative with respect to the vector field $\l$ that becomes tangential to $\partial \Omeg...

We study a degenerate oblique derivative problem in Sobolev spaces W2,p(Ω),∀p>1, for uniformly elliptic operators with Lipschitz continuous coefficients. The vector field prescribing the boundary condition becomes tangential to ∂Ω at the points of a non-empty set and is of emergent type.

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, pr...

We improve the results from [10] on strong solvability and uniqueness for the oblique derivative problem $$ \left\{ \begin{gathered} a^{ij} \left( x \right)D_{ij} u + b^i \left( x \right)D_i u + c\left( x \right)u = f\left( x \right) a.a. in \Omega , \hfill \\ \partial u/\partial \ell + \sigma \left( x \right)u = \varphi \left( x \right) on \partia...

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 Hlder) regularity of strong solutions to higher-order elliptic and parabolic equations with VMO coefficients.

Let Q(T) be a cylinder in Rn+1 and x = (x', t) epsilon R-n x R. It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator {u(t) - Sigma(i,j = 1)(n)D a(ij)(x) D(ij)u = f(x) a.e. in Q(T), {u(x) = 0 on partial derivativeQ(T), in the Morrey spaces W-p, lambda(2, 1)(Q(T)), p epsilon (1, infinity), lambda epsilon (0, n + 2), supposi...

Strong solvability in the Sobolev space W2, p(Ω) is proved for the oblique derivative problem∂u/∂ℓ+σ(x)u=ϕ(x)in the trace sense on ∂Ωin the case when the vector field ℓ(x) has a contact of infinite order with ∂Ω at the points of some non-empty subset E ⊂ ∂Ω.

solvability theory;regularity theory;second order elliptic;parabolic equations

The degenerate Neumann problem \[ \begin{cases} \ \displaystyle \sum_{i,j=1}^{n}a^{ij}(x)\frac{\partial^{2}u}{\partial x_i\partial x_j}=f(x,u,Du) & \text{in}\ \Omega ,\\ \ a(x)\dfrac{\partial u}{\partial v}+b(x)u=\varphi(x) & \text{on}\ \Gamma \end{cases} \] is studied in the case where $a(x)$ and $b(x)$ are non-negative functions on $\Gamma$ such...

This article presents a study of the regular oblique derivative problem $$ displaylines{ sum_{i,j=1}^n a^{ij}(x) frac{partial^2 u }{partial x_ipartial x_j} =f(x) cr frac{partial u }{partial ell(x)}+ sigma(x) u = varphi(x),. }$$ Assuming that the coefficients $a^{ij}$ belong to the Sarason's class of functions with vanishing mean oscillation, we sho...

Well-posedness is proved in the space W2, p, λ(Ω)∩W1, p0(Ω) for the Dirichlet problem -- EQUATION OMITTED -- if the principal coefficients aij(x) of the uniformly elliptic operator belong to VMO∩L∞(Ω).

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 a...

Strong solvability in Sobolev spaces W2,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.

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

. Classical solvability and uniqueness in the Holder space C 2+ff (OmegaGamma is proved for the oblique derivative problem ae a ij (x)D ij u + b(x; u; Du) = 0 inOmega ; @u=@` = '(x) on @Omega in the case when the vector field `(x) = (` 1 (x); : : : ; ` n (x)) is tangential to the boundary @Omega at the points of some non-empty set S ae @Omega ; and...

Strong solvability and uniqueness in the Sobolev space W 2, q (Ω), q > n , are proved for the oblique derivative problem assuming the coefficients of the quasilinear elliptic operator to be Carathéodory functions, a ij ∈ VMO ∩ L ∞ with respect to x , and b to grow at most quadratically with respect to the gradient.

Classical solvability and uniqueness in the Hölder space C 2+α (Ω ¯) is proved for the oblique derivative problem ∑ i,j=1 n a ij (x)D ij u+b(x,u,Du)=0inΩ,∂u/∂l≡∑ i=1 n l i (x)D i u=φ(x)on∂Ω in the case when the vector field l(x)=(l 1 (x),⋯,l n (x)) is tangential to the boundary ∂Ω at the points of some non-empty set S⊂∂Ω, and the nonlinear term b(x...

A priori estimates and strong solvability results in Sobolev space W 2,p (), 1 < p < ∞ are proved for the regular oblique derivative problem n i,j=1 a ij (x) ∂ 2 u ∂xi∂xj = f(x) a.e. ∂u ∂ℓ + σ(x)u = ϕ(x) on ∂ when the principal coefficientsa ij are VMO ∩ L ∞ functions.

Existence and uniqueness results are proved for a degenerate boundary value problem for second order elliptic operators. The vector field that prescribes the boundary operator is allowed to be tangential to the boundary, and it is of neutral type.

Global existence and uniqueness of strong solutions to nonlinear elliptic partial differential equations are proved. The nonlinear operator under consideration is defined by Carathéodory’s functions, and it is elliptic in sense of Campanato. The main tools of our investigations are both Leray–Schauder fixed point theorem and Aleksandrov–Pucci maxim...

Strong solvability and uniqueness in Sobolev space $ {W^{2,n}}(\Omega )$ are proved for the Dirichlet problem $\displaystyle \left\{ {_{u = \varphi \quad {\text{on}}\partial \Omega .}^{{a^{i... ...{D_{ij}}u + b(x,u,Du) = 0\quad {\text{a}}{\text{.e}}{\text{.}}\Omega }} \right.$ It is assumed that the coefficients of the quasilinear elliptic operator...

Global solvability and uniqueness results are established for Dirichlet's problem for a class of nonlinear differential equations on a convex domain in the plane, where the nonlinear operator is elliptic in sense of Campanato. We prove existence by means of the Leray-Schauder fixed point theorem, using Alexandrov-Pucci maximum principle in order to...

The present article is a continuation of our papers [ibid. 43, No. 1, 17- 20 (1990; Zbl 0711.35070); ibid. 44, No. 1, 7-10 (1991; Zbl 0754.35075)]. We are concerned with the existence and uniqueness questions for the classical solutions of the tangential oblique derivative problem for quasilinear parabolic equations of second order (see P. R. Popiv...

O. - Introduction. In the present article we consider the solutions of a second order parabolic equation 2~u = f in the cylinder Q. More precisely, we will concerned with the existence and uniqueness questions for such solutions which satisfy a boundary condition in term of directional derivative au/al = r on the lateral boundary S of Q, and an ini...