Convergent approximations in parabolic variational inequalities II: Hamilton-jacobi inequalities

In this paper we consider two-sided parabolic inequalities of the form (li) $$\psi _1 \leqslant u \leqslant \psi _2 , in{\mathbf{ }}Q;$$ (lii) $$\left[ { - \frac{{\partial u}}{{\partial t}} + A(t)u + H(x,t,u,Du)} \right]e \geqslant 0, in{\mathbf{ }}Q,$$ for alle in the convex support cone of the solution given by $$K(u) = \left\{ {\lambda (\upsilon - u):\psi _1 \leqslant \upsilon \leqslant \psi _2 ,\lambda > 0} \right\}{\mathbf{ }};$$ (liii) $$\left. {\frac{{\partial u}}{{\partial v}}} \right|_\Sigma = 0, u( \cdot ,T) = \bar u$$ where $$Q = \Omega \times (0,T), \sum = \partial \Omega \times (0,T).$$ Such inequalities arise in the characterization of saddle-point payoffsu in two person differential games with stopping times as strategies. In this case,H is the Hamiltonian in the formulation. A numerical scheme for approximatingu is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence tou of orderO(h ^1/2) is demonstrated in theL ²(0,T; H ¹(Ω)) norm, whereh is the maximum diameter of a given triangulation.     

Nonlinear elliptic equations of order 2m and subdifferentials

LetA be an operator of the calculus of variations of order 2m onW ^m,p (Ω) andj a normal convex integrand. Forf ∈L ^p (Ω), the equation $$\mathcal{A}u + \partial j(x,u) \ni f, in \Omega , u - \phi \in W_0^{m,p} (\Omega ),$$ may have no strong solutions whenm>1, even ifj is independent ofx and φ=0. However, we obtain existence results whenj is everywhere finite and $$\int_\Omega {j(x,\phi ) dx< + \infty ,} $$ by the study of the subdifferential of the function $$\upsilon \mapsto \int_\Omega {j(x,\upsilon + \phi ) dx on W_0^{m,p} (\Omega ).} $$      

An integral equation and a representation for a Green's function

The final step in the mathematical solution of many problems in mathematical physics and engineering is the solution of a linear, two-point boundary-value problem such as $$\begin{gathered} \ddot u - q(t)u = - g(t), 0< t< x \hfill \\ (0) = 0, \dot u(x) = 0 \hfill \\ \end{gathered} $$ Such problems frequently arise in a variational context. In terms of the Green's functionG, the solution is $$u(t) = \int_0^x {G(t, y, x)g(y) dy} $$ It is shown that the Green's function may be represented in the form $$G(t,y,x) = m(t,y) - \int_y^x {q(s)m(t, s) m(y, s)} ds, 0< t< y< x$$ wherem satisfies the Fredholm integral equation $$m(t,x) = k(t,x) - \int_0^x k (t,y) q(y) m(y, x) dy, 0< t< x$$ and the kernelk is $$k(t, y) = min(t, y)$$      

Necessary and sufficient conditions for optimal control of quasilinear partial differential systems

First-order necessary and sufficient conditions are obtained for the following quasilinear distributed-parameter optimal control problem: $$max\left\{ {J(u) = \int_\Omega {F(x,u,t) d\omega + } \int_{\partial \Omega } {G(x,t) \cdot d\sigma } } \right\},$$ subject to the partial differential equation $$A(t)x = f(x,u,t),$$ wheret,u,G are vectors andx,F are scalars. Use is made of then-dimensional Green's theorem and the adjoint problem of the equation. The second integral in the objective function is a generalized surface integral. Use of then-dimensional Green's theorem allows simple generalization of single-parameter methods. Sufficiency is proved under a concavity assumption for the maximized Hamiltonian $$H^\circ (x,\lambda ,t) = \max \{ H(x,u,\lambda ,t):u\varepsilon K\} $$ .     

Minimal decompositions of graphs into mutually isomorphic subgraphs

SupposeG _n={G ₁, ...,G _k } is a collection of graphs, all havingn vertices ande edges. By aU-decomposition ofG _n we mean a set of partitions of the edge setsE(G _t ) of theG _i , sayE(G _t )== \(\sum\limits_{j = 1}^r {E_{ij} } \) E _ij , such that for eachj, all theE _ij , 1≦i≦k, are isomorphic as graphs. Define the functionU(G _n) to be the least possible value ofr aU-decomposition ofG _n can have. Finally, letU _k (n) denote the largest possible valueU(G) can assume whereG ranges over all sets ofk graphs havingn vertices and the same (unspecified) number of edges. In an earlier paper, the authors showed that $$U_2 (n) = \frac{2}{3}n + o(n).$$ In this paper, the value ofU _k (n) is investigated fork>2. It turns out rather unexpectedly that the leading term ofU _k (n) does not depend onk. In particular we show $$U_k (n) = \frac{3}{4}n + o_k (n),k \geqq 3.$$      

Regularity property of Donsker's delta function

LetL be the space of rapidly decreasing smooth functions on ? andL ^* its dual space. Let (L ²)⁺ and (L ²)^? be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise spaceL ^* with standard Gaussian measure. The Donsker delta functionδ(B(t)?x) is in (L ²)^? and admits the series representation $$\delta (B(t) - x) = (2\pi t)^{ - 1/2} \exp ( - x^2 /2t)\sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} H_n (x/\sqrt {2t} )} \times H_n (B(t)/\sqrt {2t} )$$ , whereH _n is the Hermite polynomial of degreen. It is shown that forφ in (L ²)⁺,g _t,φ(x)≡〈δ(B(t)?x), φ〉 is inL and the linear map takingφ intog _t,φ is continuous from (L ²)⁺ intoL. This implies that forf inL ^* is a generalized Brownian functional and admits the series representation $$f(B(t)) = (2\pi t)^{ - 1/2} \sum\limits_{n = 0}^\infty {(n!2^n )^{ - 1} \langle f,\xi _{n, t} \rangle } H_n (B(t)/\sqrt {2t} )$$ , whereξ _n,t is the Hermite function of degreen with parametert. This series representation is used to prove the Ito lemma forf inL ^*, $$f(B(t)) = f(B(u)) + \int_u^t {\partial _s^ * } f'(B(s)) ds + (1/2)\int_u^t {f''} (B(s)) ds$$ , where? _s ^* is the adjoint of \(\dot B(s)\) -differentiation operator? _s .     

A counterexample to the strong subadditivity of extremal plurisubharmonic functions

A simple example is given which shows that one way have $$h_E (z^0 ) + h_F (z^0 ) > h_{E \cup F} (z^0 ) + h_{E \cap F} (z^0 )$$ for some pointz ⁰∈ω, where $$h_E (z) = \sup \{ u(z):u \in PSH (\Omega ),u \leqslant 0 on E,u \leqslant 1 in \Omega \} ,z \in \Omega ,$$ is the extremal function often studied in complex analysis.     

On a multiplicative function on the set of shifted primes

It is proved that if ?(n) is a multiplicative function taking a valueζ on the set of primes such thatζ ³ = 1,ζ ≠ 1 and? ³(p ^r)=1 forr≥2, then there exists aθ ∈ (0, 1), for which $$|\sum\limits_{p \leqslant x} f (p + 1)| \leqslant \theta \pi (x)$$ , where $$\pi (x) = \sum\limits_{p \leqslant x} 1$$ .     

An analog of the weyl decomposition of the spaceL q (ω, ℝ n ) for a first-order differential operatorfor a first-order differential operator

We prove the following theorem: Suppose the function f(x) belongs toL _q (ω, ? ⁿ ), ω ? ? ^m , q∈(1, ∞), and satisfies the inequality $$|\int\limits_\omega {(f(x),{\mathbf{ }}v(x)){\mathbf{ }}dx| \leqslant \mu ||} v||'_q ,{\mathbf{ }}\tfrac{1}{q} + \tfrac{1}{{q'}} = 1,$$ for all n-dimensional vector-valued functions in the kernel of a scalar-valued first-order differential operator £ for which the second-order operatorLL ^* is elliptic. Then there exists a function p(x)∈W _q ¹ (ω) such that $$||f(x) - \mathfrak{L}^* p(x)||q \leqslant C_q \mu .$$ Bibliography: 6 titles.     

Time-optimal control under state-dependent control restraints

We consider the control system $$\dot x = Ax + Bu,$$ subject to the state-dependent control restraint $$u(t) \in \Omega \cap (\mathcal{H}{\text{ + }}Cx(t)\} ,$$ where Ω is a compact convex set, ? is a subspace, andC is a matrix. The existence of time-optimal maximal controls is proven. A natural application of this result is in solving time-optimal problems under the control restraintu(t)∈Ω and the requirement that the outputy=Sx be maintained at zero during the transfer, whereS is a given matrix. An example is provided.     

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.     

On weak solutions of linear coercive integral equations in spacesL 2(X; ℋ k )

Пусть (X,A, μ) - полное про странство с σ-конечно й мерой, и пусть \(\overline {\mu \times \mu } \) . - замык ание меры μ×μ. Пусть далееg: X×X→C - квадратично интегрируемая функц ия по мере \(\overline {\mu \times \mu } \) . Рассматривается лин ейное интегральное у равнение (слабого) типа (1) (1) $$u(t) + A(\mathop \smallint \limits_x g(t,s)u(s)d\mu ) = f(t)\Pi .B.B\,X,$$ гдеА - максимальное р асширение L _k ^′ (в простр анстве ХëрмандераH ₁=B₂к) соотв ествующего линейного (псевдодиф ференциального) опер атораL: S→S; иS обозначает класс Щварца функций Rⁿ→-C. Уст анавливается сущест вование (слабых) решений (1) при н екотором условии коэрпитивно сти на оператор (2) (2) $$(L\Psi )(t) = \Psi (t) + \int\limits_x {g(t,s)L(\Psi (s))d\mu ,} $$ где Ψ принадлежит про странстувуD(Х, S) всех конечно-значных функ ций изX→S. Далее, изучается обобщенна я обратимость максим ального расширения оператора L. Наконец, пр иводится некоторое алгебраическое усло вие, обеспечивающее коэрцитивность L.     

Maximum principle of optimal periodic control for functional differential systems

This paper discusses the optimal periodic control problem to minimize the cost function $$J(u) = \int_0^1 {g(t,x(t),u(t))dt} $$ subject to the functional differential system $$dx(t)/dt = f(t,x_t ,u(t)),x_1 = x_0 $$ andu(·) εU _ad. The maximum principle as a necessary condition of optimal control is proved under the assumption that Eq. (4) and its adjoint equation (5) both have no nontrivial periodic solution with period of 1. In this paper, the control domainU is an arbitrary set inR ^m.     

On the average number of direct factors of a finite Abelian group

Let \(T(x) = \sum\limits_{ord(G) \leqq x} {t(G),} \) , wheret(G) define the number of direct factors of a finite Abelian group.E. Krätzel ([5]) defined a remainderΔ ₁(x) in the asymptotic ofT(x) and proved $$\Delta _1 (x)<< x^{{5 \mathord{\left/ {\vphantom {5 {12}}} \right. \kern-\nulldelimiterspace} {12}}} \log ^4 x.$$ Using two different methods to estimate a special three-dimensional exponential sum we get the better results $$\Delta _1 (x)<< x^{{{282} \mathord{\left/ {\vphantom {{282} {683}}} \right. \kern-\nulldelimiterspace} {683}}} \log ^4 x$$ and $$\Delta _1 (x)<< x^{{{45} \mathord{\left/ {\vphantom {{45} {109}}} \right. \kern-\nulldelimiterspace} {109}} + \varepsilon } (\varepsilon > 0).$$      

Finite-difference method for solving the first boundary-value problem for a second-order nonlinear ordinary differential equation with a divergent principal part

The approximation is studied of the first boundary-value problem for the equation (1) $$- \frac{d}{{dx}}K(x,\frac{{du}}{{dx}}) + f(x,u) = 0,0< x< 1,$$ with boundary conditions (2) $$u(0) = u(1) = 0$$ by difference boundary-value problems of form (3) $$- \left[ {a(x,w_{\bar x} )} \right]_x + \varphi (x,w) = 0,x \in w_r ,$$ (4) $$w(0) = w(1) = 0.$$ Theorems are established on the solvability of problem (3), (4). Theorems are proved on uniform convergence and on the order of uniform convergence. Here, as usual, boundedness is not assumed, but just the summability of the corresponding derivatives of the solutions of problem (1), (2). Also considered are singular boundary-value problems of form (1), (2), where uniform convergence with order h is proved under assumption of piecewise absolute continuity of the functionf(x,u(x)).     

On the divergence of Lagrange interpolation processes on a set of second category

В статье даны полные д оказательства следу ющих утверждений. Пустьω — непрерывная неубывающая полуадд итивная функций на [0, ∞),ω(0)=0 и пусть M?[0, 1] — матрица узл ов интерполирования. Если $$\mathop {\lim sup}\limits_{n \to \infty } \omega \left( {\frac{1}{n}} \right)\log n > 0$$ то существует точкаx ₀∈[0,1] и функцияf ∈ С[0,1] таки е, чтоω(f, δ)=О(ω(δ)), для которой $$\mathop {\lim sup}\limits_{n \to \infty } |L_n (\mathfrak{M},f,x_0 ) - f(x_0 )| > 0$$ Если же $$\mathop {\lim sup}\limits_{n \to \infty } \omega \left( {\frac{1}{n}} \right)\log n = \infty$$ , то существуют множес твоE второй категори и и функцияf ∈ С[0,1],ω(f, δ)=o(ω(δ)) та кие, что для всехx∈E $$\mathop {\lim sup}\limits_{n \to \infty } |L_n (\mathfrak{M},f,x)| = \infty$$ . Исправлена погрешно сть, допущенная автор ом в [5], и отмеченная в работе П. Вертеши [9].     

Strong approximation by Fourier series and differentiability properties of functions

ПустьΦ —N-функция Юнг а со свойствами $$\Phi (x)x^{ - 1} \downarrow 0, \exists \alpha > 1 \Phi (x)x^{ - \alpha } \uparrow (x \downarrow 0),$$ илиΦ(х)=х, {λ_k} — положи тельная, неубывающая последовательность и $$S_\Phi \{ \lambda \} = \left\{ {f:\left\| {\sum\limits_{k = 0}^\infty \Phi (\lambda _k |f - s_k |)} \right\|_\infty< \infty } \right\}.$$ В работе найдены необ ходимые и достаточны е условия для вложений $$S_\Phi \{ \lambda \} \subset W^r F(r \geqq 0),$$ , гдеF=C, L ^∞, Lip α (0<α≦1). С этой то чки зрения рассматриваются и др угие классы (например, \(W^r H^\omega ,\tilde W^r F\) ).     

A property of an autonomous minimax optimal control problem

A control system \(\dot x = f\left( {x,u} \right)\) ,u) with cost functional $$\mathop {ess \sup }\limits_{T0 \leqslant t \leqslant T1} G\left( {x\left( t \right),u\left( t \right)} \right)$$ is considered. For an optimal pair \(\left( {\bar x\left( \cdot \right),\bar u\left( \cdot \right)} \right)\) ,ū(·)), there is a maximum principle of the form $$\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right) = \mathop {\max }\limits_{u \in \Omega \left( t \right)} \eta \left( t \right)f\left( {\bar x\left( t \right),u} \right).$$ By means of this fact, it is shown that \(\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right)\) is equal to a constant almost everywhere.