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## Über dieses Buch

The new series, International Mathematical Series founded by Kluwer / Plenum Publishers and the Russian publisher, Tamara Rozhkovskaya is published simultaneously in English and in Russian and starts with two volumes dedicated to the famous Russian mathematician Professor Olga Aleksandrovna Ladyzhenskaya, on the occasion of her 80th birthday.

O.A. Ladyzhenskaya graduated from the Moscow State University. But throughout her career she has been closely connected with St. Petersburg where she works at the V.A. Steklov Mathematical Institute of the Russian Academy of Sciences.

Many generations of mathematicians have become familiar with the nonlinear theory of partial differential equations reading the books on quasilinear elliptic and parabolic equations written by O.A. Ladyzhenskaya with V.A. Solonnikov and N.N. Uraltseva.

Her results and methods on the Navier-Stokes equations, and other mathematical problems in the theory of viscous fluids, nonlinear partial differential equations and systems, the regularity theory, some directions of computational analysis are well known. So it is no surprise that these two volumes attracted leading specialists in partial differential equations and mathematical physics from more than 15 countries, who present their new results in the various fields of mathematics in which the results, methods, and ideas of O.A. Ladyzhenskaya played a fundamental role.

Nonlinear Problems in Mathematical Physics and Related Topics I presents new results from distinguished specialists in the theory of partial differential equations and analysis. A large part of the material is devoted to the Navier-Stokes equations, which play an important role in the theory of viscous fluids. In particular, the existence of a local strong solution (in the sense of Ladyzhenskaya) to the problem describing some special motion in a Navier-Stokes fluid is established. Ladyzhenskaya's results on axially symmetric solutions to the Navier-Stokes fluid are generalized and solutions with fast decay of nonstationary Navier-Stokes equations in the half-space are stated. Application of the Fourier-analysis to the study of the Stokes wave problem and some interesting properties of the Stokes problem are presented. The nonstationary Stokes problem is also investigated in nonconvex domains and some Lp-estimates for the first-order derivatives of solutions are obtained. New results in the theory of fully nonlinear equations are presented. Some asymptotics are derived for elliptic operators with strongly degenerated symbols. New results are also presented for variational problems connected with phase transitions of means in controllable dynamical systems, nonlocal problems for quasilinear parabolic equations, elliptic variational problems with nonstandard growth, and some sufficient conditions for the regularity of lateral boundary.

Additionally, new results are presented on area formulas, estimates for eigenvalues in the case of the weighted Laplacian on Metric graph, application of the direct Lyapunov method in continuum mechanics, singular perturbation property of capillary surfaces, partially free boundary problem for parametric double integrals.

## Inhaltsverzeichnis

### Area Formulas for σ-Harmonic Mappings

Abstract
The goal of the present paper is two-fold. First, we review some recent progress concerning generalizations of various classical results, such as sufficient conditions to guarantee univalence of harmonic mappings in dimension two, to certain pairs of elliptic partial differential equations with measurable coefficients. Second, we apply these results to prove new area formulas which are valid for a large class of mappings arising as solutions of these pairs of elliptic partial differential equations. Finally, we briefly discuss some applications to homogenized constants in the context of G-closure problems. To Professor Olga A. Ladyzhenskaya with our deep admiration
Giovanni Alessandrini, Vincenzo Nesi

### On a Variational Problem Connected with Phase Transitions of Means in Controllable Dynamical Systems

Abstract
We study the optimization of the integral of a given smooth function along the distribution determined by a density bounded by given functions from above and from below. Phase transitions are nonsmooth dependence of optimal means, as well as optimal strategies, on parameters. In this variational problem and, in particular, in the case of functions of even number of variables, they necessarily appear by topological reasons, which leads to logarithmic singularities. Two-dimensional variational problems in hydrodynamics and in magnetohydrodynamics are also considered. In these problems, as in the case studied in this paper, singularities are caused by topological reasons which I discussed with great pleasure with O. A. Ladyzhenskaya in Dilizhan in 1973.

### A Priori Estimates for Starshaped Compact Hypersurfaces with Prescribed mth Curvature Function in Space Forms

Abstract
We obtain a priori bounds for solutions of the nonlinear second-order elliptic equation of the geometric problem consisting in finding a compact starshaped hypersurface in a space form whose mth elementary symmetric function of principal curvatures is a given function.
J. Lucas M. Barbosa, Jorge H. S. Lira, Vladimir I. Oliker

### Elliptic Variational Problems with Nonstandard Growth

Abstract
For a bounded Lipschitz domain a minimization problem is considered over functions of the Orlicz-Sobolev space generated by an N-function A (with Δ2-property) that have prescribed trace u 0. Regularity results are established. In the vector case N > 1, partial C 1,α-regularity is proved without any additional structural conditions. The results are easily extended to the case of locally minimizing mappings. In the scalar case, the results obtained cover the case of (double) obstacles. Under an additional assumption, the regularity results can be improved (cf. Theorem 3 below which admits the anisotropic two-dimensional vector case).
Michael Bildhauer, Martin Fuchs

### Existence and Regularity of Solutions of dω = f with Dirichlet Boundary Conditions

Abstract
Given a bounded open set Ω ⊂ ℝ n and a (k + l)-form f satisfying some compatibility conditions, we solve the problem (in Hölder spaces)
$$d\omega = f\;in\;\Omega, \quad \omega = 0\;on\;\partial \Omega$$
.
Bernard Dacorogna

### A Singular Perturbation Property of Capillary Surfaces

Abstract
A discontinuous reversal of limiting behavior for capillary surfaces is considered, that was established for a particular configuration in [1]. A class of configurations is postulated, for which it is conjectured that an analogous behavior will be observed. We establish the conjecture in a particular case. We show also that the result of the conjecture, if correct, could not be significantly improved, in the sense that it can be made to fail under an arbitrary small change in the configuration at points distinct from the limiting set.
Robert Finn

### On Solutions with Fast Decay of Nonstationary Navier—Stokes System in the Half-Space

Abstract
The Navier—Stokes initial-value problem in the half-space is studied. Employing the asymptotic expansion of solutions, as well as the idea of [1], we specify a class of solutions which decay in time more rapidly than observed in general. The class is described in terms of moments and correlations of velocity fields. The existence of such solutions is proved. The same initial-value problem with the Neumann boundary condition as in [2] is considered. A class of solutions with fast decay is specified, also in terms of conditions on moments and correlations which, however, are complementary to those on solutions to the standard Navier—Stokes system.
Yoshiko Fujigaki, Tetsuro Miyakawa

### Strong Solutions to the Problem of Motion of a Rigid Body in a Navier—Stokes Liquid under the Action of Prescribed Forces and Torques

Abstract
This paper is devoted to the motion of a rigid body in an infinite Navier-Stokes liquid under the action of external forces and torques. For sufficiently regular data, we prove the existence of a local strong solution to the corresponding initial-boundary-value problem for the system body-liquid.
Giovanni P. Galdi, Ana L. Silvestre

### The Partially Free Boundary Problem for Parametric Double Integrals

Abstract
We prove the existence of conformally paramaterized minimizers for parametric two-dimensional variational problems subject to partially free boundary conditions. We establish regularity of class $$H_{loc}^{2,2} \cap C^{1,\alpha } ,0 < \alpha < 1$$, up to the free boundary under the assumption that there exists a perfect dominance function in the sense of Morrey.
Stefan Hildebrandt, Heiko von der Mosel

### On Evolution Laws Forcing Convex Surfaces to Shrink to a Point

Abstract
A general approach to evolution laws forcing convex surfaces to shrink to a point in a finite time is presented. The nontriviality of such a generalization is illustrated by examples.
Nina M. Ivochkina

### Existence of a Generalized Green Function for Integro-Differential Operators of Fractional Order

Abstract
It is a great pleasure and honor for both authors of this paper to contribute to this volume. Throughout their education in analysis, especially in the field of partial differential equations, the authors were in close contact with the ideas and works of O. A. Ladyzhenskaya, in particular through the well-known monographs [1]–[3] written by O. A. Ladyzhenskaya and her former students. The first author spent the academic year 1993/1994 as a graduate student in St.-Petersburg and thereby became acquainted with the famous school of analysis headed by O. A. Ladyzhenskaya.
Moritz Kassmann, Mark Steinhauer

### Lq-Estimates of the First-Order Derivatives of Solutions to the Nonstationary Stokes Problem

Abstract
For a solution to the nonstationary Stokes problem in the half-space $$\mathbb{R}_{ + }^3$$ with the external force $$f = \nabla \; \cdot \;F,\;F\; \in \;{L_q}\left( {\mathbb{R}_{ + }^3\; \times \;\left( {0,\;T} \right)} \right)$$, we establish the L q -estimates for the first-order derivatives of the vector field of velocities and prove that the pressure does not belong to the space $${L_q}\left( {\mathbb{R}_{ + }^3\; \times \;\left( {0,\;T} \right)} \right)$$.
Herbert Koch, Vsevolod A. Solonnikov

### Two Sufficient Conditions for the Regularity of Lateral Boundary for the Heat Equation

Abstract
The one-dimensional heat equation in the domain x > x(t), t ≥ 0, is considered. We prove the following fact: if the lateral boundary is “Hölder” regular for the heat equation u t = v 2 u xx for at least one v > 0, then it is regular for the equation with any v > 0. The proof is based on another condition of regularity somewhat close to the exterior cone condition for Laplace’s equation.
Nicolai V. Krylov

### Bound State Asymptotics for Elliptic Operators with Strongly Degenerated Symbols

Abstract
We study the rate of accumulation of eigenvalues at the edge of the essential spectrum of Schrödinger-type operators $${\left| {P\left( {i\nabla } \right)} \right|^{\gamma }} - V(x)$$V(x), where γ is a positive number, on L2(ℝ d ) in the case where the kinetic energy strongly degenerates at some nontrivial minimal Fermi surface P(ξ) = 0.
Ari Laptev, Oleg Safronov, Timo Weidl

### Nonlocal Problems for Quasilinear Parabolic Equations

Abstract
We study a class of quasilinear parabolic equations with nonlocal initial conditions. The initial conditions are a generalization of periodicity with respect to time and include conditions studied by other authors, which can be used to study inverse problems and problems arising in reactor theory.
Gary M. Lieberman

### Boundary Feedback Stabilization of a Vibrating String with an Interior Point Mass

Abstract
We study the boundary stabilization of a vibrating string with an interior point mass, zero Dirichlet condition at the left end and velocity feedback at the right end. Assuming finite energy initially, we show that the energy to the right of the point mass decays like C/t while that of the point mass decays like C/√t. The energy to the left of the point mass approaches zero but at no specific rate.
Walter Littman, Stephen W. Taylor

### On Direct Lyapunov Method in Continuum Theories

Abstract
Let S b be a basic motion. We consider two aspects of the direct Lyapunov method of stability theory. The first one is related to the control of perturbations of S b in terms of the data (stability in mean), and the second one is related to an asymptotic decay to zero for perturbation. First, for a Lyapunov functional we take the difference between the total energy of a given flow and that of the basic flow. An algorithm for computing the norm of perturbation (in a certain space) is demonstrated by three examples. We also propose the useful technique based on the general variational formulation. The algorithm consists in the choice of a test function. Precisely, we note that different test functions can be used for the same formulation and provide us with different informations. We show how to choose the test function in three examples.

### The Fourier Coefficients of Stokes’ Waves

Abstract
It is common to formulate the Stokes wave problem as Nekrasov’s nonlinear integral equation to be satisfied by a periodic function θ which gives the angle between the tangent to the wave and the horizontal. The function θ is odd for symmetric waves. In that case, numerical calculations using spectral methods reveal the coefficients in the sine series of θ to form a sequence of positive terms that converges monotonically to zero. In this paper, we prove that the Fourier sine coefficients of θ form a log-convex sequence that converges monotonically to zero. In harmonic analysis there are many very beautiful theorems about the behavior of functions whose Fourier sine series form a convex monotone sequence tending to zero.
Pavel I. Plotnikov, John F. Toland

### A Geometric Regularity Estimate via Fully Nonlinear Elliptic Equations

Abstract
We prove that integral n-varifolds μ in codimension 1 with $$H{\;_{\mu }}\; \in \;L_{{loc}}^p\left( \mu \right),\;p\; > \;n$$, p > n, p ≥ 2, have quadratic tilt-excess decay tiltexμ(x, ϱ, T x μ) = O x 2) for μ-almost all x. This regularity estimate is used to establish a general convergence procedure for hypersurfaces Σ j with interior E j whose mean curvatures are given by the trace of ambient Sobolev functions $$\overrightarrow {{H_{{{\Sigma_j}}}}} = {u_j}{\nu_{{{E_j}}}}$$ on Σ j , where ν Ej denotes the inner normal of Σ j .
Reiner Schätzle

### On Eigenvalue Estimates for the Weighted Laplacian on Metric Graphs

Abstract
It is shown that the eigenvalues of the equation — ⋋Δu = V u on a graph G of finite total length |G|, where V ∈ L1(G) is nonnegative, under appropriate boundary conditions satisfy the inequality n 2n ≤ |G| G Vdx independently of geometry of a given graph. Applications and generalizations of this result are also discussed.
Michael Solomyak

### Potential Theory for the Nonstationary Stokes Problem in Nonconvex Domains

Abstract
Based on the theory of nonstationary hydrodynamic potentials, we construct a solution to the nonstationary Stokes problem in a bounded domain or in an exterior domain with C 2 -boundary. For the kernel of the main hydrodynamic potential we take the matrix of “Poisson’s kernels” for the exterior of a ball.
Vsevolod A. Solonnikov

### Stability of Axially Symmetric Solutions to the Navier—Stokes Equations in Cylindrical Domains

Abstract
Using the result of O. A. Ladyzhenskaya [1], we establish the existence of a global solution, close to axially symmetric solutions, for the Navier-Stokes equations in a cylinder with boundary slip conditions. The solution belongs to a weight Sobolev space and possesses the property that the angular component of velocity, as well as the angular derivatives of cylindrical components of velocity and pressure, is sufficiently small. The uniqueness theorem is also established.
Wojciech M. Zajączkowski

### Backmatter

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