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}.     

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 $$k(t, y) = min(t, y)$$      

On normal control processes

Control processes of the form \(\dot x - A(t) x = B(t) u(t)\) , which are normal with respect to the unit ballB _{p′, r′} of the control spaceL ^p′([τ, T]),l _m ^r ′ are characterized in terms ofH(t)=X(T)X ^?1(t),B(t),X(t) any fundamental matrix solution of \(\dot x - A(t)x = 0\) , and directly in terms ofA, B, when bothA andB are independent oft.     

The parabolic logistic equation with blow-up initial and boundary values

In this article, we investigate the parabolic logistic equation with blow-up initial and boundary values $${u_t} - \Delta u = a(x,t)u - b(x,t){u^p}in\Omega \times (0,T),$$ $$u = \infty on\partial \Omega \times (0,T) \cup \overline \Omega \times \{ 0\} ,$$ where ?? is a smooth bounded domain, T > 0 and p > 1 are constants, and a and b are continuous functions, b > 0 in ?? × [0, T) and b(x, T) ?? 0. We study the existence and uniqueness of positive solutions and their asymptotic behavior near the parabolic boundary. We show that under the extra condition that $b(x,t) \ge c{(T - t)^\theta }d{(x,\partial \Omega )^\beta } on \Omega \times \left[ {0,T} \right)$ for some constants c > 0, ?? > 0, and ?? > ?2, such a solution stays bounded in any compact subset of ?? as t increases to T, and hence solves the equation up to t = T.     

Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote ${\Delta_g=-{\rm div}_g\nabla}$ the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation $$\Delta_gu(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)}$$ on (M, g). Here, p, q ≥ 0, A(x), B(x) and h(x) are smooth functions on (M, g). We also derive the Harnack differential inequality for the positive solutions of $$u_t(x,t)+\Delta_gu(x,t)+h(x)u(x,t)=A(x)u^p(x,t)+\frac{B(x)}{u^q(x,t)}$$ on (M, g) with initial data u(x, 0) > 0.     

Types and criteria of nonoscillatory solutions for some second order

In this paper we discuss the following NFDE $$[r(t)[x(t) - cx(t - \tau )']' + \smallint _a^b p(t,\xi )x[g(t,\xi )]d\sigma (\xi ) = 0$$ where τ>0, 1>c≥0, 0≤g(t, ε)≤t,r(t)>0,p(t, ε)>0, and some sufficient and necessary conditions are given, under which there are three types of nonoscillatory solutions for the above equation.     

A Hardy type inequality for W m,1(0, 1) functions

In this paper, we consider functions ${u\in W^{m,1}(0,1)}$ where m ≥ 2 and u(0) = Du(0) = · · · = D ^m-1 u(0) = 0. Although it is not true in general that ${\frac{D^ju(x)}{x^{m-j}} \in L^1(0,1)}$ for ${j\in \{0,1,\ldots,m-1\}}$ , we prove that ${\frac{D^ju(x)}{x^{m-j-k}} \in W^{k,1}(0,1)}$ if k ≥ 1 and 1 ≤ j + k ≤ m, with j, k integers. Furthermore, we have the following Hardy type inequality, $$\left\|{D^k\left({\frac{D^ju(x)}{x^{m-j-k}}}\right)}\right\|_{L^1(0,1)} \leq \frac {(k-1)!}{(m-j-1)!} \|{D^mu}\|_{L^1(0,1)},$$ where the constant is optimal.     

On constant-sign solutions of a system of discrete equations

We consider the following system of discrete equations $$u_i (k) = \sum\limits_{\ell = 0}^N {g_i (k,\ell )fi(\ell ,u_1 (\ell )} ,u_2 (\ell ), \cdots ,u_n (\ell )), k \in \{ 0,1, \cdots ,T\} ,$$ 1≤i≤n whereT≥N>0, 1≤i≤n. Existence criteria for single, double and multiple constant-sign solutions of the system are established. To illustrate the generality of the results obtained, we include applications to several well known boundary value problems. The above system is also extended to that on {0, 1,…} $$u_i (k) = \sum\limits_{\ell = 0}^\infty {g_i (k,\ell )fi(\ell ,u_1 (\ell )} ,u_2 (\ell ), \cdots ,u_n (\ell )), k \in \{ 0,1, \cdots \} ,1 \leqslant i \leqslant n$$ for which the existence of constant-sign solutions is investigated.     

An asymmetric inequality for non-homogeneous ternary quadratic forms

LetQ(x,y,z) be an indefinite ternary quadratic form of type (2,1) and determinantD(<0). Let 0≤t≤1/3 and \(f(t) = \frac{4}{{(1 + t)^2 (1 + 5t)}}\) . Then given any real numbersx ₀,y ₀,z ₀ there exist integersx,y,z satisfying $$ - t(f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}< Q (x + x_0 ,y + y_0 ,z + z_0 ) \leqslant (f(t)|D|)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} $$ All the cases when equality holds are also obtained.     

Degenerate differential operators in weighted Hölder spaces

A differential operator ?, arising from the differential expression $$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$ , and system of boundary value conditions $$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$ is considered in a Banach space E. Herev ^[k](t)=(a(t) d/dt) ^k v(t)a(t) being continuous fort?0, α(t) >0 for t > 0 and \(\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}\) the operator A is strongly positive in E. The estimates , are obtained for ?: n even, λ varying over a half plane.     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$      

Moduli of continuity defined by zero continuation of functions andK-functionals with restrictions

We consider the followingK-functional: $$K(\delta ,f)_p : = \mathop {\sup }\limits_{g \in W_{p U}^r } \left\{ {\left\| {f - g} \right\|_{L_p } + \delta \sum\limits_{j = 0}^r {\left\| {g^{(j)} } \right\|_{L_p } } } \right\}, \delta \geqslant 0,$$ where ? ∈L _p :=L _p [0, 1] andW _p,U ^r is a subspace of the Sobolev spaceW _p ^r [0, 1], 1≤p≤∞, which consists of functionsg such that $\int_0^1 {g^{(l_j )} (\tau ) d\sigma _j (\tau ) = 0, j = 1, ... , n} $ . Assume that 0≤l _l ≤...≤l _n ≤r-1 and there is at least one point τ _j of jump for each function σ _j , and if τ _j =τ _s forj ≠s, thenl _j ≠l _s . Let $\hat f(t) = f(t)$ , 0≤t≤1, let $\hat f(t) = 0$ ,t<0, and let the modulus of continuity of the functionf be given by the equality $$\hat \omega _0^{[l]} (\delta ,f)_p : = \mathop {\sup }\limits_{0 \leqslant h \leqslant \delta } \left\| {\sum\limits_{j = 0}^l {( - 1)^j \left( \begin{gathered} l \hfill \\ j \hfill \\ \end{gathered} \right)\hat f( - hj)} } \right\|_{L_p } , \delta \geqslant 0.$$ We obtain the estimates $K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 ]} (\delta ,f)_p $ and $K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 + 1]} (\delta ^\beta ,f)_p $ , where β=(pl _l + 1)/p(l ₁ + 1), and the constantc>0 does not depend on δ>0 and ? ∈L _p . We also establish some other estimates for the consideredK-functional.     

On the convergence to infinity of Fourier series along dense subsequences of numbers

В статье доказываетс я Теорема.Какова бы ни была возрастающая последовательность натуральных чисел {H _k } _{k = 1} ^∞ c $$\mathop {\lim }\limits_{k \to \infty } \frac{{H_k }}{k} = + \infty$$ , существует функцияf∈L(0, 2π) такая, что для почт и всех x∈(0, 2π) можно найти возраст ающую последовательность номеров {n_k(x)} _k=1 ^∞ ,удовлетворяющую усл овиям 1) $$n_k (x) \leqq H_k , k = 1,2, ...,$$ 2) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t} (x)} (x,f) = + \infty ,$$ 3) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t - 1} (x)} (x,f) = - \infty$$ .     

Eine Verschärfung der Abschätzung des Restes Taylorscher Näherungspolynome

I begin with a new short proof of: (I) LetP(t) inR ^d be a function oft havingn continuous derivatives fora≤t≤x. ThenP(x)∈ convK, where $$K = \left\{ {\sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}} P^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}P^{(n)} (t),a \leqslant t \leqslant x} \right\}.$$ for applying (I) let bef(t) a real function such that the point ((t?a) ⁿ⁺¹,f(t)) fulfills the conditions of (I). Then (I) gives a sharper estimate of then ^th remainder term off(x) than the Lagrange remainder formula. Iff( ⁿ )(t) is also convex ina≤t≤x, thenf(x)∈[c,d], where $$\begin{gathered} c = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}f^{(n)} \left( {\frac{{na + x}}{{n + 1}}} \right)} , \hfill \\ d = \sum\limits_{j = 0}^{n - 1} {\frac{{(x - a)^j }}{{j!}}f^{(j)} (a) + \frac{{(x - a)^n }}{{n!}}} \frac{{nf^{(n)} (a) + f^{(n)} (x)}}{{n + 1}}. \hfill \\ \end{gathered} $$      

Weak and generic bang-bang properties for continuous evolution inclusions and Baire’s method

We consider evolution inclusions, in a separable and reflexive Banach space ${\mathbb{E}}$ , of the form ${(\ast) x'(t) \in Ax(t) + F(t, x(t)), x(t_0) = c}$ and ${(**) x'(t) \in Ax(t) + {\rm ext} F(t,x(t)), x(t_0) = c}$ , where A is the infinitesimal generator of a C ₀-semigroup, F is a continuous and bounded multifunction defined on ${[t_0, t_1] \times \mathbb{E}}$ with values F(t, x) in the space of all closed convex and bounded subsets of ${\mathbb{E}}$ with nonempty interior, and ext F(t, x(t)) denotes the set of the extreme points of F(t, x(t)). For (*) and (**) we prove a weak form of the bang-bang property, namely, the closure of the set of the mild solutions of (**) contains the set of all internal solutions of (*). The proof is based on the Baire category method. This result is used to prove the following generic bang-bang property, that is, if A is the infinitesimal generator of a compact C ₀-semigroup then for most (in the sense of the Baire categories) continuous and bounded multifunctions, with closed convex and bounded values ${F(t, x) \subset \mathbb{E}}$ , the bang-bang property is actually valid, that is, the closure of the the set of the mild solutions of (**) is equal to the set of the mild solutions of (*).     

Convergence rates in the law of large numbers for random variables on partially ordered sets

Let (A, ≤) be a partially ordered set, {X _α} a collection of i. i. d. random variables, indexed byA. Let \(S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta \) , |α|=card {β∈A, β∈α}. We study the convergence rates ofS _α/|α|. We derive for a large class of partially ordered sets theorems, like the following one: For suitabler, t with 1/2< <r/t≤1:E|X| ^t M (|X| ^t/r )<∞ andEX=μ if and only if $$S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta $$ for all ε>0, where \(M(x) = \sum _{j< x} d(j)\) withd(j)=card {α∈A, |α|=j}.     

An abstract functional differential equation and a related nonlinear Volterra equation

We study the existence, uniqueness, regularity and dependence upon data of solutions of the abstract functional differential equation 1 $$\frac{{du}}{{dt}} + Au \ni G(u) (0 \leqq t \leqq T), u(0) = x,$$ , whereT>0 is arbitrary,A is a givenm-accretive operator in a real Banach spaceX, and \(G:C([0,T]; \overline {D(A)} ) \to L^1 (0, T; X)\) is a given mapping. This study provides simple proofs of generalizations of results by several authors concerning the nonlinear Volterra equation 2 $$u(t) + b * Au(t) \ni F(t) (0 \leqq t \leqq T),$$ , for the case in which X is a real Hilbert space. In (2) the kernelb is real, absolutely continuous on [0,T],b*g(t)=∫ ₀ ¹ (t?s)g(s)ds, andf∈W ^1,1(0,T;X).     

Weak real integral characterizations for exponential stability of semigroups in reflexive spaces

Let (t _n ) be a sequence of nonnegative real numbers tending to ∞, such that 1≤t _n+1?t _n ≤α for all natural numbers n and some positive α. We prove that a strongly continuous semigroup {T(t)} _t≥0, acting on a Hilbert space H, is uniformly exponentially stable if $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, y\bigr\rangle\bigr|\bigr)<\infty, $$ for all unit vectors x, y in H. We obtain the same conclusion under the assumption that the inequality $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, x^\ast\bigr\rangle\bigr|\bigr)<\infty, $$ is fulfilled for all unit vectors x∈X and x ^?∈X ^?, X being a reflexive Banach space. These results are stated for functions φ belonging to a special class of functions, such as defined in the second section of this paper. We conclude our paper with a Rolewicz’s type result in the continuous case on Hilbert spaces.