Biharmonic elliptic problems involving the 2nd Hessian operator

In this paper we will study the equation $$\begin{aligned} \Delta ^2 u=S_2(D^2u),\quad \Omega \subset \mathbb {R}^N, \end{aligned}$$ with \(N=3,\) where \( S_2(D^2u)(x)=\sum _{1\le i , being \(\lambda _i,\) the solutions to the equation $$\begin{aligned} \mathrm{det}\left( \lambda I-D^2u(x)\right) =0, \end{aligned}$$ \(i=1,\dots ,N,\) and \(\Omega \) is a bounded domain with smooth boundary. We deal with several boundary conditions looking for the appropriate framework to get existence and multiplicity of nontrivial solutions. This kind of equation is related to some models of growth, and for this reason it is natural to study the effect of zero order local reaction terms of the type \(F_{\lambda }(x,u)=\lambda |u|^{p-1}u\) , with \(\lambda \in \mathbb {R}\) , \(\lambda >0\) , and \(0 , and also the solvability of the boundary problems with a source term \(f\) satisfying some integrability hypotheses.     

Rainbow Connection Number, Bridges and Radius

Let G be a connected graph. The notion of rainbow connection number rc(G) of a graph G was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph G with radius r, ${rc(G)\leq r(r+2)}$ and the bound is tight. In this paper, we show that for a connected graph G with radius r and center vertex u, if we let D ^r  = {u}, then G has r?1 connected dominating sets ${ D^{r-1}, D^{r-2},\ldots, D^{1}}$ such that ${D^{r} \subset D^{r-1} \subset D^{r-2} \cdots\subset D^{1} \subset D^{0}=V(G)}$ and ${rc(G)\leq \sum_{i=1}^{r} \max \{2i+1,b_i\}}$ , where b _i is the number of bridges in E[D ⁱ , N(D ⁱ )] for ${1\leq i \leq r}$ . From the result, we can get that if ${b_i\leq 2i+1}$ for all ${1\leq i\leq r}$ , then ${rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)}$ ; if b _i  > 2i + 1 for all ${1\leq i\leq r}$ , then ${rc(G)= \sum_{i=1}^{r}b_i}$ , the number of bridges of G. This generalizes the result of Basavaraju et al. In addition, an example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the radius of G, and another example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the number of bridges in G.     

A Matrix Differential Harnack Estimate for a Class of Ultraparabolic Equations

Let u be a positive solution of the ultraparabolic equation $$\partial _{t} u=\sum\limits_{i=1}^{n} \partial _{x_{i}}^{2} u+\sum\limits_{i=1}^{k} x_{i}\partial _{x_{n+i}}u \hspace {8mm} \text {on} \hspace {4mm} \mathbb {R}^{n+k}\times (0,T),$$ where 1 ≤ k ≤ n and 0 < T ≤ + ∞. Assume that u and its derivatives (w.r.t. the space variables) up to the second order are bounded on any compact subinterval of (0, T). Then the difference H(log u) ? H (log f) of the Hessian matrices of log u and of log f (both w.r.t. the space variables) is non-negatively definite, where f is the fundamental solution of the above equation with pole at the origin (0, 0). The estimate in the case n = k = 1 is due to Hamilton. As a corollary we get that \(\Delta l+\frac {n+3k}{2t}+\frac {6k}{t^{3}}\geq 0\) , where l = log u, and \(\Delta =\sum _{i=1}^{n+k} \partial _{x_{i}}^{2} \) .     

Existence and uniqueness of nontrivial solutions for eigenvalue boundary value problem of nonlinear fractional differential equation

In this paper, we study the existence and uniqueness of a nontrivial solution to eigenvalue problems for the following nonlinear fractional differential equation of the form $$\begin{aligned} \left\{ \begin{array}{l} -D^{\alpha }_{0^{+}}u(t)=\lambda [f(t, u(t), D^{\beta }_{0^{+}}u(t))+g(t)],~~ 0 where \(\lambda \) is a parameter, \(D^{\alpha }_{0^{+}},D^{\beta }_{0^{+}}\) are two standard Riemann–Liouville fractional derivatives, \(0<\beta <1<\alpha \le 2,\alpha -\beta >1,f: [0,1]\times {\mathbb{R }}\times {\mathbb{R }}\rightarrow {\mathbb{R }}\) is continuous, and \(g(t): (0, 1)\rightarrow [0, +\infty )\) is Lebesgue integrable. We obtain several sufficient conditions of the existence and uniqueness of nontrivial solution of the above eigenvalue problems when \(\lambda \) is in some interval. Our approach is based on the Leray–Schauder nonlinear alternative. In addition, some examples are included to demonstrate the main result.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data

This article considers the estimation for bivariate distribution function (d.f.) \(F_0(t, z)\) of survival time \(T\) and covariate variable \(Z\) based on bivariate data where \(T\) is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator \(\hat{F}_n(t,z)\) for \(F_0(t,z)\) , which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of \(\hat{F}_n(t,z)\) include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under \(\hat{F}_n(t,z)\) , the conditional d.f. of \(T\) given \(Z\) is of the same form as the Kaplan–Meier estimator for the univariate case, and that the marginal d.f. \(\hat{F}_n(\infty ,z)\) coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, \(\hat{F}_n(t,z)\) coincides with the bivariate empirical d.f. For discrete covariate \(Z\) , the strong consistency and weak convergence of \(\hat{F}_n(t,z)\) are established. Some simulation results are presented.     

On the infimum attained by the reflected fractional Brownian motion

Let {B _H (t):t≥0} be a fractional Brownian motion with Hurst parameter \(H\in (\frac {1}{2},1)\) . For the storage process \(Q_{B_{H}}(t)=\sup _{-\infty \le s\le t}\) \(\left (B_{H}(t)-B_{H}(s)-c(t-s)\right )\) we show that, for any T(u)>0 such that \(T(u)=o(u^{\frac {2H-1}{H}})\) , $$\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_{H}}(s)>u)\sim\mathbb P(Q_{B_{H}}(0)>u),$$ as \(u\to \infty \) . This finding, known in the literature as the strong Piterbarg property, goes in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.     

On coefficients of null-series and on sets of uniqueness of trigonometric and Walsh systems

В работе доказываютс я следующие утвержде ния. Теорема I.Пусть ? _n ↓0u \(\sum\limits_{n = 0}^\infty {\varepsilon _n^2 = + \infty } \) .Тогд а существует множест во Е?[0, 1]с μЕ=0 такое что:1. Существует ряд \(\sum\limits_{n = 0}^\infty {a_n W_n } (t)\) с к оеффициентами ¦а _n ¦≦{in¦n¦, который сх одится к нулю всюду вне E и ε∥a_n∥>0.2. Если b _n ¦=о(ε _n )и ряд \(\sum\limits_{n = 0}^\infty {b_n W_n (t)} \) сх одится к нулю всюду вн е E за исключением быть может некоторого сче тного множества точе к, то b _n =0для всех п. Теорема 3.Пусть ? _n ↓0u \(\mathop {\lim \sup }\limits_{n \to \infty } \frac{{\varepsilon _n }}{{\varepsilon _{2n} }}< \sqrt 2 \) Тогд а существует множест во E?[0, 1] с υ E=0 такое, что:

1. Существует ряд \(\sum\limits_{n = - \infty }^{ + \infty } {a_n e^{inx} ,} \sum\limits_{n = - \infty }^{ + \infty } {\left| {a_n } \right|} > 0,\) кот орый сходится к нулю в сюду вне E и ¦a_n≦¦n¦ для n=±1, ±2, ...
2. Если ряд \(\sum\limits_{n = - \infty }^{ + \infty } {b_n e^{inx} } \) сходится к нулю всюду вне E и ¦b_v¦=о(ε ¦n¦), то b_n=0 для всех я. Теорема 5. Пусть послед овательности S⁽¹⁾={ε ₀ ⁽¹⁾ , ε ₁ ⁽¹⁾ , ε ₂ ⁽¹⁾ , ...} u S²=ε ₀ ⁽²⁾ , ε ₁ ⁽²⁾ . ε ₂ ⁽²⁾ монотонно стремятся к нулю, \(\mathop {\lim \sup }\limits_{n \to \infty } \varepsilon ^{(i)} /\varepsilon _{2n}^{(i)}< 2,i = 1,2\) , причем \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n^{(2)} /\varepsilon _n^{(i)} = + \infty \) . Тогда для каждого ε>O н айдется множество Е? [-π,π], μE >2π — ε, которое является U(S¹), но не U(S¹) — множеством для тригонометричес кой системы. Аналог теоремы 5 для си стемы Уолша был устан овлен в [7].

     

Blow-up rates and uniqueness of large solutions for elliptic equations with nonlinear gradient term and singular or degenerate weights

This paper deals with the blow-up rate and uniqueness of large solutions of the elliptic equation ${\Delta u = b(x)f(u)+c(x)g(u)|\nabla u|^q}$ in ${\Omega \subset \mathbb{R}^N}$ , where q > 0, f(u) and g(u) are regularly varying functions at infinity, and the weight functions ${b(x),\,c(x) \in C^\alpha(\Omega,\,\mathbb{R}^+)}$ , 0 < α < 1, may be singular or degenerate on the boundary ${\partial\Omega}$ . Combining the regular variation theoretic approach of Cîrstea–R?dulescu and the systematic approach of Bandle–Giarrusso, we are able to improve and generalize most of the previously available results in the literature.     

Stability of the Derivative of a Canonical Product

With each sequence \(\alpha =(\alpha _n)_{n\in \mathbb{N }}\) of pairwise distinct and non-zero points which are such that the canonical product $$\begin{aligned} P_\alpha (z) := \lim _{r\rightarrow \infty }\prod _{|\alpha _n|\le r}\left( 1-\frac{z}{\alpha _n}\right) \end{aligned}$$ converges, the sequence $$\begin{aligned} \alpha ^{\prime } := \bigl (P_\alpha ^{\prime }(\alpha _n)\bigr )_{n\in \mathbb{N }} \end{aligned}$$ is associated. We give conditions on the difference \(\beta -\alpha \) of two sequences which ensure that \(\beta ^{\prime }\) and \(\alpha ^{\prime }\) are comparable in the sense that $$\begin{aligned} \exists \,c,C>0:\quad c|\alpha ^{\prime }_n| \le |\beta ^{\prime }_n| \le C|\alpha ^{\prime }_n|, \quad n\in \mathbb{N }. \end{aligned}$$ The values \(\alpha ^{\prime }_n\) play an important role in various contexts. As a selection of applications we present: an inverse spectral problem, a class of entire functions and a continuation problem.     

Extinction profile of the logarithmic diffusion equation

Let ${N \geq 3}$ and u be the solution of u _t = Δ log u in ${\mathbb{R}^N \times (0, T)}$ with initial value u ₀ satisfying ${B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}$ for some constants k ₁ > k ₂ > 0 where ${B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}$ is the Barenblatt solution for the equation and ${u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 if ${N \geq 4}$ . We give a new different proof on the uniform convergence and ${L^1(\mathbb{R}^N)}$ convergence of the rescaled function ${\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}$ , on ${\mathbb{R}^N}$ to the rescaled Barenblatt solution ${\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}$ for some k ₀ > 0 as ${s \rightarrow \infty}$ . When ${N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}$ in ${\mathbb{R}^N}$ , and ${|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L ¹ space in ${\mathbb{R}^N}$ of the rescaled solution ${\tilde{u}(x, s)}$ to ${\tilde{B}_{k_0}(x)}$ as ${s \rightarrow \infty}$ .     

Existence results of fractional differential equations with irregular boundary conditions and p-Laplacian operator

In this paper, we discuss the existence of solutions for irregular boundary value problems of nonlinear fractional differential equations with p-Laplacian operator $$\left \{ \begin{array}{l} {\phi}_p(^cD_{0+}^{\alpha}u(t))=f(t,u(t),u'(t)), \quad 0< t<1, \ 1< \alpha \leq2, \\ u(0)+(-1)^{\theta}u'(0)+bu(1)=\lambda, \qquad u(1)+(-1)^{\theta}u'(1)=\int_0^1g(s,u(s))ds,\\ \quad \theta=0,1, \ b \neq \pm1, \end{array} \right . $$ where \(^{c}D_{0+}^{\alpha}\) is the Caputo fractional derivative, ? _p (s)=|s| ^p?2 s, p>1, \({\phi}_{p}^{-1}={\phi}_{q}\) , \(\frac {1}{p}+\frac{1}{q}=1\) and \(f: [0,1] \times\mathbb{R} \times\mathbb {R} \longrightarrow\mathbb{R}\) . Our results are based on the Schauder and Banach fixed point theorems. Furthermore, two examples are also given to illustrate the results.     

Long-time asymptotics for fully nonlinear homogeneous parabolic equations

We study the long-time asymptotics of solutions of the uniformly parabolic equation $$ u_t + F(D^2u) = 0 \quad{\rm in}\, {\mathbb{R}^{n}}\times \mathbb{R}_{+},$$ for a positively homogeneous operator F, subject to the initial condition u(x, 0) =  g(x), under the assumption that g does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution Φ⁺ and negative solution Φ^?, which satisfy the self-similarity relations $$\Phi^\pm (x,t) = \lambda^{\alpha^\pm}\Phi^\pm ( \lambda^{1/2} x,\lambda t ).$$ We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to ${\Phi^+}$ ( ${\Phi^-}$ ) locally uniformly in ${\mathbb{R}^{n} \times \mathbb{R}_{+}}$ . The anomalous exponents α⁺ and α^? are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in ${\mathbb{R}^{n}}$ .     

A weak generalized localization criterion for multiple Walsh-Fourier series with J k -lacunary sequence of rectangular partial sums

We obtain a criterion for the validity of weak generalized localization almost everywhere on an arbitrary set of positive measure \(\mathfrak{A}\) , \(\mathfrak{A} \subset \mathbb{I}^N = \{ x \in \mathbb{R}^N :0 \leqslant x_j < 1,j = 1,2, \ldots ,N\}\) , N ≥ 3 (in terms of the structure and geometry of the set \(\mathfrak{A}\) ), for multiple Walsh-Fourier series (summed over rectangles) of functions f in the classes \(L_p (\mathbb{I}^N )\) , p > 1 (i.e., necessary and sufficient conditions for the convergence almost everywhere of the Fourier series on some subset of positive measure \(\mathfrak{A}_1\) of the set \(\mathfrak{A}\) , when the function expanded in a series equals zero on \(\mathfrak{A}\) ), in the case when the rectangular partial sums S _n (x; f) of this series have indices n = (n ₁, …, n _N ) ∈ ? ^N in which some components are elements of (single) lacunary sequences.     

On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators

We consider processes of the form [s,T]?t?u(t,X _t ), where (X,P _s,x ) is a multidimensional diffusion corresponding to a uniformly elliptic divergence form operator. We show that if $u\in{\mathbb{L}}_{2}(0,T;H_{\rho }^{1})$ with $\frac{\partial u}{\partial t} \in{\mathbb{L}}_{2}(0,T;H_{\rho }^{-1})$ then there is a quasi-continuous version $\tilde{u}$ of u such that $\tilde{u}(t,X_{t})$ is a P _s,x -Dirichlet process for quasi-every (s,x)∈[0,T)×? ^d with respect to parabolic capacity, and we describe the martingale and the zero-quadratic variation parts of its decomposition. We also give conditions on u ensuring that $\tilde{u}(t,X_{t})$ is a semimartingale.     

Sur un problème de capitulation du corps ℚ\sqrt {p_1 p_2 } , i dont le 2-groupe de classes est élémentaire

Let p ₁ ≡ p ₂ ≡ 1 (mod 8) be primes such that \(\left( {\tfrac{{p_1 }} {{p_2 }}} \right) = - 1\) and \(\left( {\tfrac{2} {{a + b}}} \right) = - 1\) , where p ₁ p ₂ = a ²+b ². Let \(i = \sqrt { - 1} \) , d = p ₁ p ₂, \(\Bbbk = \mathbb{Q}(\sqrt {d,} i),\Bbbk _2^{(1)} \) be the Hilbert 2-class field and \(\Bbbk ^{(*)} = \mathbb{Q}(\sqrt {p_1 } ,\sqrt {p_2 } ,i)\) be the genus field of \(\Bbbk \) . The 2-part \(C_{\Bbbk ,2} \) of the class group of \(\Bbbk \) is of type (2, 2, 2), so \(\Bbbk _2^{(1)} \) contains seven unramified quadratic extensions \(\mathbb{K}_j /\Bbbk \) and seven unramified biquadratic extensions \(\mathbb{L}_j /\Bbbk \) . Our goal is to determine the fourteen extensions, the group \(C_{\Bbbk ,2} \) and to study the capitulation problem of the 2-classes of \(\Bbbk \) .     

Degenerate homogeneous parabolic equations associated with the infinity-Laplacian

We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem ${u_t = \Delta_{\infty}^{h} u}$ , where ${\Delta_{\infty}^{h}}$ is the h-homogeneous operator associated with the infinity-Laplacian, ${\Delta_{\infty}^{h} u = |Du|^{h-3} \langle D^{2}uDu, Du \rangle}$ , and h > 1. We also derive the asymptotic behaviour of u for the problem posed in the whole space, and for the Dirichlet problem posed in a bounded domain with zero boundary conditions.     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

Conic volume ratio of the packing cone associate to a convex body

Given a convex body ${K\subset\mathbb{R}^n}$ (n??? 1) which contains o in its interior and ${{\bf u} \in S^{n-1}}$ , we introduce conic volume ratio r(K, u) of K in the direction of u by $$r(K, {\bf u})=\frac{vol(cone(K,{\bf u})\cap B_2^n)}{vol(B_2^n)},$$ where cone(K, u) is the packing cone of K in the direction of u. We prove that if K is an o-symmetric convex body in ${\mathbb{R}^n}$ and r(K, u) is a constant function of u, then K must be a Euclidean ball.