Stability analysis of abstract systems of Timoshenko type

We consider an abstract system of Timoshenko type

$$\begin{aligned} \left\{ {\begin{array}{l} \rho_1{{\ddot \varphi}} + a A^{\frac12}(A^{\frac12}\varphi + \psi) =0\\\rho_2{{\ddot \psi}} + b A \psi + a (A^{\frac12}\varphi + \psi) -\delta A^\gamma {\theta} = 0\\\rho_3{{\dot \theta}} + c A\theta + \delta A^\gamma {{\dot \psi}} =0 \end{array}} \right. \end{aligned}$$

where the operator \({A}\) is strictly positive selfadjoint. For any fixed \({\gamma \in {\mathbb{R}}}\), the stability properties of the related solution semigroup \({S(t)}\) are discussed. In particular, a general technique is introduced in order to prove the lack of exponential decay of \({S(t)}\) when the spectrum of the leading operator \({A}\) does not consist of eigenvalues only.

     

On the stability of a generalization of Jensen functional equation

Using the fixed point method, we investigate the stability of a generalization of Jensen functional equation

$$ \sum_{k=0}^{n-1} f(x+ b_{k}y)=nf(x),$$

where \({n \in \mathbb{N}_{2}}\), \({b_{k}=\exp(\frac{2i\pi k}{n})}\) for \({0\leq k \leq n-1}\), in Banach spaces. Also, we prove the hyperstability results of this equation by the fixed point method.

     

On the equation $${{\rm det}\,\nabla{u}=f}$$ with no sign hypothesis

We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying

$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$

where k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfies

$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$

with no sign hypothesis on f.

     

On some invariance of the quotient mean with respect to Makó–Páles means

Given a continuous strictly monotone function \(\varphi \) defined on an open real interval I and a probability measure \(\mu \) on the Borel subsets of [0, 1], the Makó–Páles mean is defined by

$$\begin{aligned} {\mathcal {M}}_{\varphi ,\mu }(x,y):=\varphi ^{-1}\left( \int ^1_0\varphi (tx+(1-t)y)\, d\mu (t)\right) ,\quad x,y\in I. \end{aligned}$$

Under some conditions on the functions \(\varphi \) and \(\psi \) defined on I, the quotient mean is given by

$$\begin{aligned} Q_{\varphi ,\psi }(x,y):=\left( \frac{\varphi }{\psi }\right) ^{-1}\left( \frac{\varphi (x)}{\psi (y)}\right) , \quad x,y\in I. \end{aligned}$$

In this paper, we study some invariance of the quotient mean with respect to Makó–Páles means.

     

Classification of minimal mass blow-up solutions for an $${L^{2}}$$ critical inhomogeneous NLS

We establish the classification of minimal mass blow-up solutions of the \({L^{2}}\) critical inhomogeneous nonlinear Schrödinger equation

$$i\partial_t u + \Delta u + |x|^{-b}|u|^{\frac{4-2b}{N}}u = 0,$$

thereby extending the celebrated result of Merle (Duke Math J 69(2):427–454, 1993) from the classic case \({b=0}\) to the case \({0< b< {\rm min} \{2,N\} }\), in any dimension \({N \geqslant 1}\).

     

P-means and the Solution of a Functional Equation Involving Cauchy Differences

Solutions to the functional equation

$$f(x + y) - f(x) - f(y) = 2f(\Phi (x, y)), x, y > 0, \qquad\qquad (1)$$

are sought for the admissible pairs \({(f, \Phi)}\) constituted by a strictly monotonic function f and a strictly increasing in both variables mean \({\Phi}\). A related class of means, P-means, is introduced, studied and then employed in solving (1) under additional hypotheses on \({\Phi}\). For instance, Ger has proved that the unique P-mean which is also quasiarithmetic is the geometric mean \({G(x,y)=\sqrt{xy}}\). An elementary proof to this result is given in this paper. Moreover, as a consequence of a fundamental result on the uniqueness of representation of P-means it is proved that the geometric mean G is the unique homogeneous P-mean.

     

A note on functional equations connected with the Cauchy mean value theorem

The aim of this paper is to describe the solution (f, g) of the equation

$$\begin{aligned}{}[f(x)-f(y)]g'(\alpha x+(1-\alpha )y)= [g(x)-g(y)]f'(\alpha x+(1-\alpha )y),\ x,y\in I, \end{aligned}$$

where \(I\subset \mathbb {R}\) is an open interval, \(f,g:I\rightarrow \mathbb {R}\) are differentiable, \(\alpha \) is a fixed number from (0, 1).

     

Convergence of the solutions of the discounted equation: the discrete case

We derive a discrete version of the results of Davini et al. (Convergence of the solutions of the discounted Hamilton–Jacobi equation. Invent Math, 2016). If M is a compact metric space, \(c : M\times M \rightarrow \mathbb {R}\) a continuous cost function and \(\lambda \in (0,1)\), the unique solution to the discrete \(\lambda \)-discounted equation is the only function \(u_\lambda : M\rightarrow \mathbb {R}\) such that

$$\begin{aligned} \forall x\in M, \quad u_\lambda (x) = \min _{y\in M} \lambda u_\lambda (y) + c(y,x). \end{aligned}$$

We prove that there exists a unique constant \(\alpha \in \mathbb {R}\) such that the family of \(u_\lambda +\alpha /(1-\lambda )\) is bounded as \(\lambda \rightarrow 1\) and that for this \(\alpha \), the family uniformly converges to a function \(u_0 : M\rightarrow \mathbb {R}\) which then verifies

$$\begin{aligned} \forall x\in X, \quad u_0(x) = \min _{y\in X}u_0(y) + c(y,x)+\alpha . \end{aligned}$$

The proofs make use of Discrete Weak KAM theory. We also characterize \(u_0\) in terms of Peierls barrier and projected Mather measures.

     

Local boundedness of variational solutions to evolutionary problems with non-standard growth

We prove the local boundedness of variational solutions and parabolic minimizers to evolutionary problems, where the integrand f is convex and satisfies a non-standard p, q-growth condition with

$$1 < p \leq q \leq p \tfrac{n+2}{n}.$$

A function \({u\colon \Omega_T := \Omega \times (0,T) \to \mathbb{R}}\) is called parabolic minimizer if it satisfies the minimality condition

$$\int_{\Omega_T} u \cdot \partial_t \varphi +f(x, Du) {\rm d} z \leq \int_{\Omega_T} f(x, Du + D \varphi) {\rm d}z$$

for every \({\varphi \in C^\infty_0(\Omega_T)}\). Moreover, we will show local boundedness for parabolic minimizers, if f satisfies an anisotropic growth condition.

     

A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $${\mathbb R^{N}}$$

In this work we study the following class of problems in \({\mathbb R^{N}, N > 2s}\)

$$\varepsilon^{2s}(-\Delta)^{s}u + V(z)u = f(u), \,\,\,u(z) > 0$$

where \({0 < s < 1}\), \({(-\Delta)^{s}}\) is the fractional Laplacian, \({\varepsilon}\) is a positive parameter, the potential \({V : \mathbb{R}^N \to \mathbb{R}}\) and the nonlinearity \({f : \mathbb R \to \mathbb R}\) satisfy suitable assumptions; in particular it is assumed that \({V}\) achieves its positive minimum on some set \({M.}\) By using variational methods we prove existence and multiplicity of positive solutions when \({\varepsilon \to 0^{+}}\). In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the “topological complexity” of the set \({M}\).

     

About the Uniform Hölder Continuity of Generalized Riemann Function

In this paper, we study the uniform Hölder continuity of the generalized Riemann function \({R_{\alpha,\beta} \,\,{\rm (with}\,\, \alpha > 1 \,\,{\rm and}\,\, \beta > 0}\)) defined by

$$R_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty} \frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x \in \mathbb{R},$$

using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of \({R_{\alpha,\beta} \,\,{\rm as}\,\, \beta}\) tends to infinity.

     

Limit problems for a Fractional p-La placian as $${ p\to\infty}$$

The purpose of this work is the analysis of the solutions to the following problems related to the fractional p-Laplacian in a Lipschitzian bounded domain \({\Omega \subset \mathbb{R}^N}\),

$$\left\{\begin{array}{lll}-\int_{\mathbb{R}^N}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{\alpha p}}\;dy=f(x,u)\;\;&x\in \Omega,\\ u=g(x) &x\in\mathbb{R}^N\setminus \Omega,\end{array}\right.$$

where \({\alpha\in(0,1)}\) and the exponent p goes to infinity. In particular we will analyze the cases:

1. (i)
   \({f=f(x).}\)
    
2. (ii)
   \({f=f(u)=|u|^{\theta(p)-1} u \, {\rm with} \, 0 < \theta(p) < p -1 \, {\rm and} \, \lim_{p\to\infty}\frac{\theta(p)}{p-1}=\Theta < 1 \, {\rm with} \, g \geq 0.}\)
    

We show the convergence of the solutions to certain limit as \({p\to\infty}\) and identify the limit equation. In both cases, the limit problem is closely related to the Infinity Fractional Laplacian:

$$\mathcal{L}_\infty v(x)=\mathcal{L}_\infty^+ v(x)+\mathcal{L}_\infty^- v(x),$$

where

$$\mathcal{L}_\infty^+ v(x)=\sup_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}, \quad \mathcal{L}_\infty^- v(x)=\inf_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}.$$

     

Random approximation and the vertex index of convex bodies

We prove that there exists an absolute constant \({\alpha > 1}\) with the following property: if K is a convex body in \({{\mathbb R}^n}\) whose center of mass is at the origin, then a random subset \({X\subset K}\) of cardinality \({{\rm card}(X)=\lceil\alphan\rceil }\) satisfies with probability greater than \({1-e^{-c_1n}}\)

$$K\subseteq c_2n\, {\rm conv}(X),$$

where \({c_1, c_2 > 0}\) are absolute constants. As an application we show that the vertex index of any convex body K in \({{\mathbb R}^n}\) is bounded by \({c_3n^2}\), where \({c_3 > 0}\) is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.

     

Fixed points and additive $${\rho}$$-functional equations

In this paper, we solve the additive \({\rho}\)-functional equations

$$\begin{aligned} f(x+y)-f(x)-f(y)= & {} \rho(2f(\frac{x+y}{2})-f(x)-f(y)), \\ 2f(\frac{x+y}{2})-f(x)-f(y)= & {} \rho(f(x+y)-f(x)-f(y)), \end{aligned}$$

where \({\rho}\) is a fixed non-Archimedean number or a fixed real or complex number with \({\rho \neq 1}\). Using the fixed point method, we prove the Hyers–Ulam stability of the above additive \({\rho}\)-functional equations in non-Archimedean Banach spaces and in Banach spaces.

     

On estimation of delay location

Assume that we observe a stationary Gaussian process X(t), \({t \in [-r, T]}\) , which satisfies the affine stochastic delay differential equation

$d X(t) = \int\limits_{[-r,0]}X(t+u)\, a_\vartheta (du)\,dt +dW(t), \quad t\ge 0,$

where W(t), t ≥ 0, is a standard Wiener process independent of X(t), \({t\in [-r, 0]}\) , and \({a_\vartheta}\) is a finite signed measure on [?r, 0], \({\vartheta\in\Theta}\) . The parameter \({\vartheta}\) is unknown and has to be estimated based on the observation. In this paper we consider the case where \({\Theta=(\vartheta_0,\vartheta_1)}\) , \({-\infty\,<\,\vartheta_0 <0 \,<\,\vartheta_1\,<\,\infty}\) , and the measures \({a_\vartheta}\) are of the form

$a_\vartheta = a+b_\vartheta-b,$

where a and b are finite signed measure on [?r, 0] and \({b_\vartheta}\) is the translate of b by \({\vartheta}\) . We study the limit behaviour of the normalized likelihoods

$Z_{T,\vartheta}(u) = \frac{dP_T^{\vartheta+\delta_T u}}{dP_T^\vartheta}$

as T→ ∞, where \({P_T^\vartheta}\) is the distribution of the observation if the true value of the parameter is \({\vartheta}\) . A necessary and sufficient condition for the existence of a rescaling function δ _T such that \({Z_{T,\vartheta}(u)}\) converges in distribution to an appropriate nondegenerate limiting function \({Z_{\vartheta}(u)}\) is found. It turns out that then the limiting function \({Z_{\vartheta}(u)}\) is of the form

$Z_\vartheta(u)=\exp\left(B^H(u) - E[B^H(u)]^2/2\right),$

where \({H\in[1/2,1]}\) and B ^H (u), \({u\in\mathbb{R}}\) , is a fractional Brownian motion with index H, and δ _T  = T ^?1/(2H) ?(T) with a slowly varying function ?. Every \({H\in[1/2,1]}\) may occur in this framework. As a consequence, the asymptotic behaviour of maximum likelihood and Bayes estimators is found.

     

Sharp weighted bounds for the Hardy–Littlewood maximal operators on Musielak–Orlicz spaces

Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytönen et al. in [J. Funct. Anal. 263:3883–3899, 2012].     

A semilinear parabolic problem with singular term at the boundary

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary.

More precisely, we consider the problem

$$\left\{\begin{array}{l@{\quad}l}u_t-\Delta u=\lambda \frac{u^p}{d^2}&\text{ in }\,\Omega_{T}\equiv\Omega \times (0,T), \\u>0 &\text{ in }\,{\Omega_T}, \\u(x,0)=u_0(x)>0 &\text{ in }\,\Omega, \\u=0 &\text{ on }\partial \Omega \times (0,T),\end{array}\right.$$

(P)

where Ω is a bounded regular domain of \({\mathbbm{R}^N}\), \({d(x)=\text{dist}(x,\partial\Omega)}\), \({p > 0}\), and \({\lambda > 0}\) is a positive constant.

We prove that

1. 1.
   If \({0 < p < 1}\), then (P) has no positive very weak solution.
    
2. 2.
   If \({p=1}\), then (P) has a positive very weak solution under additional hypotheses on \({\lambda}\) and \({u_0}\).
    
3. 3.
   If \({p > 1}\), then, for all \({\lambda > 0}\), the problem (P) has a positive very weak solution under suitable hypothesis on \({u_0}\).
    

Moreover, we consider also the concave–convex-related case.

     

Zeta functions and asymptotic additive bases with some unusual sets of primes

Fix \(\delta \in (0,1]\), \(\sigma _0\in [0,1)\) and a real-valued function \(\varepsilon (x)\) for which \(\varlimsup _{x\rightarrow \infty }\varepsilon (x)\leqslant 0\). For every set of primes \(\mathcal {P}\) whose counting function \(\pi _\mathcal {P}(x)\) satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl (x^{\sigma _0+\varepsilon (x)}\bigr ), \end{aligned}$$

we define a zeta function \(\zeta _\mathcal {P}(s)\) that is closely related to the Riemann zeta function \(\zeta (s)\). For \(\sigma _0\leqslant \frac{1}{2}\), we show that the Riemann hypothesis is equivalent to the non-vanishing of \(\zeta _\mathcal {P}(s)\) in the region \(\{\sigma >\frac{1}{2}\}\).

For every set of primes \(\mathcal {P}\) that contains the prime 2 and whose counting function satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl ((\log \log x)^{\varepsilon (x)}\bigr ), \end{aligned}$$

we show that \(\mathcal {P}\) is an exact asymptotic additive basis for \(\mathbb {N}\), i.e. for some integer \(h=h(\mathcal {P})>0\) the sumset \(h\mathcal {P}\) contains all but finitely many natural numbers. For example, an exact asymptotic additive basis for \(\mathbb {N}\) is provided by the set

$$\begin{aligned} \{2,547,1229,1993,2749,3581,4421,5281\ldots \}, \end{aligned}$$

which consists of 2 and every hundredth prime thereafter.

     

Boundary behavior of <Emphasis Type="Italic">p</Emphasis>-harmonic functions in the Heisenberg group

In this paper we study the boundary behavior of nonnegative p-harmonic functions in a bounded domain in the Heisenberg group \({\mathbb H^n}\). Under suitable geometric assumptions on the ground domain \({\Omega\subset \mathbb H^n}\) our main contributions can be summarized as follows: (1) In Theorem 1.1 we obtain an estimate from above stating that any such function should vanish at most linearly like the sub-Riemannian distance from the boundary:

$\frac{u(g)}{u(A_r(g_0))}\leq C^{-1}\frac{d(g,\partial \Omega)}{r},$

where \({A_r(g_0)\in \Omega}\) is a non-tangential point relative to \({g_0\in \partial \Omega}\). (2) In Theorem 1.2 we establish an estimate from below which states that, away from the characteristic set of Ω, the order of vanishing is exactly linear, i.e.:

$\frac{u(g)}{u(A_r(g_0))}\ge C \frac{d(g,\partial \Omega)}{r}.$

     

Blow-up phenomena for a nonlocal <Emphasis Type="Italic">p</Emphasis>-Laplace equation with Neumann boundary conditions

This paper is concerned with the blow-up of solutions to the following nonlocal p-Laplace equation:

$$u_t-\mathrm{div}(|\nabla{u}|^{p-2}\nabla{u})=|u|^{q-1}u-\frac{1}{|\Omega|} \int\limits_\Omega{|u|^{q-1}u}dx,\quad x\in\Omega,\quad 0 < t < T,$$

under homogeneous Neumann boundary conditions in a bounded smooth domain \({\Omega\subset\mathrm{R}^N}\). For all \({p > 2, q > p-1}\), a blow-up result for the solutions to the above equation with positive initial energy is established. This result improves a recent result by Qu and Liang (Abstr Appl Anal 3:551–552, 2013) which asserts the blow-up of solutions for \({p-1 < q\leq\frac{Np}{(N-p)_+}-1}\).