Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type

In this paper, we examine, in a systematic fashion, some ill-posed problems arising in the theory of heat conduction. In abstract terms, letH be a Hilbert space andA: D (A)?H→H be an unbounded normal operator, we consider the boundary value problemü(t)=Au(t), 0<t<∞,u(0)=u ₀∈D(A), \(\mathop {\lim }\limits_{t \to 0} \left\| {u\left( t \right)} \right\| = 0\) . The problem of recoveringu ₀ whenu(T) is known for someT>0 is not well-posed. Suppose we are given approximationsx ₁,x ₂,…,x _N tou(T ₁),…,u(T _N) with 0<T, <…<T _N and positive weightsP _i,i=1,…,n, \(\sum\limits_{i = 1}^N {P_i = 1} \) such that \(Q_2 \left( {u_0 } \right) = \sum\limits_{i = 1}^N {P_i } \left\| {u\left( {T_i } \right) - x_i } \right\|^2 \leqslant \varepsilon ^2 \) . If ‖u _t(0)‖≤E for some a priori constantE, we construct a regularized solution ν(t) such that \(Q\left( {\nu \left( 0 \right)} \right) \leqslant \varepsilon ^2 \) while \(\left\| {u\left( 0 \right) - \nu \left( 0 \right)} \right\| = 0\left( {ln \left( {E/\varepsilon } \right)} \right)^{ - 1} \) and \(\left\| {u\left( t \right) - \nu \left( t \right)} \right\| = 0\left( {\varepsilon ^{\beta \left( t \right)} } \right)\) where 0<β(t)<1 and the constant in the order symbol depends uponE. The function β(t) is larger thant/m whent _k andk is the largest integer such that \((\sum\limits_{k = 1}^N {P_i (T_i )} )< (\sum\limits_{k = 1}^N {P_i (T_i )} = m\) , which β(t)=t/m on [T _k, m] and β(t)=1 on [m, ∞). Similar results are obtained if the measurement is made in the maximum norm, i.e.,Q _∞(u ₀)=max{‖u(T _i)?x _i‖, 1≤i≤N}.     

Scanning control of a vibrating string

We construct scanning feedback controls {γ _i (t)} for the vibrating string equation $$\begin{gathered} y_{tt} (x,t) = y_{xx} (x,t) + Ry(x,t) + \sum\limits_{i = 1}^N {\phi (x - \gamma _i } (t))y(x,t), \hfill \\ 0< x< 1,y = 0 at x = 0,1. \hfill \\ \end{gathered} $$ so that (y, y _t ) → (0,0) ast → ∞ in the weak topology ofH ₀ ¹ (0,1) ×L ₂ (0,1). In particular we show that ifφ is an even polynomial of degreeN with nonpositive coefficients that forR <π ² we can find such stabilizingγ _i (t), i=1,?,N.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness

In this paper we study viscosity solutions to the system $$\begin{array}{ll} \min \{ -\mathcal{H}u_i(x,t)-\psi _i(x,t),u_i(x,t) - \max_{j \neq i} (-c_{i ,j} (x,t) + u_j (x,t)) \} = 0,\\ u_i(x,T)=g_i (x), \, i \in \{1,\ldots , d \},\end{array}$$ where \({(x,t)\in{\mathbb{R}}^{N} \times [0,T]}\) . Concerning \({{\mathcal{H}}}\) , we assume that \({{\mathcal{H}}={\mathcal{L}}+{\mathcal{I}}}\) where \({{\mathcal{L}}}\) is a linear, possibly degenerate, parabolic operator of second order and \({{\mathcal{I}}}\) is a non-local integro-partial differential operator. A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems when the dynamics of the underlying state variables is described by an N-dimensional Levy process. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the data, i.e., on the operator \({{\mathcal{H}}}\) and on the continuous functions \({\psi_i}\) , c _i,j , and g _i . Using the comparison principle we prove the existence of a unique viscosity solution (u ₁, . . . , u _d ) to the system by Perron’s method. Our main contribution is that we establish existence and uniqueness of viscosity solutions, in the setting of Levy processes and non-local operators, with no sign assumption on the switching costs {c _{i, j} } and allowing c _{i, j} to depend on x as well as t.     

An inverse problem of the three-dimensional wave equation for a general annular vibrating membrane with piecewise smooth boundary conditions

This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian inR ³. The asymptotic expansion of the trace of the wave operator $\widehat\mu (t) = \sum\limits_{\upsilon = 1}^\infty {\exp \left( { - it\mu _\upsilon ^{1/2} } \right)} $ for small ?t? and $i = \sqrt { - 1} $ , where $\{ \mu _\nu \} _{\nu = 1}^\infty $ are the eigenvalues of the negative Laplacian $ - \nabla ^2 = - \sum\limits_{k = 1}^3 {\left( {\frac{\partial }{{\partial x^k }}} \right)} ^2 $ in the (x ¹,x ²,x ³), is studied for an annular vibrating membrane Ω inR ³ together with its smooth inner boundary surfaceS ₁ and its smooth outer boundary surfaceS ₂. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth componentsS ^* _i(i=1, …,m) ofS ₁ and on the piecewise smooth componentsS ^* _i(i=m+1, …,n) ofS ₂ such that $S_1 = \bigcup\limits_{i = 1}^m {S_i^* } $ and $S_2 = \bigcup\limits_{i = m + 1}^n {S_i^* } $ are considered. The basic problem is to extract information on the geometry of the annular vibrating membrane ω from complete knowledge of its eigenvalues by analyzing the asymptotic expansions of the spectral function $\widehat\mu (t)$ for small ?t?.     

Mean value theorem and a maximum principle for Kolmogorov's equation

For an equation of the form $$\begin{gathered} \frac{{\partial u}}{{\partial t}} - \sum\nolimits_{ij = 1}^n {{\text{ }}\alpha ^{ij} } \frac{{\partial ^2 u}}{{\partial x^i \partial x^j }} + \sum\nolimits_{ij = 1}^n {\beta _j^i x^i } \frac{{\partial u}}{{\partial x^i }} = 0, \hfill \\ {\text{ }}x \in R^n ,{\text{ }}t \in R^1 , \hfill \\ \end{gathered}$$ where α=(α^ij) is a constant nonnegative matrix andΒ=(Β _i ⁱ ) is a constant matrix, subject to certain conditions, we construct a fundamental solution, similar in its structure to the fundamental solution of the heat conduction equation; we prove a mean value theorem and show that u(x₀, t₀) can be represented in the form of the mean value of u(x, t) with a nonnegative density over a level surface of the fundamental solution of the adjoint equation passing through the point (x₀, t₀); finally, we prove a parabolic maximum principle.     

Oscillatory Properties of Solutions of Impulsive Differential Equations With Several Retarded Arguments

The impulsive differential equation $\begin{gathered} x\prime (t) + \sum\limits_{i = 1}^m {p_i (t)x(t - \tau _i ) = 0,} {\text{ }}t \ne \xi _k , \\ \Delta x(\xi _k ) = b_k x(\xi _k ) \\ \end{gathered} $ with several retarded arguments is considered, where p _i(t) ≥ 0, 1 + b _k > 0 for i = 1, ..., m, t ≥ 0, $k \in \mathbb{N}$ . Sufficient conditions for the oscillation of all solutions of this equation are found.     

Exact Mixing Times for Random Walks on Trees

We characterize the extremal structures for certain random walks on trees. Let G = (V, E) be a tree with stationary distribution π. For a vertex ${i \in V}$ , let H(π, i) and H(i, π) denote the expected lengths of optimal stopping rules from π to i and from i to π, respectively. We show that among all trees with |V| = n, the quantities ${{\rm min}_{i \in V} H(\pi, i), {\rm max}_{i \in V} H(\pi, i), {\rm max}_{i \in V} H(i, \pi)}$ and ${\sum_{i \in V} \pi_i H(i, \pi)}$ are all minimized uniquely by the star S _n = K _1,n?1 and maximized uniquely by the path P _n .