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Vladimir Maz'ya

Publications and source records attributed to Vladimir Maz'ya.

At least 55 records · Page 3Linked to original sources
arXiv2013

Second-order differentiability for solutions of elliptic equations in the plane

For a second-order elliptic equation of nondivergence form in the plane, we investigate conditions on the coefficients which imply that all strong solutions have first-order derivatives that are Lipschitz continuous or differentiable at a given point. We assume the coefficients have modulus of continuity satisfying the square-Dini condition, and obtain additional conditions associated with a dynamical system that is derived from the coefficients of the elliptic equation. Our results extend those of previous authors who assume the modulus of continuity satisfies the Dini condition.

math.AP↗
arXiv2013

Fast cubature of volume potentials over rectangular domains

In the present paper we study high-order cubature formulas for the computation of advection-diffusion potentials over boxes. 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. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order $O(h^6)$ up to dimension $10^8$.

math.NA↗
arXiv2012

Global boundedness of the gradient for a class of nonlinear elliptic systems

Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.

math.AP↗
arXiv2012

Regularity of solutions to the polyharmonic equation in general domains

The present paper establishes boundedness of $[m-\frac n2+\frac 12]$ derivatives for the solutions to the polyharmonic equation of order $2m$ in arbitrary bounded open sets of $\RR^n$, $2\leq n\leq 2m+1$, without any restrictions on the geometry of the underlying domain. It is shown that this result is sharp and cannot be improved in general domains. Moreover, it is accompanied by sharp estimates on the polyharmonic Green function.

math.AP↗
arXiv2012

On linear water wave problem in the presence of a critically submerged body

We study the problem of propagation of linear water waves in a deep water in the presence of a critically submerged body (i.e. the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of the solution which satisfies the radiation conditions at infinity as well as, additionally, at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure. Next we introduce a relevant scattering matrix and analyse its properties. Under a geometric condition introduced by Maz'ya, see \cite{M1}, we show that the method of multipliers applies to cusp singularities, thus proving a new important property of the scattering matrix, which may be interpreted as the absence of a version of "full internal reflection". This property also allows us to prove uniqueness and existence of the solution in the functional spaces $H^2_{loc}\cap L^\infty $ and $H^2_{loc}\cap L^p $, $2<p<6$, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. This description of the solution does not rely on the radiation conditions or the limiting absorption principle. This is the first result of this type known to us in the theory of linear wave problems in unbounded domains.

math.AP↗
arXiv2010

Differentiability of Solutions to Second-Order Elliptic Equations via Dynamical Systems

For a second-order elliptic equation in divergence form we investigate conditions on the coefficients which imply that all solutions are Lipschitz continuous or differentiable at a given point. We assume the coefficients have modulus of continuity satisfying the square-Dini condition, and obtain additional conditions that examples show are sharp. Our results extend those of previous authors who assume the modulus of continuity satisfies the Dini condition. Our method involves the study of asymptotic properties of solutions to a dynamical system that is derived from the coefficients of the elliptic equation.

math.AP↗
arXiv2010

Mesoscale asymptotic approximations to solutions of mixed boundary value problems in perforated domains

We describe a method of asymptotic approximations to solutions of mixed boundary value problems for the Laplacian in a three-dimensional domain with many perforations of arbitrary shape, with the Neumann boundary conditions being prescribed on the surfaces of small voids. The only assumption made on the geometry is that the diameter of a void is assumed to be smaller compared to the distance to the nearest neighbour. The asymptotic approximation, obtained here, involves a linear combination of dipole fields constructed for individual voids, with the coefficients, which are determined by solving a linear algebraic system. We prove the solvability of this system and derive an estimate for its solution. The energy estimate is obtained for the remainder term of the asymptotic approximation.

math-ph↗
arXiv2009

On the fast computation of high dimensional volume potentials

A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate approximations are used. Results of numerical experiments, which show approximation order O(h^8) for the Newton potential in high dimensions, for example, for n= 200 000, are provided. The computation time scales linearly in the space dimension. New one-dimensional integral representations with separable integrands of the potentials of advection-diffusion and heat equations are obtained.

math.NA↗
arXiv2009

A quasi-commutativity property of the Poisson and composition operators

Let $Φ$ be a real valued function of one real variable, let $L$ denote an elliptic second order formally self-adjoint differential operator with bounded measurable coefficients, and let $P$ stand for the Poisson operator for $L$. A necessary and sufficient condition on $Φensuring the equivalence of the Dirichlet integrals of $Φ\circ Ph$ and $P(Φ\circ h)$ is obtained. We illustrate this result by some sharp inequalities for harmonic functions.

math.AP↗
arXiv2009

Pathological solutions to elliptic problems in divergence form with continuous coefficients

We construct a function $u \in W^{1,1}_{\mathrm{loc}} (B(0,1))$ which is a solution to $\Div (A \nabla u)=0$ in the sense of distributions, where $A$ is continuous and $u \not \in W^{1,p}_{\mathrm{loc}} (B(0,1))$ for $p > 1$. We also give a function $u \in W^{1,1}_{\mathrm{loc}} (B(0,1))$ such that $u \in W^{1,p}_{\mathrm{loc}}(B(0,1))$ for every $p < \infty$, $u$ satisfies $\Div (A \nabla u)=0$ with $A$ continuous but $u \not \in W^{1, \infty}_{\mathrm{loc}}(B(0,1))$.

math.AP↗
arXiv2009

Tensor product approximations of high dimensional potentials

The paper is devoted to the efficient computation of high-order cubature formulas for volume potentials obtained within the framework of approximate approximations. We combine this approach with modern methods of structured tensor product approximations. Instead of performing high-dimensional discrete convolutions the cubature of the potentials can be reduced to a certain number of one-dimensional convolutions leading to a considerable reduction of computing resources. We propose one-dimensional integral representions of high-order cubature formulas for n-dimensional harmonic and Yukawa potentials, which allow low rank tensor product approximations.

math.NA↗