Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive numbers ρ<R, there exists a radially symmetric stationary solution with tumor boundary r=R and necrotic core boundary r=ρ. The system depends on a positive parameter μ, which describes the tumor aggressiveness. There also exists a sequence of values μ₂<μ₃<? for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric solution branch.     

Covering Spheres with Spheres

Given a sphere of any radius r in an n-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average number of solid spheres covering a point in a bigger sphere. For growing dimension n, we design a covering that gives the covering density of order (nln n)/2 for a sphere of any radius r>1 and a complete Euclidean space. This new upper bound reduces two times the order nln n established in the classic Rogers bound.     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

Closed-form solution of a maximization problem

According to the characterization of eigenvalues of a real symmetric matrix A, the largest eigenvalue is given by the maximum of the quadratic form 〈xA, x〉 over the unit sphere; the second largest eigenvalue of A is given by the maximum of this same quadratic form over the subset of the unit sphere consisting of vectors orthogonal to an eigenvector associated with the largest eigenvalue, etc. In this study, we weaken the conditions of orthogonality by permitting the vectors to have a common inner product r where 0 ≤ r < 1. This leads to the formulation of what appears—from the mathematical programming standpoint—to be a challenging problem: the maximization of a convex objective function subject to nonlinear equality constraints. A key feature of this paper is that we obtain a closed-form solution of the problem, which may prove useful in testing global optimization software. Computational experiments were carried out with a number of solvers. We dedicate this paper to the memory of our great friend and colleague, Gene H. Golub.     

The Diameter of a Hyperbolic Disc

Recently, Matti Vuorinen asked whether the set-theoretic diameter of a hyperbolic disc of radius r in a hyperbolic plane region Ω is 2r. The answer is affirmative if Ω is simply or doubly connected. However, there are a hyperbolic discs in the triply-punctured sphere whose set-theoretic diameter is less than twice the radius. Also, for finitely connected hyperbolic plane regions all hyperbolic discs sufficiently close to the boundary have set-theoretic diameter equal to twice the radius. Precisely, if Ω is a hyperbolic plane region of finite connectivity, then there is a compact subset K of Ω such that any hyperbolic disc which is disjoint from K has diameter equal to twice the radius.     

New formulations for the Kissing Number Problem

Determining the maximum number of D-dimensional spheres of radius r that can be adjacent to a central sphere of radius r is known as the Kissing Number Problem (KNP). The problem has been solved for two, three and very recently for four dimensions. We present two nonlinear (nonconvex) mathematical programming models for the solution of the KNP. We solve the problem by using two stochastic global optimization methods: a Multi Level Single Linkage algorithm and a Variable Neighbourhood Search. We obtain numerical results for two, three and four dimensions.     

Volumetric bounds for intersections of congruent balls in Euclidean spaces

We investigate the intersections of balls of radius r, called r-ball bodies, in Euclidean d-space. An r-lense (resp., r-spindle) is the intersection of two balls of radius r (resp., balls of radius r containing a given pair of points). We prove that among r-ball bodies of a given volume, the r-lense (resp., r-spindle) has the smallest inradius (resp., largest circumradius). In general, we upper (resp., lower) bound the intrinsic volumes of r-ball bodies of a given inradius (resp., circumradius). This complements and extends some earlier results on volumetric estimates for r-ball bodies.

     

Galois theory over rings of arithmetic power series

Let R be a domain, complete with respect to a norm which defines a non-discrete topology on R. We prove that the quotient field of R is ample, generalizing a theorem of Pop. We then consider the case where R is a ring of arithmetic power series which are holomorphic on the closed disc of radius 0<r<1 around the origin, and apply the above result to prove that the absolute Galois group of the quotient field of R is semi-free. This strengthens a theorem of Harbater, who solved the inverse Galois problem over these fields.     

No TVD Fields for 1-D Isentropic Gas Flow

We prove that isentropic gas flow does not admit non-degenerate TVD fields on any invariant set ?(r ₀, s ₀) = {r ₀ < r < s < s ₀}, where r, s are Riemann coordinates. A TVD field refers to a scalar field whose spatial variation Var _X (?(τ(t, X), u(t, X))) is non-increasing in time along entropic solutions. The result is established under the assumption that the Riemann problem defined by an overtaking shock-rarefaction interaction gives the asymptotic states in the exact solution.

Little is known about global existence of large-variation solutions to hyperbolic systems of conservation laws u _t  + f(u) _x  = 0. In particular it is not known if isentropic gas flow admits a priori BV bounds which apply to all BV data.

In the few cases where such results are available (scalar case, Temple class, systems satisfying Bakhvalov's condition, isothermal gas dynamics) there are TVD fields which play a key role for existence. Our results show that the same approach cannot work for isentropic flow.     

Embedding Theorems for Hölder Classes Defined on p-Adic Linear Spaces

We prove analogues of P. L. Ul’yanov and V. A. Andrienko results concerning embeddings of L^q Hölder spaces into Lebesgue spaces L^r or L^r Hölder spaces in the case 1 ≤ q < r ≤ 2 for functions defined on p-adic linear spaces. The conditions presented in these theorems are sharp. Also we give necessary and sufficient conditions for such embeddings in the case 1 ≤ q < r < ∞ that generalize recent results of S. S. Platonov.

     

Optimal reconstruction of the solution of the Dirichlet problem from inaccurate input data

In this paper, we consider the optimal reconstruction of the solution of the Dirichlet problem in the d-dimensional ball on the sphere of radius r from inaccurately prescribed traces of the solution on the spheres of radii R ₁ and R ₂, where R ₁ < r < R ₂. We also study the optimal reconstruction of the solution of the Dirichlet problem in the d-dimensional ball from a finite collection of Fourier coefficients of the boundary function which are prescribed with an error in the mean-square and uniform metrics.     

Time-periodic solutions of wave equations on ℝ1 and ℝ3

The classical Paiey-Wiener theorem and Hilbert space methods are used to show the existence of time-periodic solutions of the wave equation w_tt?w_rr+λw=h, 0 < r < + ∞, which are radially symmetric and have exponential decay as r → + ∞. This problem is obtained when considering a one-dimensional or three-dimensional problem, and should be thought of as a linearization of a semilinear problem in which the associated linear operator has point spectrum (? ∞, λ). When λ ? 0 there is uniqueness, otherwise there is a non-trivial finite-dimensional null space. Estimates on w, w_t, w_r are obtained, which show that in either case there is a continuous correspondence h → w, where w is a uniquely characterized solution.     

A minimal area problem in conformal mapping II

LetS denote the usual class of functionsf holomorphic and univalent in the unit diskU. For 0<r<1 andr(1+r)⁻²<b<r(1−r)⁻², letS(r, b) be the subclass of functionsf∈S such that |f(r)|=b. In Theorem 1, we solve the problem of minimizing the Dirichlet integral inS(r, b). The first main ingredient of the solution is the establishment of sufficient regularity of the domains onto whichU is mapped by extremal functions, and here techniques of symmetrization and polarization play an essential role. The second main ingredient is the identification of all Jordan domains satisfying a certain kind of functional equation (called “quadrature identities”) which are encountered by applying variational techniques. These turn out to be conformal images ofU by mappings of a special form involving a logarithmic function. In Theorem 2, this aspect of our work is generalized to encompass analogous minimal area problem when a larger number of initial data are prescribed. The third author thanks for its hospitality the Mittag-Leffler Institute of Royal Swedish Academy of Sciences where this work was finalized. This author was supported in part by the Swedish Institute and by the Russian Fund for Fundamental Research, grant no. 97-01-00259.     

Parameter-dependent operator equations of the Riccati type with application to transport theory

We obtain an existence result for global solutions to initial-value problems for Riccati equations of the form R′(t) + TR(t) + R(t)T = T^ρ A(t)T^1?ρ + T^ρ B(t)T^1?ρ R(t) + R(t)T^ρC(t) T^1?ρ + R(t)T^ρD(t)T^1?ρ R(t), R(0)=R₀, where 0 ? ρ ? 1 and where the functions R and A through D take on values in the cone of non-negative bounded linear operators on L¹ (0, W; μ). T is an unbounded multiplication operator. This problem is of particular interest in case ρ = 1 since it arisess in the theories of particle transport and radiative transfer in a slab. However, in this case there are some serious difficulties associated with this equation, which lead us to define a solution for the case ρ = 1 as the limit of solutions for the cases 0 < ρ < 1.     

Indefinite higher Riesz transforms

Stein’s higher Riesz transforms are translation invariant operators on L ²(R ⁿ ) built from multipliers whose restrictions to the unit sphere are eigenfunctions of the Laplace–Beltrami operators. In this article, generalizing Stein’s higher Riesz transforms, we construct a family of translation invariant operators by using discrete series representations for hyperboloids associated to the indefinite quadratic form of signature (p,q). We prove that these operators extend to L ^r -bounded operators for 1<r<∞ if the parameter of the discrete series representations is generic.     

On the stationary quasi-Newtonian flow obeying a power-law

We investigate in this paper existence of a weak solution for a stationary incompressible Navier-Stokes system with non-linear viscosity and with non-homogeneous boundary conditions for velocity on the boundary. Our concern is with the viscosity obeying the power-law dependence ν(ξ) = ∣Tr(ξξ*)∣^r/2?1, r < 2, on shear stress ξ. It is corresponding to most quasi-Newtonian flows with injection on the boundary. Since for r ? 2 the inertial term precludes any a priori estimate in general, we suppose the Reynolds number is not too large. Using the specific algebraic structure of the Navier-Stokes system we prove existence of at least one approximate solution. The constructed approximate solution turns out to be uniformly bounded in W^1,r (Omega;)ⁿ and using monotonicity and compactness we successfully pass to the limit for r ≥ 3n/(n + 2). For 3n/(n + 2) > r > 2n/(n + 2) our construction gives existence of at least one very weak solution. Furthermore, for r ≥ 3n/(n + 2) we prove that all weak solutions lying in the ball in W of radius smaller than critical are equal. Finally, we obtain an existence result for the flow through a thin slab.     

Perturbations of Hemivariational Inequalities with Constraints and Applications

The aim of the present paper is to discuss the influence which certain perturbations have on the solution of the eigenvalue problem for hemivariational inequalities on a sphere of given radius. The perturbation results in adding a term of the type >g ⁰(x, u(x); v(x)) to the hemivariational inequality, where g is a locally Lipschitz nonsmooth and nonconvex energy functional. Applications illustrate the theory.     

ASYMPTOTICS OF THE NUMBER OF RESONANCES IN THE TRANSMISSION PROBLEM

We consider the resonances for the transmission problem associated with a strictly convex transparent obstacle. Under some natural assumptions we show that there is a free of resonances region in the complex upper half plane given by {C ≤ Im λ ≤ C ₁|λ|^1/3 ? C ₂}, where C, C ₁ and C ₂ are positive constants. Moreover, we obtain asymptotics for the number of resonances counted with multiplicities in the region {0 < Im λ ≤ C, 0 < Re λ ≤ r} as r → ∞, where C > 0 is the same constant as above.

     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997