Vladimir Maz'ya

Differentiable functions on bad domains

Part 1 Sobolev spaces for good or general domains: density theorems, extension theorems and Poincares inequality for Sobolev functions imbedding theorems of Sobolev type and their applications imbedding theorems and isoperimetric inequalities. Part 2 Sobolev spaces for domains singularly depending on parameters: extension of functions defined on parameter dependent domains traces of functions with first derivatives on parameter dependent components of a boundary. Part 3 Sobolev spaces for domains with cusps: extension of functions to the exterior of a domain with the vertex of a peak on the boundary imbedding theorems for Sobolev spaces in domains with cusps traces of functions in Sobolev spaces on the boundary of a domain with a cusp.

The Schrödinger operator on the energy space: boundedness and compactness criteria

We characterize the class of measurable functions (or, more generally, realor complexvalued distributions) V such that the Schr5dinger operator H=-A§ maps the energy o space L~(R n) to its dual L~-I(Rn). Similar results are obtained for the inhomogeneous Sobolev space W~(Rn). In other words, we give a complete solution to the problem of the relative form-boundedness of the potential energy operator V with respect to the Laplacian --A, which is fundamental to quantum mechanics. Relative compactness criteria for the corresponding quadratic forms are established as well. We also give analogous boundedness and compactness criteria for Sobolev spaces on domains f t C R n under mild restrictions on 012. One of the main goals of the present paper is to give necessary and sufficient conditions for the classical inequality

Global Lipschitz Regularity for a Class of Quasilinear Elliptic Equations

The Lipschitz continuity of solutions to Dirichlet and Neumann problems for nonlinear elliptic equations, including the p-Laplace equation, is established under minimal integrability assumptions on the data and on the curvature of the boundary of the domain. The case of arbitrary bounded convex domains is also included. The results have new consequences even for the Laplacian.

Criteria of solvability for multidimensional Riccati equations

We study the solvability problem for the multidimensional Riccati equation −∇u=|∇u|q+ω, whereq>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation −Δu−ωu=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions onRn in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type−Lu=f(x, u, ∇u)+ω where, andL is a uniformly elliptic operator.

Pathological solutions to elliptic problems in divergence form with continuous coefficients

We construct a function u is an element of W-loc(1.1) (B(0, 1)) which is a solution to div(A del u) = 0 in the sense of distributions, where A is continuous and u is not an element of W-loc(1.p) (B(0, 1)) for p > 1. We also give a function u is an element of W-loc(1.1)(B(0, 1)) such that u is an element of W-loc(1.p) (B(0, 1)) for every p < infinity, u satisfies div(A del u) = 0 with A continuous but u is not an element of W-loc(1.infinity) (B(0, 1)). This answers questions raised by H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335-338). To cite this article: T Jin et al., C. R. Acad. Sci. Paris., Ser. 1 347 (2009)

On Quasi-Interpolation with Non-uniformly Distributed Centers on Domains and Manifolds

The paper studies quasi-interpolation by scaled shifts of a smooth and rapidly decaying function. The centers are images of a smooth mapping of the hZ^n-lattice in R^s, s>=n, and the scaling parameters are proportional to h. We show that for a large class of generating functions the quasi-interpolants provide high order approximations up to some prescribed accuracy. Although in general the approximants do not converge as h tends to zero, the remaining saturation error is negligible in numerical computations if a scalar parameter is suitably chosen. The lack of convergence is compensated for by a greater flexibility in the choice of generating functions used in numerical methods for solving operator equations.

Jacques Hadamard, a universal mathematician

Part I. Hadamards Life: Prologue The beginning The turn of the century Mature years After the Great War Le Maitre In the thirties World War II After eighty Part II. Hadamards Mathematics: Analytic function theory Number theory Analytical mechanics and geometry Calculus of variations and functionals Miscellaneous topics Elasticity and hydrodynamics Partial differential equations Hadamards last works Epilogue Principal dates in Hadamards life A Hadamard collection Bibliography of Jacques Hadamard Publications about Jacques Hadamard and his work General bibliography Archival material Index.

Gradient regularity via rearrangements for

A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.

p

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 neighbor. 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.

-Laplacian type elliptic boundary value problems

Non-linear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented