arXiv subjects

Vladimir G. Maz'ya

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

6 recordsLinked to original sources

arXiv2015

Pointwise Inequalities for Elliptic Boundary Value Problems

We introduce a new approach to obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator being considered satisfies a certain type of weighted integral inequalities. The method is illustrated on several examples, including a scalar second-order elliptic equation, the 3D Lamé system, and a scalar higher-order elliptic equation. The techniques can be extended to other elliptic boundary value problems provided that the corresponding weighted integral inequalities are satisfied.

math.AP↗

arXiv2012

Quasilinear elliptic equations and weighted Sobolev-Poincaré inequalities with distributional weights

We introduce a class of weak solutions to the quasilinear equation $-Δ_p u = σ|u|^{p-2}u$ in an open set $Ω\subset\mathbf{R}^n$. Here $p>1$, and $Δ_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $σ$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-Δ_p v = (p-1)|\nabla v|^p + σ$, under the same conditions on $σ$. Our results for this latter equation will allow us to characterize the class of distributions $σ$ which satisfy the Sobolev-Poincare inequality, thereby extending earlier results on the form boundedness problem for the Schrödinger operator to $p\neq 2$.

math.AP↗

arXiv2012

Existence and regularity of positive solutions to elliptic equations of Schrödinger type

We prove the existence of positive solutions with optimal local regularity to homogeneous elliptic equations of Schrödinger type, under only a form boundedness assumption on $σ\in D'(Ω)$ and ellipticity assumption on $\mathcal{A}\in L^\infty(Ω)^{n\times n}$, for an arbitrary open set $Ω\subseteq \mathbf{R}^n$. We demonstrate that there is a two way correspondence between the form boundedness and the existence of positive solutions to this equation, as well as weak solutions to certain elliptic equations with quadratic nonlinearity in the gradient. As a consequence, we obtain necessary and sufficient conditions for both the form-boundedness (with a sharp upper form bound) and the positivity of the quadratic form of the Schrödinger type operator with arbitrary distributional potential $σ\in D'(Ω)$, and give examples clarifying the relationship between these two properties.

math.AP↗

arXiv2009

Wiener Type Regularity of a Boundary Point for the 3D Lamé System

In this paper, we study the 3D Lamé system and establish its weighted positive definiteness for a certain range of elastic constants. By modifying the general theory developed in Maz'ya (2002), we then show, under the assumption of weighted positive definiteness, that the divergence of the classical Wiener integral for a boundary point guarantees the continuity of solutions to the Lamé system at this point.

math.AP↗

arXiv2004

Lp estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains

A mixed boundary value problem for the Stokes system in a polyhedral domain is considered. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the sides of the polyhedron. The authors prove the existence of solutions in (weighted and non-weighted) $L_p$ Sobolev spaces and obtain regularity assertions for weak solutions. The results are based on point estimates of Green's matrix.

math-ph↗

arXiv2004

Pointwise estimates for Green's kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone

The paper deals with a mixed boundary value problem for the Stokes system in a polyhedral cone. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the sides of the polyhedron. The authors obtain regularity results for weak solutions in weighted $L_2$ Sobolev spaces and prove point estimates of Green's matrix.

math-ph↗