Fixed point iterations for certain classes of nonlinear mappings

Let X = L_p or L_p, 2≤p<∞, and let K be a nonempty closed convex bounded subset of X. It is proved that for some classes of nonlinear mappings T:K → K (more precisely, for T P₂ or C in the terminology of F.E. Browder and W.V. Pretryshyn; and B.E. Rhoades), the iteration process: x₁ ?K,X_n+1 = (1-C_n)x_n+C_n Tx_n, n ≥1,under suitable conditions on K and the real sequence {C_n}_n=1 ^∞ converges strongly to a fixed point of T. While our thorems generalize serveral known results, our method is also of independent interest     

Variational approach to contact problems in nonlinear elasticity

We use variational methods to study problems in nonlinear 3-dimensional elasticity where the deformation of the elastic body is restricted by a rigid obstacle. For an assigned variational problem we first verify the existence of constrained minimizers whereby we extend previous results. Then we rigorously derive the Euler-Lagrange equation as necessary condition for minimizers, which was possible before only under strong smoothness assumptions on the solution. The Lagrange multiplier corresponding to the obstacle constraint provides structural information about the nature of frictionless contact. In the case of contact with, e.g., a corner of the obstacle, we derive a qualitatively new contact condition taking into account the deformed shape of the elastic body. By our analysis it is shown here for the first time rigorously that energy minimizers really solve the mechanical contact problem. Received: 20 October 2000 / Accepted: 7 June 2001 / Published online: 5 September 2002     

Slow mappings of finite distortion

We examine mappings of finite distortion from Euclidean spaces into Riemannian manifolds. We use integral type isoperimetric inequalities to obtain Liouville type growth results under mild assumptions on the distortion of the mappings and the geometry of the manifolds.     

The boundary distortion and extremal problems in certain classes of univalent functions

We consider the class S₁(τ), 0<τ<1, of functions f(z)=rz+a₂z²+... that are regular and univalent in the unit disk U and have |f(z)|<1. We obtain sharp estimates for the 1-measure of the sets {θ: |f(e^iθ)|=1}. As a corollary, for the familiar class S we find Kolmogorov-type estimates for the sets {θ: |f(e^iθ)|>M}, M>1, and prove inequalities for the harmonic measure, which are similar to those by Carleman-Milloux and Baernstein. We also consider problems on distortion of fixed systems of boundary arcs in the classes of functions that are regular (or meromorphic) and univalent in the disk or circular annulus. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 115–142. Translated by A. Yu. solynin.     

Partial regularity for certain classes of polyconvex functionals related to nonlinear elasticity

Consider the variational integral where Ω⊂ℝ ⁿ andp≥n≥2. H: (0, ∞)→[0, ∞) is a smooth convex function such that . We approximateJ by a sequence of regularized functionalsJ _δ whose minimizers converge strongly to anJ-minimizing function and prove partial regularity results forJ _δ-minimizers.     

Compactness theory and mappings with finite length distortion

The present paper is devoted to the study of mappings with finite length distortion introduced in 2004 by O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov. It is proved that the locally uniform limit of homeomorphisms with finite length distortion is a homeomorphism or a constant provided that the so-called inner dilatations of the sequence of homeomorphisms are almost everywhere (a.e.) majorized by a locally integrable function. In particular, it is studied the pointwise behavior of the so-called outer dilatations. For these dilatations, the pointwise semicontinuity and semicontinuty in the mean are proved. It is also proved some theorems on the convergence of matrix dilatations.     

On the theory of mappings with finite area distortion

It is shown that mappings in ℝⁿ with finite distortion of area in all dimensions 1 ≤ k ≤ n − 1 satisfy certain modulus inequalities in terms of inner and outer dilatations of the mappings; in particular, generalizations of the well-known Poletskii inequality for quasiregular mappings are proved. The theory developed is applicable, for example, to the class of finitely bi-Lipschitz mappings, which is a natural generalization of the bi-Lipschitz mappings, as well as isometries and quasi-isometries in ℝⁿ.     

On the weak limit of mappings with finite distortion

We give a new proof that the limit of a weakly convergent sequence of mappings with finite distortion also has finite distortion. The result has been recently proved by Gehring and Iwaniec using the biting convergence of Jacobians. We present a different proof using simply the lower semi-continuity of quasiconvex functionals.

     

On a certain differential property of mappings with bounded distortion

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 6, pp. 199–200, November–December, 1992.     

On the lower order of mappings with finite length distortion

We study the problem of the so-called lower order for one kind of mappings with finite distortion, actively investigated in the recent 15–20 years.We prove that mappings with finite length distortion f: D → ? ⁿ , n ≥ 2, whose outer dilatation is integrable to the power α > n ? 1 with finite asymptotic limit have lower order bounded from below.     

On one modulus inequality for mappings with finite length distortion

The V?is?l? inequality, which is well known in the theory of quasilinear mappings, is extended to the class of mappings with finite length distortion.     

Variational analysis in dynamical problems of linear elasticity

The initial–boundary value problems in the linear theory of elasticity is considered. Based on the method of integrodifferential relations (MIDR) two dynamical variational principles is proposed and discussed. It is shown that the Hamilton principle as well as the corresponding complementary principle stated for dynamic boundary value problems follow out the variational formulations proposed. To minimize the nonnegative functional under algebraic and differential constraints a regular finite element algorithm is worked out. The algorithm allows us to estimate explicitly the local and integral quality of numerical solutions obtained. A 3D problem of lateral motions of a rectilinear elastic prism with a rectangular cross section are considered. The numerical results are presented and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive solutions to certain classes of singular nonlinear second order boundary value problems

This paper is devoted to the problem of existence of solutions to the nonlinear singular two point boundary value problem , withy satisfying either mixed boundary datay(1)=Lim_y0+p(t)y(t)=0 or dirichlet boundary datay(0)=y(1)=0. Throughout our nonlinear termqf is allowed to be singular att=0,t=1,y=0 and/orpy=0.