arXiv subjects

V. G. Maz'ya

Publications and source records attributed to V. G. Maz'ya.

7 recordsLinked to original sources

arXiv2019

Accretivity and form boundedness of second order differential operators

Let $\mathcal{L}$ be the general second order differential operator with complex-valued distributional coefficients $A=(a_{jk})_{j, k=1}^n$, $\vec{b}=(b_{j})_{j=1}^n$, and $c$ in an open set $Ω\subseteq \mathbb{R}^n$ ($n \ge 1$), with principal part either in the divergence form, $\mathcal{L} u= {\rm div} \, (A \nabla u) + \vec{b} \cdot\nabla u + c \, u$, or non-divergence form, $ \mathcal L u= \sum_{j, \, k=1}^n \, a_{jk} \, \partial_j \partial_k u + \vec{b} \cdot\nabla u + c \, u $. We give a survey of the results by the authors which characterize the following two properties of $\mathcal{L}$: (1) $-\mathcal{L}$ is accretive, i.e., ${\rm Re} \, \langle -\mathcal L u, \, u\rangle \ge 0$; (2) $\mathcal L$ is form bounded, i.e., $\vert \langle \mathcal L u, u \rangle \vert \le C \, \Vert \nabla u \Vert_{L^2(Ω)}^2$, for all complex-valued $u \in C^\infty_0(Ω)$.

math.AP↗

arXiv2018

Accretivity of the general second order linear differential operator

For the general second order linear differential operator $$\mathcal L_0 = \sum_{j, \, k=1}^n \, a_{jk} \, \partial_j \partial_k + \sum_{j=1}^n \, b_{j} \, \partial_j + c$$ with complex-valued distributional coefficients $a_{jk}$, $b_{j}$, and $c$ in an open set $Ω\subseteq \mathbf{R}^n$ ($n \ge 1$), we present conditions which ensure that $-\mathcal L_0$ is accretive, i.e., ${\rm Re} \, \langle -\mathcal L_0 ϕ, ϕ\rangle \ge 0$ for all $ϕ\in C^\infty_0(Ω).$

math.AP↗

arXiv2016

Eigenvalue problem in a solid with many inclusions: asymptotic analysis

We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplace's operator inside a domain containing a cloud of small rigid inclusions. The separation of the small inclusions is characterised by a small parameter which is much larger compared with the nominal size of inclusions. Remainder estimates for the approximations to the first eigenvalue and associated eigenfield are presented. Numerical illustrations are given to demonstrate the efficiency of the asymptotic approach compared to conventional numerical techniques, such as the finite element method, for three-dimensional solids containing clusters of small inclusions.

math-ph↗

arXiv2006

Mixed boundary value problems for the Navier-Stokes system in polyhedral domains

Mixed boundary value problems for the Navier-Stokes system in a polyhedral domain are considered. Different boundary conditions (in particular, Dirichlet, Neumann, slip conditions) are prescribed on the faces of a polyhedron. The authors obtain regularity results for weak solutions in weighted (and non-weighted) L_p Sobolev and Hölder spaces.

math-ph↗

arXiv2004

Form boundedness of the general second order differential operator

We give explicit necessary and sufficient conditions for the boundedness of the general second order differential operator L with real- or complex-valued distributional coefficients acting from the Sobolev space W^{1,2}(R^n) to its dual W^{-1,2}(R^n). This enables us to obtain analytic criteria for the fundamental notions of relative form boundedness, compactness, and infinitesimal form boundedness of L with respect to the Laplacian on L^2(R^n). In particular, we establish a complete characterization of the form boundedness of the Schroedinger operator (i \nabla + a)^2 + q with magnetic vector potential a \in L^2_{loc} and q \in D'(\R^n).

math.AP↗

arXiv2004

Infinitesimal form boundedness and Trudinger's subordination for the Schrödinger operator

We give explicit analytic criteria for two problems associated with the Schrödinger operator $H = -Δ+ Q$ on $L^2(\R^n)$ where $Q\in D'(\R^n)$ is an arbitrary real- or complex-valued potential. First, we obtain necessary and sufficient conditions on $Q$ so that the quadratic form $ $ has zero relative bound with respect to the Laplacian. For $Q\in L^1_{\rm loc}(\R^n)$, this property can be expressed in the form of the integral inequality: $$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq ε||\nabla u||^2_{L^2(\R^n)} + C(ε) ||u||^2_{L^2(\R^n)}, \quad \forall u \in C^\infty_0(\R^n), $$ for an arbitrarily small $ε>0$ and some $C(ε)> 0$. Secondly, we characterize Trudinger's subordination property where $C(ε)$ in the above inequality is subject to the condition $C(ε) \le c {ε^{-β}}$ ($β>0$) as $ε\to +0$. Such quadratic form inequalities can be understood entirely in the framework of Morrey--Campanato spaces, using mean oscillations of $\nabla (1-Δ)^{-1} Q$ and $(1-Δ)^{-1} Q$ on balls or cubes. As a consequence, we characterize the class of those $Q$ which satisfy a multiplicative quadratic from inequality of Nash's type.

math.FA↗

arXiv2003

The form boundedness criterion for the relativistic Schrödinger operator

We establish necessary and sufficient conditions for the boundedness of the relativistic Schrödinger operator $\mathcal{H} = \sqrt{-Δ} + Q$ from the Sobolev space $W^{1/2}_2 (\R^n)$ to its dual $W^{-1/2}_2 (\R^n)$, for an arbitrary real- or complex-valued potential $Q$ on $\R^n$. %Analogous results for %$\mathcal{H}_m = \sqrt{-Δ+ m^2} - m + Q$, as well as %the corresponding compactness criteria are obtained. In other words, we give a complete solution to the problem of the domination of the potential energy by the kinetic energy in the relativistic case characterized by the inequality $$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq \text{const} ||u||^2_{W_2^{1/2}}, \quad u \in C^\infty_0(\R^n), $$ where the ``indefinite weight'' $Q$ is a locally integrable function (or, more generally, a distribution) on $\R^n$. Along with necessary and sufficient results, we also present new broad classes of admissible potentials $Q$ in the scale of Morrey spaces of negative order, and discuss their relationship to well-known $L_p$ and Fefferman-Phong conditions.

math-ph↗