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## Über dieses Buch

This volume includes several invited lectures given at the International Workshop "Analysis, Partial Differential Equations and Applications", held at the Mathematical Department of Sapienza University of Rome, on the occasion of the 70th birthday of Vladimir G. Maz'ya, a renowned mathematician and one of the main experts in the field of pure and applied analysis.

The book aims at spreading the seminal ideas of Maz'ya to a larger audience in faculties of sciences and engineering. In fact, all articles were inspired by previous works of Maz'ya in several frameworks, including classical and contemporary problems connected with boundary and initial value problems for elliptic, hyperbolic and parabolic operators, Schrödinger-type equations, mathematical theory of elasticity, potential theory, capacity, singular integral operators, p-Laplacians, functional analysis, and approximation theory.

Maz'ya is author of more than 450 papers and 20 books. In his long career he obtained many astonishing and frequently cited results in the theory of harmonic potentials on non-smooth domains, potential and capacity theories, spaces of functions with bounded variation, maximum principle for higher-order elliptic equations, Sobolev multipliers, approximate approximations, etc. The topics included in this volume will be particularly useful to all researchers who are interested in achieving a deeper understanding of the large expertise of Vladimir Maz'ya.

## Inhaltsverzeichnis

### On the Occasion of the 70th Birthday of Vladimir Maz’ya

This volume includes a selection of lectures given at the International Workshop “Analysis, Partial Differential Equations and Applications”, held at the Mathematical Department of Sapienza University (Rome, June 30th–July 3rd, 2008), on the occasion of the 70th birthday of Vladimir Maz’ya.
Alberto Cialdea, Flavia Lanzara, Paolo E. Ricci

### Boundary Trace for BV Functions in Regions with Irregular Boundary

The aim of this work is to generalize some results of [2] by Yu. Burago and V. Maz’ya for a wider class of regions with irregular boundaries.
Yuri Burago, Nikolay N. Kosovsky

### Dirac Equation as a Special Case of Cosserat Elasticity

We suggest an alternative mathematical model for the electron in which the dynamical variables are a coframe (field of orthonormal bases) and a density. The electron mass and external electromagnetic field are incorporated into our model by means of a Kaluza-Klein extension. Our Lagrangian density is proportional to axial torsion squared. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. We prove that in the special case with no dependence on the third spatial coordinate our model is equivalent to the Dirac equation. The crucial element of the proof is the observation that our Lagrangian admits a factorisation.
James Burnett, Olga Chervova, Dmitri Vassiliev

### Hölder and Lipschitz Estimates for Viscosity Solutions of Some Degenerate Elliptic PDE’s

We report here on some recent results, obtained in collaboration with F. Leoni and A. Porretta [7] concerning Hölder and Lipschitz regularity and the solvability of the Dirichlet problem for degenerate quasilinear elliptic equations of the form
$$- Tr(A(x)D^2 u) + |Du|^p + \lambda u = f(x),x \in \Omega .$$
The research presented here is partly motivated by a paper by J.M. Lasry and P.L. Lions [12]. Our results can be regarded as extensions to the degenerate elliptic case of some of those contained in that paper.
Italo Capuzzo Dolcetta

### Criteria for the L p -dissipativity of Partial Differential Operators

In this paper we present a survey of some results concerning the L p -dissipativity of partial differential operators. These results were obtained in joint works with V. Maz’ya
Alberto Cialdea

### Sharp Estimates for Nonlinear Potentials and Applications

A sharp estimate for the decreasing rearrangement of the nonlinear potential of a function in terms of the rearrangement of the function itself is presented. As a consequence, boundedness properties of nonlinear potentials in rearrangement invariant spaces are characterized. In particular, the case of Orlicz and Lorentz spaces is discussed. Applications to rearrangement estimates for local solutions to quasilinear elliptic PDE’s and for their gradients are given.
Andrea Cianchi

### Solvability Conditions for a Discrete Model of Schrödinger’s Equation

Let ω be a Borel measure on ℝ n , and let Q denote the dyadic cubes in ℝ n . For a sequence s= |s Q } Q∈Q of nonnegative scalars, we consider an operator T defined by $$Tu(x) = \int_{\mathbb{R}^n } {K(x,y)u(y)dw(y)}$$ with kernel $$K(x,y) = \sum\nolimits_{Q \in \mathcal{Q}} {s_Q \omega } (Q)^{ - 1} \chi _Q (x)\chi _Q (y)$$. We obtain conditions for the existence of a solution u to the inhomogeneous equation u=Tu+α, which serves as a discrete model for an inhomogeneous, time-independent Schrödinger equation on ℝ n . Define a discrete Carleson norm
$$||s||_\omega = \mathop {\sup }\limits_{Q \in \mathcal{Q}:\omega (Q) \ne 0} \omega (Q)^{ - 1} \sum\limits_{P \in \mathcal{Q}:P \subseteq Q} {|s_P |\omega (P),}$$
, and let $$A_Q (x) = \sum\nolimits_{P \in \mathcal{Q}:P \subseteq Q} {s_P \chi _P (x)}$$. If $$||s||_\omega < \tfrac{1} {{12}}$$, and
$$\int_{\mathbb{R}^n } {\sum\limits_{Q \in \mathcal{Q}} {\frac{{s_Q }} {{|Q|_\omega }}e^6 (A_Q (x) + A_Q (y))_{\chi Q(x)\chi Q(y)|\alpha (y)|d\omega (y) < + \infty } } }$$
dω-a.e., then there exists u satisfying u=Tu+α. Other sufficient conditions are derived. In the converse direction, if α≥ 0 and the equation u=Tu+α has a solution u≥0, then $$||s||_\omega \leqslant 1$$ and
$$\int_{\mathbb{R}^n } {\sum\limits_{Q \in \mathcal{Q}} {\frac{{s_Q }} {{|Q|_\omega }}e^{\tfrac{1} {2}} (A_Q (x) + A_Q (y))_{\chi Q(x)\chi Q(y)|\alpha (y)|d\omega (y) < + \infty } } }$$
dω-a.e. These results are obtained from bilateral estimates for the kernel of the Neumann series $$\sum\nolimits_{j = 0}^\infty {T^j }$$.
Michael Frazier, Igor Verbitsky

### An Algebra of Shift-invariant Singular Integral Operators with Slowly Oscillating Data and Its Application to Operators with a Carleman Shift

The paper is devoted to studying Banach algebras of shift-invariant singular integral operators with slowly oscillating coefficients and their extensions by shift operators associated with iterations of a slowly oscillating Carleman shift generating a finite cyclic group. Both algebras are contained in the Banach algebra of bounded linear operators on a weighted Lebesgue space with a slowly oscillating Muckenhoupt weight over a composed slowly oscillating Carleson curve. By applying the theory of Mellin pseudodifferential operators, Fredholm symbol calculi for these algebras and Fredholm criteria and index formulas for their elements are established in terms of their Fredholm symbols.
Yu. I. Karlovich

### Frozen History: Reconstructing the Climate of the Past

The ice caps on Greenland and Antarctica are huge memory banks: the temperature of the past is preserved deep in the ice. In this paper we present a mathematical model for the reconstruction of past temperatures from records of the present ones in a drilled hole.
Christer O. Kiselman

### Multidimensional Harmonic Functions Analogues of Sharp Real-part Theorems in Complex Function Theory

In the present paper, the sharp multidimensional analogues of Lindelöf inequality and similar estimates for analytic functions are considered. Using a sharp inequality for the gradient of a bounded or semibounded harmonic function in a ball, one arrives at improved estimates (compared with the known ones) for the gradient of harmonic functions in an arbitrary subdomain of ℝ n . A representation of the sharp constant in a pointwise estimate of the gradient of a harmonic function in a half-space is obtained under the assumption that function’s boundary values belong to L p . This representation is realized in the three-dimensional case and the values of sharp constants are explicitly given for p=1, 2,∞.
Gershon Kresin

### Cubature of Integral Operators by Approximate Quasi-interpolation

In this paper we report on some recent results concerning Hermite quasi-interpolation on uniform grids with interesting applications to the approximation of solutions to elliptic PDE, quasi-interpolation on nonuniform grids and the cubature of convolutions with radial kernel functions based on an approximation method proposed by V. Maz’ya.
Flavia Lanzara, Gunther Schmidt

### Pointwise Estimates for the Polyharmonic Green Function in General Domains

In the present paper we establish sharp estimates on the polyharmonic Green function and its derivatives in an arbitrary bounded open set.

### On Elliptic Operators in Nondivergence and in Double Divergence Form

This work surveys results on the existence and asymptotic behavior of the fundamental solution for an elliptic operator L in nondivergence form, including recent results for operators whose coefficients are continuous with mild conditions on the modulus of continuity: if the square of the modulus of continuity satisfies the Dini condition, then there is an integral invariant which controls the behavior of solutions of L*u=0 and whether there is a fundamental solution for L that is asymptotic to the fundamental solution for the associated constant coefficient operator.
Robert McOwen

### On the Well-posedness of the Dirichlet Problem in Certain Classes of Nontangentially Accessible Domains

We prove that if Ω ⊂ ℝ n is a bounded NTA domain (in the sense of Jerison and Kenig) with an Ahlfors regular boundary, and which satisfies a uniform exterior ball condition, then the Dirichlet problem
$$\Delta u = 0 in \Omega , u|_{\partial \Omega } = f \in L^p (\partial \Omega , d\sigma ),$$
(1)
, has a unique solution for any p ∈ (1,∞). This solution satisfies natural nontangential maximal function estimates and can be represented as
$$u(y) = - \int_{\partial \Omega } {\partial _{\nu (x)} G(x,y)f(x) d\sigma (x),} y \in \Omega .$$
(1)
. Above, ν denotes the outward unit normal to Ω and G(·,·) stands for the Green function associated with Ω.
Dorina Mitrea, Marius Mitrea

### On Negative Spectrum of Schrödinger Type Operators

The classical Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities in their general form allow one to estimate the number of negative eigenvalues N = # i < 0} and the sums S γ = ∑|λ i | γ for a wide class of Schrödinger operators. We will present here some new examples (Anderson Hamiltonian, operators on lattices, quantum graphs and groups). In some cases below, the parabolic semigroup has an exponential fractional decay at t→∞. This makes it possible to consider potentials decaying very slowly (logarithmical) at infinity. We also will discuss the case of small local dimension of the underlying manifold, which is usually not covered by the general theory.
S. Molchanov, B. Vainberg

### ACL-homeomorphisms in the Plane

We study planar homeomorphisms $$f:\Omega \subset \mathbb{R}^2 \underrightarrow {onto} \Omega ' \subset \mathbb{R}^2$$, f=(u, v), which are absolutely continuous on lines parallel to the axes (ACL) together with their inverse f −1. The main result is that u and v have almost the same critical points. This generalizes a previous result ([8]) and extends investigation of ACL-solutions to non-trivial first-order systems of PDE’s. The main ingredients are (N) and co-(N) properties of such mappings that we call ACL-homeomorphisms.
G. Moscariello, A.Passarelli di Napoli, C. Sbordone

### Crack Problems for Composite Structures

We investigate three-dimensional interface crack problems (ICP) for metallic-piezoelectric composite bodies with regard to thermal effects. We give a mathematical formulation of the physical problem when the metallic and piezoelectric bodies are bonded along some proper parts of their boundaries where interface cracks occur. By potential methods the ICP is reduced to an equivalent strongly elliptic system of pseudodifferential equations on manifolds with boundary. We study the solvability of this system in different function spaces and prove uniqueness and existence theorems for the original ICP. We analyze the regularity properties of the corresponding thermomechanical and electric fields near the crack edges and near the curves where the boundary conditions change. In particular, we characterize the stress singularity exponents, which essentially depend on the material parameters, and show that they can be explicitly calculated with the help of the principal homogeneous symbol matrices of the corresponding pseudodifferential operators.
David Natroshvili, Zurab Tediashvili

### On Positive Solutions of p-Laplacian-type Equations

Let Ω be a domain in ℝ d , d ≥ 2, and 1 < p < ∞. Fix V loc (Ω). Consider the functional Q and its Gâteaux derivative Q′ given by
$$Q(u): = \tfrac{1} {p}\int_\Omega {(|\nabla u|^p + V|u|^p )dx, Q'(u): = - \nabla \cdot (|\nabla u|^{p - 2} \nabla u) + V|u|^{p - 2} u.}$$
In this paper we discuss a few aspects of relations between functional-analytic properties of the functional Q and properties of positive solutions of the equation Q′ (u)=0.
Yehuda Pinchover, Kyril Tintarev

### Mixed Boundary Value Problems for Stokes and Navier-Stokes Systems in Polyhedral Domains

The paper deals with a boundary value problem for the stationary Stokes and Navier-Stokes systems, where different boundary conditions (in particular, Dirichlet, Neumann, slip conditions) are prescribed on the faces of a polyhedral domain. Various regularity results in weighted and nonweighted Sobolev and Hölder spaces are given here. Furthermore, the paper contains a maximum modulus estimate for the velocity.
Jürgen Rossmann

### On Some Classical Operators of Variable Order in Variable Exponent Spaces

We give a survey of a selection of recent results on weighted and non-weighted estimations of classical operators of Harmonic Analysis in variable exponent Lebesgue, Morrey and Hölder spaces, based on the talk presented at International Conference Analysis, PDEs and Applications on the occasion of the 70th birthday of Vladimir Maz’ya, Rome, June 30–July 3, 2008. We touch both the Euclidean case and the general setting within the frameworks of quasimetric measure spaces. Some of the presented results are new.
Stefan Samko

### Irregular Conductive Layers

In this survey some singular homogenization results are described. This approach leads to the spectral convergence of a sequence of weighted second-order elliptic partial differential operators to a singular elliptic operator with a fractal term.
Maria Agostina Vivaldi

### On the Double Layer Potential

Neumann’s classical integral equation with the double layer potential operator is considered on different spaces of boundary charges, such as continuous data, L2(Γ) and energy trace spaces. Corresponding known results for different classes of boundaries are collected and discussed in view of their consequences for collocation or Galerkin boundary element methods.
W. L. Wendland
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