Downloads & Free Reading Options - Results

ICTP Postgraduate Diploma Course in Mathematics - Lectures on Partial Differential Equations -- NOTE: This course was recorded automatically in slots of one hour and processed without human intervention. Lectures are split between videos; their starting time may not coincide with the beginning of videos and intervals were not removed.

We consider Poisson's equation with a finite number of weighted Dirac masses as a source term, together with its discretization by means of conforming finite elements. For the error in fractional Sobolev spaces, we propose residual-type a posteriori estimators with a specifically tailored oscillation and show that, on two-dimensional polygonal domains, they are reliable and locally efficient. In numerical tests, their use in an adaptive algorithm leads to optimal error decay rates.

In this paper, we study the well-posedness of classical solutions to the nonlinear unsteady Prandtl equations with Robin boundary condition in half space in weighted Sobolev spaces. We firstly investigate the monotonic shear flow with Robin boundary condition and the linearized Prandtl-type equations with Robin boundary condition in weighted Sobolev spaces. Due to the degeneracy of the Prandtl equations and the loss of regularity, we apply the Nash-Moser-Hormander iteration scheme to prove the existence of classical solutions to the nonlinear Prandtl equations with Robin boundary condition when the initial data is a small perturbation of a monotonic shear flow satisfying Robin boundary condition. The uniqueness and stability are also proved in the weighted Sobolev spaces. The nonlinear Prandtl equations with Robin boundary arise in the inviscid limit of incompressible Navier-Stokes equations with Navier-slip boundary condition for which the slip length is square root of viscosity. Our results are also valid for the Dirichlet boundary case.

We extend in this article the classical Sobolev inequalities for the module of continuity for the functions belonging to the integer order Sobolev's space on the Sobolev-Bilateral Grand Lebesgue spaces. As a consequence, we deduce the fractional Orlicz-Sobolev imbedding theorems and investigate the rectangle module of continuity of non-Gaussian multiparameter random fields.

Let $\Gamma$ be a graph endowed with a reversible Markov kernel $p$, and $P$ the associated operator, defined by $Pf(x)=\sum_y p(x,y)f(y)$. Denote by $\nabla$ the discrete gradient. We give necessary and/or sufficient conditions on $\Gamma$ in order to compare $\Vert \nabla f \Vert_{p}$ and $\Vert (I-P)^{1/2}f \Vert_{p}$ uniformly in $f$ for $1 2$. The proofs rely on recent techniques developed to handle operators beyond the class of Calder\'on-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation results for Sobolev spaces in this context, which are of independent interest.

Let $\Gamma$ be a graph endowed with a reversible Markov kernel $p$, and $P$ the associated operator, defined by $Pf(x)=\sum_y p(x,y)f(y)$. Denote by $\nabla$ the discrete gradient. We give necessary and/or sufficient conditions on $\Gamma$ in order to compare $\Vert \nabla f \Vert_{p}$ and $\Vert (I-P)^{1/2}f \Vert_{p}$ uniformly in $f$ for $1 2$. The proofs rely on recent techniques developed to handle operators beyond the class of Calder\'on-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation results for Sobolev spaces in this context, which are of independent interest.

This paper establishes isomorphisms for the Laplace operator in weighted Sobolev spaces (WSS). These spaces are similar to standard Sobolev spaces, but they are endowed with weights prescribing functions growth or decay at infinity. Although well established in the whole space, these weighted results do not apply in the specific hypothesis of periodicity. This kind of problem appears when studying singularly perturbed domains (roughness, sieves, porous media, etc). When zooming on a single perturbation pattern, one often ends with a periodic problem set on an infinite strip. We present a unified framework that enables a systematic treatment of such problems. We provide existence and uniqueness of solutions in our WSS. This gives a refined description of solutions behavior at infinity which is of importance in the mutli-scale context. These isomorphism results hold for any weight exponent and any regularity index. We then identify these solutions with the convolution of a Green function (specific to periodical infinite strips) and the given data. This identification is valid again for any weight and any regularity index modulo some harmonic polynomials.

We proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs mainly bases on the contraction mapping argument using Strichartz estimate. We also apply the technique of Christ-Colliander- Tao in \cite{ChristCollianderTao} to prove the ill-posedness for (NLHW) in some cases of the super-critical range.

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces.

We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of almost conservation law and I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. Our proof also relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type.

Agnieszka Kałamajska (University of Warsaw, Poland)

We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^2$ or $\bT^2$. Higher regularity of mild solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a corollary, we obtain a global existence result for the critical 2D quasi-geostrophic equations with periodic $\dot H^1$ data. Some decay in time estimates are also provided.

We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^2$ or $\bT^2$. Higher regularity of mild solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a corollary, we obtain a global existence result for the critical 2D quasi-geostrophic equations with periodic $\dot H^1$ data. Some decay in time estimates are also provided.

In this paper we study the Gelfand and Kolmogorov numbers of Sobolev embeddings between weighted function spaces of Besov and Triebel-Lizorkin type with polynomial weights. The sharp asymptotic estimates are determined in the so-called non-limiting case.

This paper studies the space of $BV^2$ planar curves endowed with the $BV^2$ Finsler metric over its tangent space of displacement vector fields. Such a space is of interest for applications in image processing and computer vision because it enables piecewise regular curves that undergo piecewise regular deformations, such as articulations. The main contribution of this paper is the proof of the existence of the shortest path between any two $BV^2$-curves for this Finsler metric. Such a result is proved by applying the direct method of calculus of variation to minimize the geodesic energy. This method applies more generally to similar cases such as the space of curves with $H^k$ metrics for $k\geq 2$ integer. This space has a strong Riemannian structure and is geodesically complete. Thus, our result shows that the exponential map is surjective, which is complementary to geodesic completeness in infinite dimensions. We propose a finite element discretization of the minimal geodesic problem, and use a gradient descent method to compute a stationary point of the energy. Numerical illustrations show the qualitative difference between $BV^2$ and $H^2$ geodesics.

We study persistence properties of solutions to some canonical dispersive models, namely the semi-linear Schr\"odinger equation, the $k$-generalized Korteweg-de Vries equation and the Benjamin-Ono equation, in weighted Sobolev spaces $H^s(\R^n)\cap L^2(|x|^ldx),\;s,\,l>0$

We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for H\"{o}rmander vector fields.

The purpose of this paper is to study the smoothing properties (in $L^p$ Sobolev spaces) of operators of the form $f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt$, where $\gamma_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in\mathbb{R}^N\times \mathbb{R}^n$, satisfying $\gamma_0(x)\equiv x$, $\psi$ is a $C^\infty$ cut-off function supported on a small neighborhood of $0\in \mathbb{R}^n$, and $K$ is a "multi-parameter fractional kernel" supported on a small neighborhood of $0\in \mathbb{R}^N$. When $K$ is a Calder\'on-Zygmund kernel these operators were studied by Christ, Nagel, Stein, and Wainger, and when $K$ is a multi-parameter singular kernel they were studied by the author and Stein. In both of these situations, conditions on $\gamma$ were given under which the above operator is bounded on $L^p$ ($1

In this paper, we study local well-posedness for the Navier-Stokes \linebreak equations with arbitrary initial data in homogeneous Sobolev spaces $\dot{H}^s_p(\mathbb{R}^d)$ for $d \geq 2, p > \frac{d}{2},\ {\rm and}\ \frac{d}{p} - 1 \leq s < \frac{d}{2p}$. The obtained result improves the known ones for $p > d$ and $s = 0$ M. Cannone and Y. Meyer (1995). In the case of critical indexes $s=\frac{d}{p}-1$, we prove global well-posedness for Navier-Stokes equations when the norm of the initial value is small enough. This result is a generalization of the one in M. Cannone (1997) in which $p = d$ and $s = 0$.

This paper proves that the approximation of pointwise derivatives of order $s$ of functions in Sobolev space $W_2^m(\R^d)$ by linear combinations of function values cannot have a convergence rate better than $m-s-d/2$, no matter how many nodes are used for approximation and where they are placed. These convergence rates are attained by {\em scalable} approximations that are exact on polynomials of order at least $\lfloor m-d/2\rfloor +1$, proving that the rates are optimal for given $m,\,s,$ and $d$. And, for a fixed node set $X\subset\R^d$, the convergence rate in any Sobolev space $W_2^m(\Omega)$ cannot be better than $q-s$ where $q$ is the maximal possible order of polynomial exactness of approximations based on $X$, no matter how large $m$ is. In particular,scalable stencil constructions via polyharmonic kernels are shown to realize the optimal convergence rates, and good approximations of their error in Sobolev space can be calculated via their error in Beppo-Levi spaces. This allows to construct near-optimal stencils in Sobolev spaces stably and efficiently, for use in meshless methods to solve partial differential equations via generalized finite differences (RBF-FD). Numerical examples are included for illustration.

Given bounded domains $\Omega_1$ and $\Omega_2$ in $\mathds{R}^N$ and an isometry $T$ from $W^{1,p}(\Omega_1)$ to $W^{1,p}(\Omega_2)$, we give sufficient conditions ensuring that $T$ corresponds to a rigid motion of the space, i.e., $Tu = \pm (u \circ \xi)$ for an isometry $\xi$, and that the domains are congruent. More general versions of the involved results are obtained along the way.

Poisson Summation: This is really a correlation between fourier transforms and fourier series

Recently Gigli developed a Sobolev calculus on non-smooth spaces using module theory. In this paper it is shown that his theory fits nicely into the theory of differentiability spaces initiated by Cheeger, Keith and others. A relaxation procedure for $L^p$-valued subadditive functionals is presented and a relationship between the module generated by a functional and the one generated by its relaxation is given. In the framework of differentiability spaces, which includes so called PI- and $RCD(K,N)$-spaces, the Lipschitz module is pointwise finite dimensional. A general renorming theorem together with the characterization above shows that the Sobolev spaces of such spaces are reflexive.

We obtain the optimal value of the constant K(n,s) in the Sobolev-Nirenberg-Gagliardo inequality $ \|\,u\,\|_{L^{\infty}(\mathbb{R}^{n})} \leq K(n,s) \,\|\, u \,\|_{L^{2}(\mathbb{R}^{n})}^{1 - n/(2s)} \|\, u \,\|_{\dot{H}^{s}(\mathbb{R}^{n})}^{n/(2s)} $ where $ s > n/2 $.

In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in earlier by the author. This allows for a unified treatment of Sobolev-Slobodeckii and little H\"older spaces. In the flat case we recover classical results for Petrowskii-parabolic Cauchy problems.

(Revised version, January 2006. S. Gouezel pointed out that, when 1

We recall some known and present several new results about Sobolev spaces defined with respect to a measure, in particular a precise pointwise description of the tangent space to this measure in dimension 1. This allows to obtain an interesting, original compactness result which stays open in R^d, d \textgreater{} 1, and can be applied to a new transport problem, with gradient penalization.

We investigate weighted Sobolev spaces on metric measure spaces $(X,d,m)$. Denoting by $\rho$ the weight function, we compare the space $W^{1,p}(X,d,\rho m)$ (which always concides with the closure $H^{1,p}(X,d,\rho m)$ of Lipschitz functions) with the weighted Sobolev spaces $W^{1,p}_\rho(X,d,m)$ and $H^{1,p}_\rho(X,d,m)$ defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that $W^{1,p}(X,d,\rho m)=H^{1,p}_\rho(X,d, m)$. We also adapt results by Muckenhoupt and recent work by Zhikov to the metric measure setting, considering appropriate conditions on $\rho$ that ensure the equality $W^{1,p}_\rho(X,d,m)=H^{1,p}_\rho(X,d,m)$.

In this paper, we characterize parabolic Besov and parabolic Sobolev spaces in ${\bf R}^{n+1}$ and ${\bf R}^{n+1}_T, \,\, T > 0$. We also, study the relation between parabolic Besov spaces in ${\bf R}^{n}_T, \,\, T > 0$ and standard Besov space in $\R$.

In this note we establish the boundedness properties of local maximal operators $M_G$ on the fractional Sobolev spaces $W^{s,p}(G)$ whenever $G$ is an open set in $\mathbb{R}^n$, $0

In this short paper the discussion of the pointwise characterization of functions $f$ in the Sobolev space $W^{m,p}(\R^n)$ given in the recent paper (Bojarski) is supplemented in \SS1 by a direct, essentially geometric, proof of the novel inequality (for $m>1$), appearing in Bojarski apparently for the first time, and involving the use of the $m$-th difference of the function $f$. Moreover in \SS2 some additional comments to the text in Bojarski are given and a natural class of Sobolev spaces in domains $G$ in $\R^n$ is defined. \SS3 contains some final remarks.

The purpose of this investigation is to extend basic equations and inequalities which hold for functions $f$ in a Bernstein space $B_\sigma^2$ to larger spaces by adding a remainder term which involves the distance of $f$ from $B_\sigma^2$. First we present a modification of the classical modulation space $M^{2,1}(\mathbb{R})$, the so-called readapted modulation space $M^{2,1}_\text{a}(\mathbb{R})$. Our approach to the latter space and its role in functional analysis is novel. In fact, we establish several chains of inclusion relations between $M^{2,1}_\text{a}(\mathbb{R})$ and the more common Lipschitz and Sobolev spaces, including Sobolev spaces of fractional order. Next we introduce an appropriate metric for describing the distance of a function belonging to one of the latter spaces from a Bernstein space. It will be used for estimating remainders and studying rates of convergence. In the main part, we present the desired extensions. Our applications include the classical Whittaker-Kotel'nikov-Shannon sampling formula, the reproducing kernel formula, the Parseval decomposition formula, Bernstein's inequality for derivatives, and Nikol'ski\u{\i}'s inequality estimating the $l^p(\mathbb{Z})$ norm in terms of the $L^p(\mathbb{R})$ norm.

The purpose of this investigation is to extend basic equations and inequalities which hold for functions $f$ in a Bernstein space $B_\sigma^2$ to larger spaces by adding a remainder term which involves the distance of $f$ from $B_\sigma^2$. First we present a modification of the classical modulation space $M^{2,1}(\mathbb{R})$, the so-called readapted modulation space $M^{2,1}_\text{a}(\mathbb{R})$. Our approach to the latter space and its role in functional analysis is novel. In fact, we establish several chains of inclusion relations between $M^{2,1}_\text{a}(\mathbb{R})$ and the more common Lipschitz and Sobolev spaces, including Sobolev spaces of fractional order. Next we introduce an appropriate metric for describing the distance of a function belonging to one of the latter spaces from a Bernstein space. It will be used for estimating remainders and studying rates of convergence. In the main part, we present the desired extensions. Our applications include the classical Whittaker-Kotel'nikov-Shannon sampling formula, the reproducing kernel formula, the Parseval decomposition formula, Bernstein's inequality for derivatives, and Nikol'ski\u{\i}'s inequality estimating the $l^p(\mathbb{Z})$ norm in terms of the $L^p(\mathbb{R})$ norm.

We prove that the zero set of a nonnegative plurisubharmonic function that solves $\det (\partial \overline{\partial} u) \geq 1$ in $\mathbb{C}^n$ and is in $W^{2, \frac{n(n-k)}{k}}$ contains no analytic sub-variety of dimension $k$ or larger. Along the way we prove an analogous result for the real Monge-Amp\`ere equation, which is also new. These results are sharp in view of well-known examples of Pogorelov and B{\l}ocki. As an application, in the real case we extend interior regularity results to the case that $u$ lies in a critical Sobolev space (or more generally, certain Sobolev-Orlicz spaces).

We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is not limited to) spaces of homogeneous type with doubling measures. Several examples and applications are given, including Poincare inequalities for graph Laplacians, Fractional Poincare inequalities of Mouhot, Russ, and Sire [16], and implications for recent work of the author and R. M. Strain on the Boltzmann collision operator [10, 11, 9].

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative $d$-torus $\mathbb{T}^d_\theta$ (with $\theta$ a skew symmetric real $d\times d$-matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincar\'e type inequality for Sobolev spaces. We also show that the Sobolev space $W^k_\infty(\mathbb{T}^d_\theta)$ coincides with the Lipschitz space of order $k$, already studied by Weaver in the case $k=1$. We establish the embedding inequalities of all these spaces, including the Besov and Sobolev embedding theorems. We obtain Littlewood-Paley type characterizations for Besov and Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms of the Poisson, heat semigroups and differences. Some of them are new even in the commutative case, for instance, our Poisson semigroup characterizations improve the classical ones. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Br\'ezis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. The same characterization implies that the Besov space $B^\alpha_{\infty,\infty}(\mathbb{T}^d_\theta)$ for $\alpha>0$ is the quantum analogue of the usual Zygmund class of order $\alpha$. We investigate the interpolation of all these spaces, in particular, determine explicitly the K-functional of the couple $(L_p(\mathbb{T}^d_\theta), \, W^k_p(\mathbb{T}^d_\theta))$, which is the quantum analogue of a classical result due to Johnen and Scherer. Finally, we show that the completely bounded Fourier multipliers on all these spaces do not depend on the matrix $\theta$, so coincide with those on the corresponding spaces on the usual $d$-torus.

The paper deals with two inverse problems for Sturm--Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$. The first one is the problem of recovering of a potential by two spectra. We associate with this problem the map $F:\, W^\theta_2\to l_B^\theta,\ F(\sigma) =\{s_k\}_1^\infty$, where $W^\theta_2 = W^\theta_2[0,\pi]$ are Sobolev spaces with $\theta\geqslant 0$, $\sigma=\int q$ is a primitive of the potential $q$ and $l_B^\theta$ are special Hilbert spaces which we construct to place in the regularized spectral data $\bold s = \{s_k\}_1^\infty$. The properties of the map $F$ are studied in details. The main result is the theorem on uniform stability. It gives uniform estimates from above and below of the norm of the difference $\|\sigma -\sigma_1\|_\theta$ by the norm of the difference of the regularized spectral data $\|\bold s -\bold s_1\|_\theta$ where the last norm is taken in $l_B^\theta$. A similar result is obtained for the second inverse problem when the potential is recovered by the spectral function of the operator $L$ generated by Dirichlet boundary conditions. The results are new for classical case $q\in L_2$ which corresponds to the value $\theta =1$.

In this paper we consider the Cauchy problem for 2D viscous shallow water system in $H^s(\mathbb{R}^2)$, $s>1$. We first prove the local well-posedness of this problem by using the Littlewood-Paley theory, the Bony decomposition, and the theories of transport equations and transport diffusion equations. Then, we get the global existence of the system with small initial data in $H^s(\mathbb{R}^2)$, $s>1$. Our obtained result improves the recent result in \cite{W}

In this work we study Hardy Sobolev spaces in the ball of $C^n$ with respect to interpolating sequences and Carleson measures. We compare them with the classical Hardy spaces of the ball and we stress analogies and differences.

The wave-Sobolev spaces $H^{s,b}$ are $L^2$-based Sobolev spaces on the Minkowski space-time $\R^{1+n}$, with Fourier weights are adapted to the symbol of the d'Alembertian. They are a standard tool in the study of regularity properties of nonlinear wave equations, and in such applications the need arises for product estimates in these spaces. Unfortunately, it seems that with every new application some estimates come up which have not yet appeared in the literature, and then one has to resort to a set of well-established procedures for proving the missing estimates. To relieve the tedium of having to constantly fill in such gaps "by hand", we make here a systematic effort to determine the complete set of estimates in the bilinear case. We determine a set of necessary conditions for a product estimate $H^{s_1,b_1} \cdot H^{s_2,b_2} \hookrightarrow H^{-s_0,-b_0}$ to hold. These conditions define a polyhedron $\Omega$ in the space $\R^6$ of exponents $(s_0,s_1,s_2,b_0,b_1,b_2)$. We then show, in space dimension $n=3$, that all points in the interior of $\Omega$, and all points on the faces minus the edges, give product estimates. We can also allow some but not all points on the edges, but here we do not claim to have the sharp result. The corresponding result for $n=2$ and $n=1$ will be published elsewhere.

Let $\sigma\in(0,1)$ with $\sigma\neq\frac{1}{2}$. We investigate the fractional nonlinear Schr\"odinger equation in $\mathbb R^d$: $$i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-\Delta)^\sigma$ is the Fourier multiplier of symbol $|\xi|^{2\sigma}$, and $\mu=\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.

We investigate elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on appropriate couples of the inner product isotropic H\"ormander spaces $H^{s,\varphi}$, which form the refined Sobolev scale. The order of differentiation for these spaces is given by the real number $s$ and positive function $\varphi$ that varies slowly at infinity in the sense of Karamata. We consider this problem for an arbitrary elliptic equation $Au=f$ on a bounded Euclidean domain $\Omega$ under the condition that $u\in H^{s,\varphi}(\Omega)$, $s

In recent years, parabolic variational inequalities (V.I.) have been intensively developed in a functional analytic setting involving many function spaces. As in the case of elliptic V.I., the tools of potential theory have also proven to be most useful for solving and interpreting parabolic V.I. Several facts exhibit a close relationship between the functional analytic and potential theoretic approaches. Among them is the result provided in this paper. Let us describe its content. Just as for the Laplacian operator, a capacity had been associated with the heat operator in order to solve various problems in potential theory. On the other hand, functional spaces - mainly Sobolev spaces, had been introduced to solve variational inequalities involving the heat operator. We prove here that this capacity can be defined in terms of the topology naturally induced by these functional spaces. This leads to interesting new results for parabolic variational inequalities. (Author)

We study elliptic equations of order $2m$ with nonlocal boundary-value conditions in plane angles and in bounded domains, dealing with the case where the support of nonlocal terms intersects the boundary. We establish necessary and sufficient conditions under which nonlocal problems are Fredholm in Sobolev spaces and, respectively, in weighted spaces with small weight exponents. We also obtain an asymptotics of solutions to nonlocal problems near the conjugation points on the boundary, where solutions may have power singularities.

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general Lizorkin--Triebel spaces based on mixed $L_p$-norms. In this context a Nikol'skij--Plancherel--Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to spaces of the quasi-homogeneous type.

Denote by $L_D$ the Sturm-Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$ with Dirichlet boundary conditions $y(0)=y(\pi)=0$. Let $\{\lambda_k\}_1^\infty$ and $\{\alpha_k\}_1^\infty$ be the sequences of the eigenvalues and norming constants of this operator. For all $\theta \geqslant 0$ we study the map $F: W_2^{\theta} \to l_D^\theta$ defined by $F(\sigma) =\{s_k\}_1^\infty$. Here $\sigma= \int q $ is the primitive of $q$, $\bold s = \{s_k\}_1^\infty$ be regularized spectral data defined by $s_{2k} =\sqrt{\lambda_k}-k,\ s_{2k-1}=\alpha_k-\pi/2$ and $l_D^\theta$ are special Hilbert spaces which are constructed in the paper as finite dimensional extensions of the usual weighted $l_2$ spaces. We give a complete characterization of the image of this nonlinear operator, show that it is locally invertible analytic map, find explicit form of its Frechet derivative. The main result of the paper are the uniform estimates of the form $\|\sigma-\sigma_1\|_\theta \asymp \|\bold s -\bold s_1\|_\theta$, provided that the spectral data $\bold s$ and $\bold s_1$ run through special convex sets in the spaces $l_D^\theta$.

In this paper we present a new characterization of Sobolev spaces on Euclidian spaces ($\mathbb{R}^n$). Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of $\mathbb{R}^n$ and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space.

In this article, we re-examine some of the classical pointwise multiplication theorems in Sobolev-Slobodeckij spaces, and along the way we cite a simple counter-example that illustrates how certain multiplication theorems fail in Sobolev-Slobodeckij spaces when a bounded domain is replaced by Rn. We identify the source of the failure, and examine why the same failure is not encountered in Bessel potential spaces. To analyze the situation carefully, we begin with a survey of the classical multiplication results stated and proved in the 1977 article of Zolesio, and we carefully distinguish between the case of spaces defined on the all of Rn and spaces defined on a bounded domain (with e.g. a Lipschitz boundary). However, the survey we give has a few new wrinkles; the proofs we include are based almost exclusively on interpolation theory rather than Littlewood-Paley theory and Besov spaces, and some of the results we give and their proofs, including the results for negative exponents, do not appear in the literature in the way presented here. We also include a particularly important variation of one of the multiplication theorems that is relevant to the study of nonlinear PDE systems arising in general relativity and other areas. The conditions for multiplication to be continuous in the case of Sobolev-Slobodeckij spaces are somewhat subtle and intertwined, and as a result, the multiplication theorems of Zolesio in 1977 have been cited (more than once) in the standard literature in slightly more generality than what is actually proved by Zolesio, and in cases that allow for the construction of counter-examples such as the one included here.

We consider the most general class of linear boundary-value problems for higher-order ordinary differential systems whose solutions and right-hand sides belong to the corresponding Sobolev spaces. For parameter-dependent problems from this class, we obtain a constructive criterion under which their solutions are continuous in the Sobolev space with respect to the parameter. We also obtain a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem. These results are applied to a new broad class of parameter-dependent multipoint boundary-value problems.

We prove that $W^{1}_{p}$ is an interpolation space between $W^{1}_{p_{1}}$ and $W^{1}_{p_{2}}$ for $p>q_{0}$ and $1\leq p_{1}

1Theory of Social Revolutions

By Brooks Adams

Brooks Adams (1848-1927), was an American historian and a critic of capitalism. He believed that commercial civilizations rise and fall in predictable cycles. First, masses of people draw together in large population centers and engage in commercial activities. As their desire for wealth grows, they discard spiritual and creative values. Their greed leads to distrust and dishonesty, and eventually the society crumbles. In The Law of Civilisation and Decay (1895), Adams noted that as new population centers emerged in the west, centers of world trade shifted from Constantinople to Venice to Amsterdam to London. He predicted in America’s Economic Supremacy (1900) that New York would become the centre for world trade. The Theory of Social Revolutions was written in 1913. (Wikipedia)

2Cattle Brands

By Andy Adams

Cattle Brands is a collection of 14 entertaining short stories depicting not only the life of cowboys in the wild, wild West, but also the harrowing skirmishes with banditos, thrilling shoot-outs, attempt at and the recapture of stolen chattel from fierce desperados, and much, much more exciting accounts that make one think it all actually happened. (Summary by Kehinde)

3Log of a Cowboy

By Andy Adams

The Log of a Cowboy is an account of a five-month drive of 3,000 cattle from Brownsville, Texas, to Montana in 1882 along the Great Western Cattle Trail. Although the book is fiction, it is firmly based on Adams's own experiences on the trail, and it is considered by many to be the best account of cowboy life in literature. Adams was disgusted by the unrealistic cowboy fiction being published in his day; The Log of a Cowboy was his response. It is still in print, and even modern reviewers consider it a compelling classic. The Chicago Herald said: "As a narrative of cowboy life, Andy Adams' book is clearly the real thing. It carries its own certificate of authentic first-hand experience on every page." (Introduction by Wikipedia)

4Education of Henry Adams

By Henry Brooks Adams

The Education of Henry Adams records the struggle of Bostonian Henry Adams (1838-1918), in early old age, to come to terms with the dawning 20th century, so different from the world of his youth. It is also a sharp critique of 19th century educational theory and practice. In 1907, Adams began privately circulating copies of a limited edition printed at his own expense. Commercial publication had to await its author's 1918 death, whereupon it won the 1919 Pulitzer Prize. (Introduction by Wikipedia)

5Making a Rock Garden

By Henry Sherman Adams

A short look at building a rock garden, right from the rocks themselves and how to arrange them, to choosing and placing the plants, touching wall and bog gardens, too. In this little monograph, the author is trying to draw the eyes of U.S. gardeners in to the intimate beauty of this neglected hobby.<br><br> (The original work has a number of attractive and useful photographs and drawings.)

6Outlet

By Andy Adams

Andy Adams worked as a cowboy on trail drives from Texas for eight years. This is an account of a drive when he was the foreman of a herd of Texas cattle being driven to Montana. Expect the same quality writing as found in other books by Adams. (Summary by Richard Kilmer)

Abigail Adams lived the American Revolution as the wife of one of its central figures--John Adams. Her family correspondence, published along with a memoir by her grandson, Charles Francis Adams, brings that era into eloquent focus. What was it like to hear the cannon's roar from your window? to face pestilence? food shortages? rampant inflation? devalued coinage? to raise four children alone--and earn the money to keep your household afloat, while your husband was engaged in politics and diplomacy miles and oceans away . . . for years at a time? It's all there in her private letters, letters that were never meant for public eyes, letters that she repeatedly asked to be burned!<br /><br /> Writing to her husband, Abigail expressed anguish at nearby fighting: June 18, 1775, "Charlestown is laid in ashes . . . How many have fallen we know not. The constant roar of the cannon is so distressing, that we cannot eat, drink, or sleep." Yet, she displayed constant fortitude: August 5, 1777, "It is almost thirteen years since we were united, but not more than half that time have we had the happiness of living together. The unfeeling world may consider it in what light they please. I consider it as a sacrifice to my country . . ." And she enlivened her writing with pithy accounts of Revolutionary doings: July 31, 1777, "You must know that there is a great scarcity of sugar and coffee . . . An eminent, wealthy, stingy merchant had a hogshead of coffee in his store, which he refused to sell . . . under six shillings per pound. A number of females . . . assembled with a cart and trucks, marched down to the warehouse, and demanded the keys, which he refused to deliver. Upon which, one of them seized him by his neck . . ." The women "then opened the warehouse, hoisted out the coffee themselves, put it into the truck and drove off. . . .A large concourse of men stood amazed, silent spectators of the whole transaction."<br /><br /> Abigail’s letters, letters of "a vigorous mind," are a fascinating read. (Introduction by Sue Anderson)

8Democracy - An American Novel

By Henry Brooks Adams

Not until after his death in 1918 was it revealed that Henry Adams was the anonymous author of Democracy, which had been published to great acclaim in 1880. Though the book avoids dates and the characters are fictitious, the setting is no doubt that of Washington in the 1870s, the age of Presidents Grant and Hayes. The young widow, Madeleine Lee, wealthy and independent, is the protagonist, who leaves her New York for Washington to turn her intelligence to politics and to see what makes her country tick. There she meets (among others) Senator Silas P. Ratcliffe of Illinois, one of the most powerful and influential (if somewhat uncultured) men of the capital, who is considering a run for the presidency, and who needs a wife to act as First Lady, a position that (he thinks) Mrs. Lee would admirably fill. Through the book Adams plays with the themes of political necessity, compromise, corruption -- particularly the kind of corporate domination of national politics that he saw becoming all too powerful. Should honest and intelligent men keep their integrity by avoiding politics? Or would that simply mean turning over the governance of the country to power-hungry, scheming, and none too honest hacks? For all the witty conversations in his novel, this was a theme that plagued Adams (a presidential grandson) in life as well as literature, and it is a theme that has by no means disappeared today. (Introduction by Nicholas Clifford)</p>

9Secret of Lonesome Cove

By Samuel Hopkins Adams

A body is found on the beach not far from a New England town one morning. Curiously, nobody recognizes the dead woman, and nobody in or near the town seems to be a suspect in a possible murder, therefore most of them assume that she simply washed ashore from a passing vessel. Only problem is vessels didn't pass that stretch of the coast because of it's peculiar tides and eddies; hence its name, Lonesome Cove. Following the finding of the body, the officials of the town start acting a bit peculiar towards how to handle the dead body. The sheriff, the attorney in the town, the village gossip, and even the town doctor all seem to want the identification of the body hushed up. And therefore the body is immediately buried, and the officials hope all will be forgotten and done with. Cause of death: drowning. Enter Chester Kent, who isn't convinced in the same way that the townspeople are, and is willing to pull at his ear lobe in an effort to determine the identity of the body, and how it came to it's demise. (Introduction by Roger Melin)

10Reed Anthony, Cowman: An Autobiography

By Andy Adams

Adams breathes life into the story of a Texas cowboy who becomes a wealthy and influential cattleman.. (Introduction by Wikipedia)

11Wanted: A Husband

By Samuel Hopkins Adams

This serious, yet witty and hilarious, romantic comedy by Samuel Hopkins Adams is a must read/listen! From the very beginning it intrigues you with the troubles and feminine woes of young Miss Darcy Cole. Being a dowdy, unkempt fledgling lady, she finds herself in a mirthful fix after telling a whopper to her room-mates in order to save face from her severely lacking personal life. Resentful and jealous of most all other women, and contemplating suicide, she turns to actress Gloria Greene for guidance on how to become more attractive. Miss Cole had no idea what she was getting herself into! Was it even possible to make herself over? How on Earth would she get through the lie of being engaged to a “titled” man, no less! She had never even met Sir Montrose Veyze, let alone be engaged to him! And then there is the charming Jack Remsen, who seems to be in a pickle himself. Darcy knows he would never fall for someone like her; in fact, no man even looks at her! But perhaps he might be coerced into unwittingly helping her out of her predicament.<br /><br /> A true Ugly Duckling tale, Darcy experiences all the emotions that most ALL women have felt at one time or another throughout life. As quoted by the Proof Listener for this work, the narrator does a superb job of “capturing the spirit of the book!” Drawing the listener in and allowing the audience to live and feel the experience along with the characters. A highly entertaining and riveting book! (summary by reader)

12Notting Hill Mystery

By Charles Warren Adams

Charles Felix was the pseudonym of Charles Warren Adams, an English Lawyer and publisher and is now known to have been the author of "The Notting Hill Mystery", thought to be the first full length detective novel in English.<br /><br /> The story first appeared as an eight part serial in a weekly magazine in 1862, and was subsequently published as a single volume novel in 1865.</p> The story deals with the then newly emerging field of 'mesmerism' which we now know as hypnotism, and its use in the planning and execution of three truly devious crimes. The novel, unusually, is written wholly in the form of a series of letters and reports gathered by the investigator from the various witnesses in the case, and the reader is left to decide themselves the guilt or otherwise of the chief suspect. (summary by Kevin Green)

This is a small volume of poems by Canadian women's rights activist and educator Mary Adams. - Summary by Carolin

14State of the Union Addresses by United States Presidents (1790 - 1816)

By John Adams, Thomas Jefferson, James Madison and George Washington

The State of the Union address is a speech presented by the President of the United States to a joint session of the United States Congress, typically delivered annually. The address not only reports on the condition of the nation but also allows the President to outline his legislative agenda (for which he needs the cooperation of Congress) and national priorities. This album contains recordings of addresses from Washington, Adams, Jefferson, and Madison. - Summary by Wikipedia and Linette Geisel

15State of the Union Addresses by United States Presidents (1817 - 1828)

By John Quincy Adams and James Monroe

The State of the Union address is a speech presented by the President of the United States to a joint session of the United States Congress, typically delivered annually. The address not only reports on the condition of the nation but also allows the President to outline his legislative agenda (for which he needs the cooperation of Congress) and national priorities. This album contains recordings of addresses from James Monroe and John Quincy Adams. (Wikipedia and Linette Geisel)

Arguably one of America's greatest writers, Washington Irving is the author of such classics as "Legend of Sleepy Hollow," "Bracebridge Hall," and "Knickerbocker's History of New York." This book is a concise and extremely entertaining biography of this unique author. <br> Note to the listener: There are a couple of typos in this text. Chapter 33 should have been numbered as chapter 32, and there are two chapter 35's. The readers have keep the typos in the reading, therefore, there is no chapter 32, and the two chapter 35's are designated at "the first" and "the second." - Summary by Greg Giordano

17One Among So Many

By Francis William Lauderdale Adams

Adams' energy and drive can be seen through his large output of written work in his short lifetime. He often wrote quickly and did little revision, living as he did on the proceeds of his own work rather than with the support of a family or sinecure. Songs of the Army of the Night has been reprinted in many editions, but the reputation of these poems ascends from their engagement with social issues, rather than their value as pure poetry for Adams was deeply sympathetic towards downtrodden races and men. - Summary by Wikipedia

18Mystery of the Chinese Ring

By Andy Adams

The Mystery of the Chinese Ring is an exotic adventure story and is set in locations such as Burma and China, with the historical and political ramifications which applied to the mid Twentieth Century and still ring true in the early 21st Century. What is the purpose of the ring? What is the significance of the letter “K”? Why the interest in a sixteen year old boy going to visit a relative in Burma? Why are family dynasties so important, and why the secrecy concerning their survival? This is an audiobook that will find eager listeners from the ages of about ten to octogenarians, male and female, and also those enjoying adventure stories with many twists and turns. The author, Andy Adams, was a well known writer of cowboy books which are thought to be the best in that genre, and his Biff Brewster books, of which he wrote three, are likewise highly regarded. As the main hero is aged sixteen, this story will find resonances in all those that are young in heart, and is an entertaining book in its own right. - Summary by Peter thomlinson

19Brazilian Gold Mine Mystery

By Andy Adams

Mystery adventure, fiction ( Peter Thomlinson). This is a very exciting and gripping story set in the jungles of Brazil and Venezuela, and the quest for the famous El Dorado gold. Incidents with dangerous wild animals, not to mention encounters with head hunters and other native tribes, plus some black magic hocus-pocus all add to the suspense. Double dealing and threats as well as actual confrontations all make life difficult for our brave heroes, and often cause the expedition problems which slows down the quest for the yellow gold. This is a good geography lesson also, and readers will learn about all those huge rivers that flow through these regions, including the huge River Amazon.

20Hawaiian Sea Hunt Mystery

By Andy Adams

Adventure series in exotic locations. This adventure is set in the exotic Island of Hawaii and involves the search for a missing sloop and the whereabouts of a valuable mine. Biff Brewster, his father and friends become entangled with ruthless criminals who are also intending to find the sunken boat and discover the map and the mine. They have kidnapped the elderly scientist who holds the key to the discovery. An exciting adventure with lots of action on land and sea.- Summary by Peter Thomlinson

21Nothing of Importance

By John Bernard Pye Adams

Fighting in France during the Great War, Bernard Adams, an officer with a Welsh battalion, was moved to chronicle what he saw and experienced: the living conditions and duties of officers and “Tommies” (enlisted men) in their dank, rat-infested trenches and behind the lines; the maiming and deaths; and the quiet periods described in official reports as “nothing of importance”. Adams relates his wounding in June, 1916 and its aftermath. The concluding chapter, which he wrote during his convalescence in “Blighty” (soldiers’ slang for England), is an impassioned reflection on war. Following several months of recuperation Adams returned to the front where, on February 26, 1917 he was wounded again. The following day he died. (Lee Smalley)

22Texas Matchmaker

By Andy Adams

As a boy Andy Adams helped with the cattle and horses on the family farm. During the early 1880s he went to Texas, where he stayed for 10 years, spending much of that time driving cattle on the western trails. A Texas Matchmaker is a narrative that describes the work of a cowboy on a large southTexas ranch during the late 1800’s. Adams is considered to be one of foremost writers of the life of the real American cowboy. (Summary by Richard Kilmer)

23Chant of the Firemen

By Francis William Lauderdale Adams

Francis William Lauderdale Adams was an essayist, poet, dramatist, novelist and journalist who produced a large volume of work in his short life. A self-professed 'Child of his Age', Adams combined in his life and work many distinctive features of both fin de siècle British culture and the Australian radical nationalism of the 1890s, including a strong sympathy with socialist and feminist movements. - Summary by Wikipedia

24Celebrated Women Travellers of the Nineteenth Century

By W. H. Davenport Adams

The nineteenth century saw the expansion in popularity of travel among the wealthy. Add to this the independence more women were enjoying as they were released from work in the home, and we have the recipe for some intrepid female travelers. Here we see the stories of just some such adventurous women. - Summary by Lynne Thompson

25Mystery of the Ambush in India

By Andy Adams

Another Biff Brewster adventure story set this time in exotic and mystical India with much of the action around and near the Himalayas. This story brings together all of the friends that Biff Brewster has made in his previous adventures plus a couple of new friends, and involves his father and Uncle Charles once again, and the other members of his family. Some of the events are spectacular, and tension is held by the involvement of an international spy and other situations including riots in Calcutta, poisonous snakes, a mad tiger, and wild bears, as well as a strange religious cult.

26Women I'm Not Married To

By Franklin Pierce Adams

This poem, by Franklin P. Adams, is the sequel to/answer to Dorothy Parker's poem, Women I'm Not Married To, with a decidedly different but equally humorous take on the matter. ( Summary by Michele Fry )

The Premier has decided that being married would be good for his image. He asks his stenographer for advice: (Premier) Good. Just jot me down a precis of the points made by your fifteen admirers when proposing - the points that specially appealed to you. (Stenographer) I'm afraid, sir, that what most appealed to me could not be expressed in words. In fact, it wasn't words. But no, sir. The subject is too sacred.... (Premier)...But you could tell me how they began. The opening address, eh? How did they lead up? (Stenographer) Most of 'em just kissed me, sir. It seemed to give them confidence. (Premier) But I couldn't possibly start like that. (Stenographer) It's always done sir.... - Summary by ToddHW <BR><BR>Cast list:<BR> Herbert Dix, The Premier's Chief Private Secretary: <a href="https://librivox.org/reader/12916">Peter Musgrove</a><br> Effie Bimm, Stenographer in the Premier's Room: <a href="https://librivox.org/reader/11084">Devorah Allen</a><br> Gregory, Chief Messenger at Parliament House: <a href="https://librivox.org/reader/7170">alanmapstone</a><br> William Power, Premier: <a href="https://librivox.org/reader/8879">Son of the Exiles</a><br> Patrick O'Reilly, A Constituent: <a href="https://librivox.org/reader/11790">TJ Burns</a><br> Edward Vyce, The Party Whip: <a href="https://librivox.org/reader/10614">Leanne Yau</a><br> Martha Callender, The Premier's Sister: <a href="https://librivox.org/reader/6281">Beth Thomas</a><br> Charles Lukin, Reporter on The Tribune: <a href="https://librivox.org/reader/10789">Tomas Peter</a><br> Helen Pretty, A widow, the owner of Wyonora Estate: <a href="https://librivox.org/reader/10179">Sonia</a><br> Mrs. Cusack, A friend of Mrs. Pretty's: <a href="https://librivox.org/reader/10087">Sandra Schmit</a><br> Vernon Harrington, Leader of the Opposition: <a href="https://librivox.org/reader/12948">MajorToast</a><br> Maid: <a href="https://librivox.org/reader/10614">Leanne Yau</a><br> Stage Directions and Editing by: <a href="https://librivox.org/reader/6754">ToddHW</a>

28Perils in the Transvaal and Zululand

By Henry Cadwallader Adams

A young man travels to South Africa to find his Mother and sister. He wants to be a clergyman and a farmer when he arrives there. This story includes accounts of the Zulu-Boer wars. - Summary by Ingrid Kennedy

Here we have a serious family drama by Australian playwright Arthur Adams about legacy and responsibilities, with some question about exactly who exactly the Wasters in society are. "No; you produce nothing, you spend everything; you do nothing, you waste everything. You're parasites, useless parasites.... Parasites, inefficients, wasters, wasters!" "We may be parasites, but you have made us parasites. We may be wasters, but you prey on us." To highlight the growing tensions, it includes comic second act scenes placed in the women's undergarment department of the family store. - Summary by ToddHW <BR><BR>Cast list:<BR> John Dangar: <a href="https://librivox.org/reader/13022">Josh Kibbey</a><br> Baby Dangar: <a href="https://librivox.org/reader/10614">Leanne Yau</a><br> Guy Dangar: <a href="https://librivox.org/reader/13072">Campbell Schelp</a><br> Butler: <a href="https://librivox.org/reader/6754">ToddHW</a><br> Anthony Clibborn: <a href="https://librivox.org/reader/12874">Aaron White</a><br> Lindsay Thong: <a href="https://librivox.org/reader/10789">Tomas Peter</a><br> Tossie Quork: <a href="https://librivox.org/reader/11084">Devorah Allen</a><br> J. Charteris Maggs: <a href="https://librivox.org/reader/8879">Son of the Exiles</a><br> Mrs. Goodsir: <a href="https://librivox.org/reader/9743">Ulrike Denis</a><br> Mrs. Megget: <a href="https://librivox.org/reader/13127">Betsy Walker</a><br> Mrs. Cronk: <a href="https://librivox.org/reader/11790">TJ Burns</a><br> Elizabeth Hebblethwaite: <a href="https://librivox.org/reader/12500">Foon</a><br> Mrs. Clibborn: <a href="https://librivox.org/reader/12823">Pauline Latournerie</a><br> Stage Directions: <a href="https://librivox.org/reader/10179">Sonia</a><br> Edited by: <a href="https://librivox.org/reader/6754">ToddHW</a>

30Nequa or The Problem of the Ages

By Jack Adams

<i>Nequa or The Problem of the Ages</i> is one of the first feminist science fiction books published in the United States. It was first serialized in the newspaper Equity, and two editions were published in Topeka, Kansas in 1900. The voyage described in Nequa is to the Arctic. The captain of the ship is amazed to find that when they reach a certain spot, which should be close to the North Pole, the compass shows that the ship is suddenly traveling south. Actually they have sailed into the interior of the earth, where they meet the Altrurians, a people who have developed into a more cooperative society than those of the outer earth. Individual Altrurians describe how the evolution of their society took place. These explanations reflect the Populist thought of the time, with definite feminist proclivities, thereby providing the tone of a political novel. <br>While the book was published with the pseudonym of Jack Adams, it was copyrighted under the names Alcanoan A. Grigsby and Mary P. Lowe, who are presumed to be the true authors. - Summary by Jeff Burke