Comonotone approximation and interpolation by entire functions

Sampling in spaces of entire functions of exponential type in [formula omitted

In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .

Here we prove two propositions providing sufficient conditions of belonging positive continuous functions in to classes and These auxiliary classes plays important role in theory of entire functions of bounded L-index in direction and bounded L-index in joint variables, where are continuous functions. They help to constuct general theory of bounded index for very wide class of entire functions, because for every entire functions with bounded multiplicities of zero points there exists a corresponding function or providing boundedness of L-index in direction or boundedness L–index in joint variables respectively. Our result requires uniform boundedness of logarithmic derivative in all variables and for belonging the function to class Q^n. Another result requires uniform boundedness of logarithmic derivative in directions and for belonging the function to class Q^n_b where is the complex conjugate vector to b.

In this paper, we investigate the existence and regularity of positive solutions for certain singular problems that involve an anisotropic ( p , q ) -Laplacian-type operator and a singular term with a variable exponent, under zero Dirichlet boundary conditions on ∂ Ω . The main equation we analyze is − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | p i − 2 ∂ i u ( x ) ) − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | q i − 2 ∂ i u ( x ) ) = f ( x ) u ( x ) γ ( x ) in Ω , where Ω is a bounded, regular domain in R N , f is a positive function belonging to a specific Lebesgue space, and γ ( x ) is a positive continuous function on Ω ¯ . In our study, we do not make comparisons between p i and q i , and as a result, we show that the solution belongs to either W 0 1 , p → ( Ω ) ∩ W 0 1 , q → ( Ω ) or W l o c 1 , p → ( Ω ) ∩ W l o c 1 , q → ( Ω ) depending on the summability of f ( x ) and the values of γ ( x ) . The results are achieved using approximation techniques that include truncation, comparison, and variational methods.

Let $\Omega$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partial\Omega$ is not necessarily bounded. We consider two fast diffusion equations $\partial_t u= \mbox{div}(|\nabla u|^{p-2}{\nabla u})$ and $\partial_t u= \Delta u^{m}$, where $1<p<2$ and $0<m<1$. Let $u=u(x,t)$ be the solution of either the initial-boundary value problem over $\Omega$, where the initial value equals zero and the boundary value is a positive continuous function, or the Cauchy problem where the initial datum equals a nonnegative continuous function multiplied by the characteristic function of the set $\mathbb R^N\setminus \Omega$. Choose an open ball $B$ in $\Omega$ whose closure intersects $\partial\Omega$ only at one point, and let $\alpha > \frac {(N+1)(2-p)}{2p}$ or $\alpha > \frac {(N+1)(1-m)}{4}$. Then, we derive asymptotic estimates for the integral of $u^\alpha$ over $B$ for short times in terms of principal curvatures of $\partial\Omega$ at the point, which tells us about the interaction between fast diffusion and geometry of domain.

The paper discusses the asymptotic behaviour of all solutions of the differential equation , t ∈ I = [t0, ∞), with a positive continuous function a, continuous functions bi, f, and n continuously differentiable unbounded lags. We establish conditions under which any solution y of this equation can be estimated by means of a solution of an auxiliary functional equation with one unbounded lag. Moreover, some related questions concerning functional equations are discussed as well.

This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type $$\begin{aligned} \triangle ^2 u-\lambda M(\Vert \nabla u\Vert ^2)\triangle u-\frac{\mu }{\vert x\vert ^4}u=\frac{h(x)}{u^\gamma }+k(x)u^\alpha , \end{aligned}$$ under Navier boundary conditions, \(u=\triangle u=0\). Here \(\varOmega \subset {\mathbf {R}}^N\), \(N\ge 1\) is a bounded \(C^4\)-domain, \(0\in \varOmega \), h(x) and k(x) are positive continuous functions, \(\gamma \in (0,1)\), \(\alpha \in (0,1)\) and \(M:{\mathbf {R}}^+\rightarrow {\mathbf {R}}^+\) is a continuous function. By using Galerkin method and sharp angle lemma, we will show that this problem has a positive solution for \(\lambda > \frac{\mu }{\mu ^*m_0}\) and \(0 0\) and h, k are Lipschitz functions, we show that this problem has a positive smooth solution. If \(h,k\in C^{2,\,\theta _0}(\overline{\varOmega })\) for some \(\theta _0\in (0,1)\), then this problem has a positive classical solution.

In this paper we perform an accurate performance analysis of the generalised random binned antenna arrays, emphasising their tie with another type of aperiodic arrays that are obtained through the density-tapering methods (density-tapered arrays). In fact, since the former share with the latter the same procedure of the array aperture subdivision, we call such random arrays also probabilistic density-tapered arrays. In essence, one of our aims is to emphasise the fact that the generalised binned arrays can play a double role: in the first one they can be used as a tool for an a priori analysis of “deterministic” density-tapered arrays, on the other hand, they can be exploited in order to synthesise aperiodic arrays by generating different sample arrays, then choosing the best one. However, the second role is feasible when the number of antenna elements is very high because in this way the variance of the array factor has sufficiently low values. Our analysis focuses mostly on symmetric generalised binned arrays, since they can be manipulated more easily from the mathematical point of view than the asymmetric ones and also because, from the point of view of density-tapering, different reference currents are even real and positive continuous functions, giving symmetric density-tapered arrays.

In this paper, we study the existence of positivesolution for the following p-Laplacain type equations with critical nonlinearity \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -\Delta_p u + V （x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad x\in\mathbb{R}^N,\\ u \in \mathcal{D}^{1,p}(\mathbb{R}^N), \end{array} \right. \end{equation*}where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u),\ 1 < p < N,\ p^* =\frac{Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish atinfinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded,nonnegative continuous function.By working in the weighted Sobolev spaces, and using variational method, weprove that the given problem has at least one positive solution.

We consider the existence of solutions of the following weighted problem: \begin{equation*} \left\{ \begin{array}{rclll}\!\!\! L_{(\sigma,\xi)}u &amp;= &amp; f(x,u)&amp; \mbox{in} \ B \\ u &amp;&gt;&amp;0 &amp;\mbox{in }B\\ u&amp;=&amp;0 &amp;\mbox{on } \partial B, \end{array} \right. \end{equation*} where \[L_{(\sigma,\xi)}u:=\!g\Bigl(\int_{B}\bigl(\sigma(x)|\nabla u|^{N}\!+\!\xi(x)|u|^{N}\bigr)dx\!\Bigr)\!\bigl[-\textmd{div} \bigl(\sigma(x)|\nabla u|^{N-2} \nabla u\bigr)\!+\!\xi(x)u^{N-1}\bigr], \] B is the unit ball of $\mathbb{R}^{N}$, $ N&gt;2$, $\sigma(x)=(\log(\frac{e}{|x|}))^{N-1}$ the singular logarithm weight with the limiting exponent $N-1$ in the Trudinger--Moser embedding, $\xi(x)$ is a positive continuous function. The Kirchoff function $g$ is positive and continuous on $(0, +\infty)$. The nonlinearities are critical or subcritical growth in view of Trudinger--Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass Theorem. In the critical case, the function of Euler--Lagrange does not fulfil the requirements of Palais--Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness. multiples of a given positive divisor with real coefficients.

We prove results on exact asymptotics as for the means and probabilities , where , , is an -dimensional Wiener process, is a positive continuous function (potential) satisfying certain conditions, , and are prescribed vectors. The results are obtained by a new method developed in this paper, the Laplace method for the occupation time of a multi-dimensional Wiener process. We consider examples of monomial and radial potentials and prove results on exact asymptotics of small deviations for the probabilities as with a fixed .

Radial continuous rotation invariant valuations on star bodies

A nonoscillation theorem for sublinear Emden–Fowler equations

In this paper we study the existence of infinitely many solutions for the second-order differential equation (φp(u′(t)))′−a(t)φq(u(t))+b(t)φr(u(t))=0, t∈(0,T). coupled with Dirichlet’s boundary conditions u (0) = u(T) = 0. Here φ(t) = |t|p-2t < q < p, 1 < r < p and a,b are positive continuous functions on [0, T]. An existence theorem is proved using the variational method and the generalized Clark’s theorem due to Liu and Wang.

Consider a spectrally positive Lévy process Z with log-Laplace exponent Ψ and a positive continuous function R on (0,∞). We investigate the entrance from infinity of the process X obtained by changing time in Z with the inverse of the additive functional η(t)=∫0tdsR(Z s). We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process X. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the Lévy process has a negative drift δ:=−γ<0, sufficient conditions over R and Ψ are found for the process to come down from infinity along the deterministic function (xt,t≥0) solution to dxt=−γR(xt)dt with x0=∞. If Ψ(λ)∼cλα as λ→0, α∈(1,2], c>0 and R is regularly varying at ∞ with index θ>α, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.