Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.     

Spectrum and the trace formula for a two-dimensional Schrödinger operator in a homogeneous magnetic field

In the space L ₂(?²), we consider the operator

$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$

. We study the spectrum of H and, for V ∈ C ₀ ² (?²), prove the trace formula

$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $

where c ₀ = π ^?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ _k ⁽ⁱ⁾ are the eigenvalues of H.

     

A Spectral Gap Estimate and Applications

We consider the Schrödinger operator

$$ \text{-} \frac{d^{2}}{d x^{2}} + V {\text{on an interval}}~~[a,b]~{\text{with Dirichlet boundary conditions}},$$

where V is bounded from below and prove a lower bound on the first eigenvalue λ ₁ in terms of sublevel estimates: if w _V (y) = |{x ∈ [a, b] : V (x) ≤ y}|, then

$$\lambda_{1} \geq \frac{1}{250} \min\limits_{y > \min V}{\left( \frac{1}{w_{V}(y)^{2}} + y\right)}.$$

The result is sharp up to a universal constant if {x ∈ [a, b] : V(x) ≤ y} is an interval for the value of y solving the minimization problem. An immediate application is as follows: let \({\Omega } \subset \mathbb {R}^{2}\) be a convex domain and let \(u:{\Omega } \rightarrow \mathbb {R}\) be the first eigenfunction of the Laplacian ? Δ on Ω with Dirichlet boundary conditions on ?Ω. We prove

$$\| u \|_{L^{\infty}({\Omega})} \lesssim \frac{1}{\text{inrad}({\Omega})} \left( \frac{\text{inrad}({\Omega})}{\text{diam}({\Omega})} \right)^{1/6} \|u\|_{L^{2}({\Omega})},$$

which answers a question of van den Berg in the special case of two dimensions.

     

One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions

We consider integrals of the form

$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$

, where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? ⁿ , and θ ∈ ? ^k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.

     

Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

An Example of an Unbounded Maximal Singular Operator

Suppose that an even integrable function Ω on the unit sphere S ¹ in R ² with mean value zero satisfies

$\mathop{\mathrm{essup}}\limits_{\xi\in \mathbf{S}^{1}}\biggl|\int_{\mathbf{S}^{1}}\Omega(\theta)\log\frac{1}{|\theta\cdot\xi|}\,d\theta\biggr|<+\infty,$

then the singular integral operator T _Ω given by convolution with the distribution p.v.?Ω(x/|x|)|x|^?2, initially defined on Schwartz functions, extends to an L ²-bounded operator. We construct examples of a function Ω satisfying the above conditions and of a continuous bounded integrable function f such that

$\limsup_{\epsilon\to 0^+}\biggl|\int_{\epsilon<|y|}\Omega(y/|y|)|y|^{-2}f(x-y)dy\biggr|=\infty\quad \hbox{a. e.}$

     

On the Bessel Heat Equation Related to the Bessel Diamond Operator

In this article, we study the equation

$\frac{\partial }{\partial t}u(x,t)=c^{2}\Diamond _{B}^{k}u(x,t)$

with the initial condition u(x,0)=f(x) for x∈? _n ⁺ . The operator ? _B ^k is named to be Bessel diamond operator iterated k-times and is defined by

$\Diamond _{B}^{k}=\bigl[(B_{x_{1}}+B_{x_{2}}+\cdots +B_{x_{p}})^{2}-(B_{x_{p+1}}+\cdots +B_{x_{p+q}})^{2}\bigr]^{k},$

where k is a positive integer, p+q=n, \(B_{x_{i}}=\frac{\partial ^{2}}{\partial x_{i}^{2}}+\frac{2v_{i}}{x_{i}}\frac{\partial }{\partial x_{i}},\) 2v _i =2α _i +1,\(\;\alpha _{i}>-\frac{1}{2}\), x _i >0, i=1,2,…,n, and n is the dimension of the ? _n ⁺ , u(x,t) is an unknown function of the form (x,t)=(x ₁,…,x _n ,t)∈? _n ⁺ ×(0,∞), f(x) is a given generalized function and c is a positive constant (see Levitan, Usp. Mat. 6(2(42)):102–143, 1951; Y?ld?r?m, Ph.D. Thesis, 1995; Y?ld?r?m and Sar?kaya, J. Inst. Math. Comput. Sci. 14(3):217–224, 2001; Y?ld?r?m, et al., Proc. Indian Acad. Sci. (Math. Sci.) 114(4):375–387, 2004; Sar?kaya, Ph.D. Thesis, 2007; Sar?kaya and Y?ld?r?m, Nonlinear Anal. 68:430–442, 2008, and Appl. Math. Comput. 189:910–917, 2007). We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel diamond heat kernel. Moreover, such Bessel diamond heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

     

Endpoint asymptotics for Müntz–Legendre polynomials

Müntz–Legendre polynomials L _n (Λ;x) associated with a sequence Λ={λ _k } are obtained by orthogonalizing the system \((x^{\lambda_{0}},x^{\lambda_{1}},x^{\lambda_{2}},\dots)\) in L ₂[0,1] with respect to the Legendre weight. Under very mild conditions on Λ, we establish the endpoint asymptotics close to x=1. The main result is

$\lim_{n\to\infty}L_n\left(1-\frac{y^2}{4S_n}\right)=J_0\big(|y|\big)$

where \(S_{n}=\sum_{k=0}^{n-1}(2\lambda_{k}+1)+\frac{2\lambda_{n}+1}{2}\) and J ₀ is the Bessel function of order 0.

     

Regularization of a three-element functional equation

In this paper we study the three-element functional equation

$(V\Phi )(z) \equiv \Phi (iz) + \Phi ( - iz) + G(z)\Phi \left( {\frac{1}{z}} \right) = g(z), z \in R,$

, subject to

$R: = \{ z:\left| z \right| < 1, \left| {\arg z} \right| < \frac{\pi }{4}\} .$

We assume that the coefficients G(z) and g(z) are holomorphic in R and their boundary values G ⁺(t) and g ⁺(t) belong to H(Γ), G(t)G(t ^?1) = 1. We seek for solutions Φ(z) in the class of functions holomorphic outside of \(\bar R\) such that they vanish at infinity and their boundary values Φ^?(t) also belong to H(Γ). Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.

     

Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Let {Q _n ^(α,β) (x)} _n=0 ^∞ denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product

$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$

where λ>0 and d μ _α,β(x)=(x?a)(1?x)^α?1(1+x)^β?1 dx, d ν _α,β(x)=(1?x) ^α (1+x) ^β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q _n ^(α,β) are obtained.

     

Least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian and potential wells

We are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian

$$\begin{array}{*{20}c} {( - \Delta )^s u(x) + \lambda V(x)u(x) = u(x)^{p - 1} ,} & {u(x) \geqslant 0,} & {x \in \mathbb{R}^N ,} \\ \end{array} $$

for sufficiently large λ, 2 < p < \(\frac{{2N}}{{N - 2s}}\) for N ≥ 2. V (x) is a real continuous function on R^N. Using variational methods we prove the existence of least energy solution u _λ(x) which localizes near the potential well int V ^?1(0) for λ large. Moreover, if the zero sets int V ?1(0) of V (x) include more than one isolated component, then u _λ(x) will be trapped around all the isolated components. However, in Laplacian case s = 1, when the parameter λ is large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrarily small in other components of int V ^?1(0). This is the essential difference with the Laplacian problems since the operator (?Δ)^s is nonlocal.

     

Uniqueness of a positive radially symmetric Solution of the Dirichlet problem for certain nonlinear differential equation of the second order in a ball

We consider the Dirichlet problem

$u_\Gamma = 0$

for the nonlinear differential equation

$\Delta u + \left| x \right|^m \left| u \right|^p = 0, x \in S,$

with constant m ≥ 0 and p > 1 in the unit ball S = {x ∈ R ⁿ : |x| < 1}(n ≥ 3) with the boundary Γ. We prove that with p ≤ m+n/n?2 this problem has a unique positive radially symmetric solution.

     

On generalized Christoffel functions

The generalized Christoffel function λ _p,q,n (dμ;x) (0<p<∞, 0≦q<∞) with respect to a measure dμ on R is defined by

$\lambda_{p,q,n}(d\mu;x)=\inf_{Q\in\mathbf{P}_{n-1},\ Q(x)=1}\int_{\mathbf{R}} \big|Q(t)\big|^p {|t-x|}^q\, d\mu(t).$

The novelty of our definition is that it contains the factor |t?x| ^q , which is of particular interest. Its properties are discussed and estimates are given. In particular, upper and lower bounds for generalized Christoffel functions with respect to generalized Jacobi weights are also provided.

     

Reconstructing a matrix from a partial sampling of Pareto eigenvalues

Let Λ={λ ₁,…,λ _p } be a given set of distinct real numbers. This work deals with the problem of constructing a real matrix A of order n such that each element of Λ is a Pareto eigenvalue of A, that is to say, for all k∈{1,…,p} the complementarity system

$x\geq \mathbf{0}_n,\quad Ax-\lambda_k x\geq \mathbf{0}_n,\quad \langle x, Ax-\lambda_k x\rangle = 0$

admits a nonzero solution x∈? ⁿ .

     

Global continuum of positive solutions for discrete <Emphasis Type="Italic">p</Emphasis>-Laplacian eigenvalue problems

We discuss the discrete p-Laplacian eigenvalue problem,

$$\left\{ \begin{gathered} \Delta (\phi _p (\Delta u(k - 1))) + \lambda a(k)g(u(k)) = 0,k \in \{ 1,2,...,T\} , \hfill \\ u(0) = u(T + 1) = 0, \hfill \\ \end{gathered} \right.$$

where T > 1 is a given positive integer and φ _p (x):= |x| ^p?2 x, p > 1. First, the existence of an unbounded continuum C of positive solutions emanating from (λ, u) = (0, 0) is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any λ > 0 and all solutions are ordered. Thus the continuum C is a monotone continuous curve globally defined for all λ > 0.

     

Bessel pairs and optimal Hardy and Hardy–Rellich inequalities

We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R ⁿ , n ≥ 1, so that the following inequalities hold for all \({u \in C_{0}^{\infty}(B)}\) :

$\label{one} \int\limits_{B}V(x)|\nabla u |^{2}dx \geq \int\limits_{B} W(x)u^2dx,$

$\label{two} \int\limits_{B}V(x)|\Delta u |^{2}dx \geq\int\limits_{B} W(x)|\nabla u|^{2}dx+(n-1)\int\limits_{B}\left(\frac{V(x)}{|x|^2}-\frac{V_r(|x|)}{|x|}\right)|\nabla u|^2dx.$

This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W)—that we call Bessel pairs—as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results—old and new—about Hardy and Hardy–Rellich type inequalities, such as those obtained by Caffarelli et al. (Compos Math 53:259–275, 1984), Brezis and Vázquez (Revista Mat. Univ. Complutense Madrid 10:443–469, 1997), Wang and Willem (J Funct Anal 203:550–568, 2003), Adimurthi et al. (Proc Am Math Soc 130:489–505, 2002), and many others.

     

Polynomiality of some hook-length statistics

We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions:

$\frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}\bigl(h_u^2 - i^2\bigr) = \frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{ r+1} \prod_{j=0}^{r} (n-j),$

where f _λ is the number of standard Young tableaux of shape λ and h _u is the hook length of the square u of the Young diagram of λ. We also obtain other similar formulas.

     

On the Surface Integral Approximation of the Second Derivatives of the Potential of a Bulk Charge Located in a Layer of Small Thickness

We consider a bulk charge potential of the form

$$u(x) = \int\limits_\Omega {g(y)F(x - y)dy,x = ({x_1},{x_2},{x_3}) \in {\mathbb{R}^3},} $$

where Ω is a layer of small thickness h > 0 located around the midsurface Σ, which can be either closed or open, and F(x ? y) is a function with a singularity of the form 1/|x ? y|. We prove that, under certain assumptions on the shape of the surface Σ, the kernel F, and the function g at each point x lying on the midsurface Σ (but not on its boundary), the second derivatives of the function u can be represented as

$$\frac{{{\partial ^2}u(x)}}{{\partial {x_i}\partial {x_j}}} = h\int\limits_\Sigma {g(y)\frac{{{\partial ^2}F(x - y)}}{{\partial {x_i}\partial {x_j}}}} dy - {n_i}(x){n_j}(x)g(x) + {\gamma _{ij}}(x),i,j = 1,2,3,$$

where the function γ_ij(x) does not exceed in absolute value a certain quantity of the order of h², the surface integral is understood in the sense of Hadamard finite value, and the n_i(x), i = 1, 2, 3, are the coordinates of the normal vector on the surface Σ at a point x.

     

A note on some results of Hua in short intervals

In this paper, we improve the previous results of the authors [G. Lü and H. Tang, On some results of Hua in short intervals, Lith. Math. J., 50(1):54–70, 2010] by proving that each sufficiently large integer N satisfying some congruence conditions can be written as

$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k},} \hfill \\ {\left| {{p_j} - \sqrt {{\frac{N}{5}}} } \right| \leqslant U,\quad \left| {p - {{\left( {\frac{N}{5}} \right)}^{\frac{1}{k}}}} \right| \leqslant U\,{N^{ - \frac{1}{2} + \frac{1}{k}}},\quad j = 1,\,2,\,\,3,\,4,} \hfill \\ \end{array} } \right. $

where U = N ^1/2?η+ε with \( \eta = \frac{1}{{2k\left( {{K^2} + 1} \right)}} \) and K = 2k ^?1, k ? 2.