Equivalence of two definitions of a generalized <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> solution to the initial-boundary value problem for the wave equation

In our previous papers, we introduced the notion of a generalized solution to the initial-boundary value problem for the wave equation with a boundary function µ(t) such that the integral ∫ ₀ ^T (T ? t)|µ(t)| ^p dt exists. Here we prove that this solution is a unique solution to the problem in L _p that satisfies the corresponding integral identity.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Variational Approximation for Fokker–Planck Equation on Riemannian Manifold

Under the bounded geometry assumption on Riemannian manifold M, a variational approximation for Fokker–Planck equation on M is constructed by the scheme of Jordan et al. in SIAM J Math Anal 29(1):1–17, 1998. Moreover, the uniqueness and global L ^p -estimate of the solution for 1 < p < dim(M)/(dim(M) ? 1) are obtained for a broad class of potential.     

Recovery of Immersions from Their Metric Tensors and Nonlinear Korn Inequalities: A Brief Survey

The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω  R~n→ E~n with prescribed metric tensor field(g ij) : Ω→ S~n, and discuss the continuity of the mapping(gij) →φ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping(gij) →φ can be obtained by means of nonlinear Korn inequalities is shown.     

Stochastic continuity equations–A general uniqueness result

We showexistence and uniqueness of L^∞-solutions to stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition. The main issue of uniqueness is proved reeling on stochastic characteristic method.     

Maximum Distance Between Slater Orders and Copeland Orders of Tournaments

Given a tournament T?=?(X, A), we consider two tournament solutions applied to T: Slater’s solution and Copeland’s solution. Slater’s solution consists in determining the linear orders obtained by reversing a minimum number of directed edges of T in order to make T transitive. Copeland’s solution applied to T ranks the vertices of T according to their decreasing out-degrees. The aim of this paper is to compare the results provided by these two methods: to which extent can they lead to different orders? We consider three cases: T is any tournament, T is strongly connected, T has only one Slater order. For each one of these three cases, we specify the maximum of the symmetric difference distance between Slater orders and Copeland orders. More precisely, thanks to a result dealing with arc-disjoint circuits in circular tournaments, we show that this maximum is equal to n(n???1)/2 if T is any tournament on an odd number n of vertices, to (n ²???3n?+?2)/2 if T is any tournament on an even number n of vertices, to n(n???1)/2 if T is strongly connected with an odd number n of vertices, to (n ²???3n???2)/2 if T is strongly connected with an even number n of vertices greater than or equal to 8, to (n ²???5n?+?6)/2 if T has an odd number n of vertices and only one Slater order, to (n ²???5n?+?8)/2 if T has an even number n of vertices and only one Slater order.     

Minimum Riesz Energy Problems for a Condenser with Touching Plates

We consider minimum energy problems in the presence of an external field for a condenser with touching plates A₁ and A₂ in \(\mathbb {R}^{n}\), \(n\geqslant 3\), relative to the α-Riesz kernel |x?y|^α?n, \(0<\alpha \leqslant 2\). An intimate relationship between such problems and minimal α-Green energy problems for positive measures on A₁ is shown. We obtain sufficient and/or necessary conditions for the solvability of these problems in both the unconstrained and the constrained settings, investigate the properties of minimizers, and prove their uniqueness. Furthermore, characterization theorems in terms of variational inequalities for the weighted potentials are established. The approach applied is mainly based on the establishment of a perfectness-type property for the α-Green kernel with \(0<\alpha \leqslant 2\) which enables us, in particular, to analyze the existence of the α-Green equilibrium measure of a set. The results obtained are illustrated by several examples.     

Homogenization of variational inequalities for a biharmonic operator with constraints on subsets <Emphasis Type="Italic">ε</Emphasis>-periodically placed along a manifold of large dimension

Behavior of solutions of variational inequalities for a biharmonic operator is studied. These inequalities correspond to one-sided constraints on subsets of a domain Ω placed ε-periodically. All possible behavior types of solutions u _ε of variational inequalities are considered for ε → 0 depending on relations between small parameters, which are the structure period ε and the contraction coefficient a _ε of subsets where one-sided constraints are posed.     

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and L^t are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.     

Problem with analogs of the Frankl’ condition on a characteristic and the degeneration segment for an equation of mixed type with a singular coefficient

For the equation (sign y)|y| ^m u _xx +u _yy ?m(2y)^?1 u _y = 0, where m > 0, considered in some mixed domain, we prove existence and uniqueness theorems for the solution of the boundary value problem with an analog of the Frankl’ condition on a characteristic and on the degeneration segment of the equation.     

Fractional Abstract Cauchy Problems

This paper is concerned with fractional abstract Cauchy problems with order \({\alpha\in(1,2)}\). The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (FACP ₀) is well-posed if and only if its coefficient operator A generates an α-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (FACP _f ).     

One-dimensional viscous diffusion equation of higher order with gradient dependent potentials and sources

In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ(s) and sources A(s). We first show the general existence and uniqueness of global classical solutions provided that the first order derivatives of bothΦ(s) and A(s) are bounded below. Such a restriction is almost necessary, namely, if one of the derivatives is unbounded from below, then the solution might blow up in a finite time. A more interesting phenomenon is also revealed for potentials or sources being unbounded from below. In fact, if either the source or the potential is dominant, then the solution will blow up definitely in a finite time. Moreover,the viscous coefficient might postpone the blow-up time. Exactly speaking, for any T 0, the solution will never blow up during the period 0 t T, so long as the viscous coefficient is large enough.     

The Turán-type inequalities for complex polynomials in <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>-metrics

In this paper we obtain inequalities for measures of trigonometric polynomials of power (P _n (e ^iφ )) and general (T _n (t)) types with the help of measures and their mth derivatives.     

Fixed point theorems in <Emphasis Type="Italic">b</Emphasis>-metric spaces with applications to differential equations

In this paper, we present some fixed point theorems for a class of contractive mappings in b-metric spaces. We verify the T-stability of Picard’s iteration and the P property for such mappings. We also give an example to support our assertions. In addition, by using our results, we obtain the existence and uniqueness of solution to some ordinary differential equations with initial value conditions. Further, we provide the precise mathematical expressions of solutions to such equations.     

Generalized variational inequalities with generalized coercivity conditions

Some results due to Fang and Peterson on generalized variational inequalities in the space ? ⁿ are extended to infinite-dimensional spaces. Theorems on the existence of solutions of such inequalities under generalized coercivity conditions are obtained.     

Generalized Ambrosetti–Rabinowitz condition for minimal period solutions of autonomous Hamiltonian systems

We show, under an iterative condition, generalizing that of Ambrosetti–Rabinowitz and using a variational method, the existence of a T-periodic solution for the autonomous superquadratic second order Hamiltonian system with even potential

$$\begin{aligned} \ddot{z} +V'(z)=0, \qquad z\in \mathbb R^N, \qquad N\in \mathbb N^* \end{aligned}$$

for any prescribed period \(T>0\). Moreover, under a certain symmetry condition, such a solution possesses T or T / 3 as its minimal period.

     

Two classes of <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated with a von Neumann algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P ₁ and P ₂ of τ-measurable operators and investigate their properties. The class P ₂ contains P ₁. If a τ-measurable operator T is hyponormal, then T lies in P ₁; if an operator T lies in P _k , then UTU* belongs to P _k for all isometries U from M and k = 1, 2; if an operator T from P ₁ admits the bounded inverse T ^?1, then T ^?1 lies in P ₁. We establish some new inequalities for rearrangements of operators from P ₁. If a τ-measurable operator T is hyponormal and T ⁿ is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P ₁ coincides with the set of all paranormal operators on H.     

Convexity Inequalities for Positive Operators

We prove pointwise convexity (Jensen-type) inequalities of the form Open image in new window where F is a convex function defined on a convex subset of some Banach space X and T is the X-valued extension of a positive operator on some function space. Examples include the pointwise Hölder inequality T(fg) ≤ (Tf ^p )^{1/ p} (Tf ^q )^{1/ q} for a positive sublinear operator T. As applications we consider vector-valued conditional expectation and a ``real'' proof of the Riesz-Thorin theorem for positive operators.     

The column-sufficiency and row-sufficiency of the linear transformation on Hilbert spaces

Given a real Hilbert space H with a Jordan product and \({\Omega\subset H}\) being the Lorentz cone, \({q\in H}\), and let T : H → H be a bounded linear transformation, the corresponding linear complementarity problem is denoted by LCP(T, Ω, q). In this paper, we introduce the concepts of the column-sufficiency and row-sufficiency of T. In particular, we show that the row-sufficiency of T is equivalent to the existence of the solution of LCP(T, Ω, q) under an operator commutative condition; and that the column-sufficiency along with cross commutative property is equivalent to the convexity of the solution set of LCP(T, Ω, q). In our analysis, the properties of the Jordan product and the Lorentz cone in H are interconnected.     

An elementary proof of the <Emphasis Type="Italic">A</Emphasis><Subscript>2</Subscript> bound

A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp A _p bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L ¹ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A ² bounds in that setting.