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Vladimir Maz'ya

Publications and source records attributed to Vladimir Maz'ya.

At least 19 recordsLinked to original sources
arXiv2023

Global second-order estimates in anisotropic elliptic problems

We deal with boundary value problems for second-order nonlinear elliptic equations in divergence form, which emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. Integrands with non polynomial growth are included in our discussion. The $W^{1,2}$-regularity of the stress-field associated with solutions, namely the nonlinear expression of the gradient subject to the divergence operator, is established under the weakest possible assumption that the datum on the right-hand side of the equation is a merely $L^2$-function. Global regularity estimates are offered in domains enjoying minimal assumptions on the boundary. They depend on the weak curvatures of the boundary via either their degree of integrability or an isocapacitary inequality. By contrast, none of these assumptions is needed in the case of convex domains. An explicit estimate for the constants appearing in the relevant estimates is exhibited in terms of the Lipschitz characteristic of the domains, when their boundary is endowed with Hölder continuous curvatures.

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arXiv2022

The Fundamental Solution of an Elliptic Equation with Singular Drift

For $n\geq 3$, we study the existence and asymptotic properties of the fundamental solution for elliptic operators in nondivergence form, ${\mathcal L}(x,\partial_x)=a_{ij}(x)\partial_i\partial_j+b_k(x)\partial_k$, where the $a_{ij}$ have modulus of continuity $ω(r)$ satisfying the square-Dini condition and the $b_k$ are allowed mild singularities of order $r^{-1}ω(r)$. A singular integral is introduced that controls the existence of the fundamental solution. We give examples that show the singular drift $b_k\partial_k$ may act as a perturbation that does not dramatically change the fundamental solution of ${\mathcal L}^o=a_{ij}\partial_i\partial_j$, or it could change an operator ${\mathcal L}^o$ that does not have a fundamental solution to one that does.

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arXiv2022

The functional dissipativity of certain systems of partial differential equations

In the present paper we consider the functional dissipativity of the Dirichlet problem for systems of partial differential operators of the form $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{hk}(x)\partial_{k})$ (${\mathop{\mathscr A}\nolimits}^{hk}$ being $m\times m$ matrices with complex valued $L^{1}_{\text{loc}}$ entries). In the particular case of the operator $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{h}(x)\partial_{h})$ (where ${\mathop{\mathscr A}\nolimits}^{h}$ are $m\times m$ matrices) we obtain algebraic necessary and sufficient conditions. We give also three different notions of functional ellipticity and investigate the relations between them and the functional dissipativity for the operators in question.

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arXiv2021

Gradient Estimate for Solutions of Second-Order Elliptic Equations

We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point x=0. In particular, we treat the case of solutions that are not Lipschitz continuous at x=0. We show that our estimate is sharp.

math.AP↗
arXiv2021

Gilbarg-Serrin Equation and Lipschitz Regularity

We discuss conditions for Lipschitz and C^1 regularity for a uniformly elliptic equation in divergence form with coefficients that were introduced by Gilbarg & Serrin. In particular, we find cases where Lipschitz or C^1 regularity holds but the coefficients are not Dini continuous, or do not even have Dini mean oscillation. The form of the coefficients also enables us to obtain specific conditions and examples for which there exists a weak solution that is not Lipschitz continuous.

math.AP↗
arXiv2021

A pointwise differential inequality and second-order regularity for nonlinear elliptic systems

A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in $\mathbb{R}^n$ are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the $p$-Laplace system, our conclusions broaden the range of the admissible values of the exponent $p$ previously known.

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arXiv2020

Friedrichs type inequalities in arbitrary domains

First and second-order inequalities of Friedrichs type for Sobolev functions in arbitrary domains are offered. The relevant inequalities involve optimal norms and constants that are independent of the geometry of the domain. Parallel inequalities for symmetric gradient Sobolev spaces of vector-valued functions are also presented. The results are derived via general criteria established in our earlier contributions [4] and [5].

math.AP↗
arXiv2020

On sharp Agmon-Miranda maximum principles

In this survey we formulate our results on different forms of maximum principles for linear elliptic equations and systems. We start with necessary and sufficient conditions for validity of the classical maximum modulus principle for solutions of second order strongly elliptic systems. This principle holds under rather heavy restrictions on the coefficients of the systems, for instance, it fails for the Stokes and Lamé systems. Next, we turn to sharp constants in more general maximum principles due to S. Agmon and C. Miranda. We consider higher order elliptic equations, Stokes and Lamé systems in a half-space as well as the system of planar deformed state in a half-plane.

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arXiv2020

Criterion for the functional dissipativity of second order differential operators with complex coefficients

In the present paper we consider the Dirichlet problem for the second order differential operator $E=\nabla(A \nabla)$,where $A$ is a matrix with complex valued $L^\infty$ entries. We introduce the concept of dissipativity of $E$ with respect to a given function $φ:R^+ \to R^+$. Under the assumption that the $Im\, A$ is symmetric, we prove that the condition $|s\, φ'(s)| \, | \langle Im\, A (x)\, ξ,ξ\rangle |\leq 2\, \sqrt{φ(s)\, [s\, φ(s)]'}\, \langle Re\, A(x) \, ξ,ξ\rangle $ (for almost every $x\inΩ\subset R^N$ and for any $s>0$, $ξ\in R^N$) is necessary and sufficient for the functional dissipativity of $E$.

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arXiv2020

Sharp pointwise estimates for solutions of weakly coupled second order parabolic system in a layer

We deal with $m$-component vector-valued solutions to the Cauchy problem for linear both homogeneous and nonhomogeneous weakly coupled second order parabolic system in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$. We assume that coefficients of the system are real and depending only on $t$, $n\geq 1$ and $T<\infty$. The homogeneous system is considered with initial data in $[L^p({\mathbb R}^n)]^m$, $1\leq p \leq \infty $. For the nonhomogeneous system we suppose that the initial function is equal to zero and the right-hand side belongs to $[L^p({\mathbb R}^{n+1}_T)]^m\cap [C^α\big (\overline{{\mathbb R}^{n+1}_T} \big )]^m $, $α\in (0, 1)$. Explicit formulas for the sharp coefficients in pointwise estimates for solutions of these problems and their directional derivative are obtained.

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arXiv2020

On meso-scale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions

In this paper we consider formal asymptotic algorithms for a class of meso-scale approximations for problems of vibration of elastic membranes, which contain clusters of small inertial inclusions distributed along contours of pre-defined smooth shapes. Effective transmission conditions have been identified for inertial structured interfaces, and approximations to solutions of eigenvalue problems have been derived for domains containing lower-dimensional clusters of inclusions.

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arXiv2019

Approximation of solutions to non-stationary Stokes system

We propose a fast method for high order approximations of the solution of the Cauchy problem for the linear non-stationary Stokes system in $R^3$ in the unknown velocity $\bf u$ and kinematic pressure $P$. The density ${\bf f}({\bf x},t)$ and the divergence vector-free initial value ${\bf g }(\bf x)$ are smooth and rapidly decreasing as $|\bf x|$ tends to infinity. We construct the vector $\bf u$ in the form $\bf u=\bf u_1+\bf u_2$ where $\bf u_1$ solves a system of homogeneous heat equations and $\bf u_2$ solves a system of non-homogeneous heat equations with right-hand side ${\bf f}-\nabla P$. Moreover $P=-{\cal L}( \nabla\cdot \bf f)$ where $\cal L$ denotes the harmonic potential. Fast semi-analytic cubature formulas for computing the harmonic potential and the solution of the heat equation based on the approximation of the data by functions with analitically known potentials are considered. In addition, the gradient $\nabla P$ can be approximated by the gradient of the cubature of $P$, which is a semi-analytic formula too. We derive fast and accurate high order formulas for the approximation of $\bf u_1,\bf u_2$, $P$ and $\nabla P$. The accuracy of the method and the convergence order $2,4,6$ and $8$ are confirmed by numerical experiments.

math.NA↗
arXiv2019

Sharp pointwise estimates for the gradients of solutions to linear parabolic second order equation in the layer

We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second order equations with real constant coefficients in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$, where $n\geq 1$ and $T<\infty$. The homogeneous equation is considered with initial data in $L^p({\mathbb R}^n)$, $1\leq p \leq \infty $. For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to $f\in L^p({\mathbb R}^{n+1}_T)\cap C^α\big (\bar{{\mathbb R}^{n+1}_T} \big ) $ , $p>n+2$ and $α\in (0, 1)$. Explicit formulas for the sharp coefficients in pointwise estimates for the length of the gradient to solutions to these problems are obtained.

math.AP↗
arXiv2019

Sobolev inequalities for the symmetric gradient in arbitrary domains

A form of Sobolev inequalities for the symmetric gradient of vector-valued functions is proposed, which allows for arbitrary ground domains in $\mathbb R ^n$. In the relevant inequalities, boundary regularity of domains is replaced with information on boundary traces of trial functions. The inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of Sobolev functions, in regular domains. Furthermore, they involve constants independent of the geometry of the domain, and hence yield novel results yet for smooth domains. Our approach relies upon a pointwise estimate for the functions in question via a Riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace.

math.FA↗
arXiv2018

Accurate computation of the high dimensional diffraction potential over hyper-rectangles

We propose a fast method for high order approximation of potentials of the Helmholtz type operator Delta+kappa^2 over hyper-rectangles in R^n. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one-dimensional integrals with separable integrands. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Numerical tests show that these formulas are accurate and provide approximations of order 6 up to dimension 100 and kappa^2=100.

math.NA↗
arXiv2018

Optimal second-order regularity for the p-Laplace system

Second-order estimates are established for solutions to the $p$-Laplace system with right-hand side in $L^2$. The nonlinear expression of the gradient under the divergence operator is shown to belong to $W^{1,2}$, and hence to enjoy the best possible degree of regularity. Moreover, its norm in $W^{1,2}$ is proved to be equivalent to the norm of the right-hand side in $L^2$. Our global results apply to solutions to both Dirichlet and Neumann problems, and entail minimal regularity of the boundary of the domain. In particular, our conclusions hold for arbitrary bounded convex domains. Local estimates for local solutions are provided as well.

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arXiv2018

Second-order regularity for parabolic p-Laplace problems

Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of $p$-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

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