Korn inequality on irregular domains

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s-John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality.     

Korn inequality for a thin rod with rounded ends

We consider an elastic rod with rounded ends and diameter proportional to a small parameter h > 0. The roundness of the ends is described by an exponent m ∈ (0,1). We derive for the rod an asymptotically sharp Korn inequality with a special weighted anisotropic norm. Weight factors with m‐dependent powers of h appear both in the anisotropic norm and the Korn inequality itself, and we discover three critical values m = 1 ∕ 4, m = 1 ∕ 2 and m = 3 ∕ 4 at which these exponents are crucially changed. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotically sharp weight Korn’s inequality for thin-walled elastic structures

An elastic junction of several thin plates is considered. All the plates, except for one, called the basic plate, are rigidly clamped along parts of lateral surfaces. We deduce an asymptotically sharp Korn inequality which is weighted and anisotropic. The constant in this Korn inequality is independent of two parameters, the thickness h ∈ (0, 1] and relative rigidity μ ∈ (0, +∞) of the supporting and basic plates. The weight factors in the Sobolev norm on the basic plate essentially depend on the parameters h, μ and on the mutual disposition of the supporting plates. Sufficient geometric and algebraic conditions for the validity of Korn inequalities with various groups of weight factors are given. We also describe special constructions that show the impossibility to improve the obtained inequalities and the necessity of the restrictions imposed on the junction structure. Bibliography: 29 titles. Illustrations: 12 figures. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 29–64.     

A nonlinear descent method for a variational inequality on a nonconvex set

In this paper, we consider a generalized variational inequality problem which involves the integrable cost mapping and a nonsmooth mapping with convex components. We propose a new gradient-type method which determines a stepsize by using the smooth part of the cost function. Thus, the method does not utilize analogs of derivatives of nonsmooth functions. We show that its convergence does not require additional assumptions.     

The isoperimetric inequality on a surface

We prove a new isoperimetric inequality which relates the area of a multiply connected curved surface, its Euler characteristic, the length of its boundary, and its Gaussian curvature. Received: 31 July 1998     

A nonlinear inequality of Moser-Trudinger type

In this note, we will prove an inequality for almost plurisubharmonic functions on any K?hler-Einstein manifolds with positive scalar curvature. This inequality generalizes the stronger version of the so called Moser-Trudinger-Onofri inequality on , which was proved in [Au], and also refines a weaker inequality found by the first author in [T2]. Received: May 27, 1997 / Accepted: June 11, 1999     

On a nonlinear inequality on the time scale

In the present paper, we consider a dynamic nonlinear integral inequality with a power-law nonlinearity. We obtain a solution of this inequality for arbitrary nonlinearity exponents exceeding unity. These results can be constructively used in the analysis of stability properties (including nonclassical stability properties) of quasilinear dynamic equations.     

Nonlinear Korn Inequalities on a Hypersurface

The authors establish several estimates showing that the distance in W~(1,p),1 p ∞,between two immersions from a domain of R~n into R~(n+1) is bounded by the distance in L~p between two matrix fields defined in terms of the first two fundamental forms associated with each immersion. These estimates generalize previous estimates obtained by the authors and P. G. Ciarlet and weaken the assumptions on the fundamental forms at the expense of replacing them by two different matrix fields.     

A Note on Weighted Korn Inequality

In this note, we show that, for domains satisfying the separation property, certain weighted Korn inequality is equivalent to the John condition. Our result generalizes previous result from Jiang–Kauranen [Calc. Var. Partial Differential Equations, 56, Art. 109,(2017)] to weighted settings.     

Notes to the proof of a weighted Korn inequality for an elastic body with peak-shaped cusps

We present the proof of the weighted anisotropic Korn inequality in a three-dimensional domain with peak-shaped cusps on the boundary. We verify the asymptotic accuracy of distribution of multipliers at the components of displacement vector and their derivatives in the corresponding weighted norm. We indicate conditions on a peak cusp under which the natural energy class is not embedded into a Sobolev or Lebesgue class. In the last case, the operator of elasticity problems possesses the continuous spectrum provoking wave processes in a finite volume (“black holes” for elastic waves). We also discuss possible generalizations of the result and open questions. Bibliography: 39 titles. Illustrations: 9 figures.     

On the isoperimetric inequality on a minimal surface

We derive an explicit formula for the isoperimetric defect of an arbitrary minimal surface ,in terms of a double integral over the surface of certain geometric quantities, together with a double boundary integral which always has the correct sign. As a by-product of these computations we show that the best known universal isoperimetric estimate, that for any minimal surface (due to L. Simon), may be improved to the universal estimate .Received: 21 June 2001, Accepted: 16 June 2002, Published online: 5 September 2002     

A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface

In this paper, a sharp form of the Moser-Trudinger inequality is established on a compact Riemannian surface via the method of blow-up analysis, and the existence of an extremal function for such an inequality is proved.

     

A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities

In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i -1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1~(**)(Σ, g)), we have■When b b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.     

Harnack's inequality for a nonlinear eigenvalue problem on metric spaces

We prove Harnack's inequality for first eigenfunctions of the p-Laplacian in metric measure spaces. The proof is based on the famous Moser iteration method, which has the advantage that it only requires a weak (1,p)-Poincaré inequality. As a by-product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains.