Sobolev spaces

We characterize the set of functions which can be approximated by smooth functions and by polynomials with the norm �f � W k, ∞ (w):= k� �f (j) �L ∞ (wj), j=0 for a wide range of (even non-bounded) weights wj’s. We allow a great deal of independence among the weights wj’s.

Final Degree Dissertation for my undergraduate in Mathematics at the University of the Basque Country. The dissertation is intended as an introduction to Sobolev spaces, with the objective of applying abstract results of Functional Analysis and Sobolev Spaces results to the study of Partial Differential Equations (PDEs).

This PhD thesis focuses on numerical and analytical methods for simulating the dynamics of volcanic ash plumes.
The study starts from the fundamental balance laws for a multiphase gas– particle mixture, reviewing the existing models and developing a new set of Partial Differential Equations (PDEs), well suited for modeling multiphase dispersed turbulence. In particular, a new model generalizing the equilibrium–Eulerian model to two-way coupled compressible flows is developed.
The PDEs associated to the four-way Eulerian-Eulerian model is studied, in- vestigating the existence of weak solutions fulfilling the energy inequalities of the PDEs. In particular, the convergence of sequences of smooth solutions to such a set of weak solutions is showed.
Having explored the well-posedness of multiphase systems, the three-dimensional compressible equilibrium–Eulerian model is discretized and numerically solved by using the OpenFOAM® numerical infrastructure. The new solver is called ASHEE, and it is verified and validated against a number of well understood benchmarks and experiments. It demonstrates to be capable to capture the key phenomena involved in the dynamics of volcanic ash plumes. Those are: turbulence, mixing, heat transfer, compressibility, preferential concentration of particles, plume entrainment.
The numerical solver is tested by taking advantage of the newest High Perfor- mance Computing infrastructure currently available.
Thus, ASHEE is used to simulate two volcanic plumes in realistic volcanological conditions. The influence of model configuration on the numerical solution is analyzed. In particular, a parametric analysis is performed, based on: 1) the kinematic decoupling model; 2) the subgrid scale model for turbulence; 3) the discretization resolution.
In a one-dimensional and steady-state approximation, the multiphase flow model is used to derive a model for volcanic plumes in a calm, stratified atmosphere. The corresponding Ordinary Differential Equations (ODEs) are written in a compact, dimensionless formulation. The six non-dimensional parameters characterizing a multiphase plume are then written. The ODEs is studied both numerically and analytically. Different regimes are analyzed, extracting the first integral of motion and asymptotic solutions. An asymptotic analytical solution approximating the model in the general regime is derived and compared with numerical results. Such a solution is coupled with an electromagnetic model providing the infrared intensity emitted by a volcanic ash plume. Key vent parameters are then retrieved by means of inversion techniques applied to infrared images measured during a real volcanic eruption.

The theme of this short article is to investigate an orthogonal decomposition of the Sobolev space W 1,2 (Ω) as W 1,2 (Ω) = A 2,2 (Ω) ⊕ D 2 W 3,2 0 (Ω) and look at some of the properties of the inner product therein and the distance defined by the inner product. We also determine the dimension of the orthogonal difference space W 1,2 (Ω) W 1,2 0 (Ω) ⊥ and show the expansion of Sobolev spaces as their regularity increases.

Physical Science focuses on recent advances of fundamental theories and principles, analytical and symbolic approaches, as well as computational techniques in nonlinear physical science and nonlinear mathematics with engineering applications.

We define a class of deformations in $W^{1,p}(\Omega,\mathbb{R}^n)$, $p>n-1$, with positive Jacobian that do not exhibit cavitation.
We characterize that class in terms of the non-negativity of the topological degree and the equality between the distributional determinant and the pointwise determinant of the gradient.
Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity.
We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in $W^{1,p}$, and the sequence of local inverses converges to the local inverse.
We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps.
We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity.

In this paper we make a survey of some recent developments of the theory of Sobolev spaces W 1,q (X, d, m), 1 < q < ∞, in metric measure spaces (X, d, m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeger's work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of m. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.

We consider shape optimization problems of the form {J(Ω) :Ω⊂ X, m(Ω)&lt; c}, where X is a metric measure space and J is a suitable shape functional. We adapt the notions of γ-convergence and weak γ-convergence to this new general abstract setting to prove the existence of an optimal domain. Several examples are pointed out and discussed.

We consider shape optimization problems with internal inclusion constraints, of the form $$\min\big\{J(\Omega)\ :\ \Dr\subset\Omega\subset\R^d,\ |\Omega|=m\big\},$$ where the set $\Dr$ is fixed, possibly unbounded, and $J$ depends on $\Omega$ via the spectrum of the Dirichlet Laplacian. We analyze the existence of a solution and its qualitative properties, and rise some open questions.

In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of an elliptic operator and we focus on the existence of an optimal domain. The known results are presented as well as a list of still open problems. Related fields as optimal partition problems, evolution flows, Cheeger-type problems, are also considered.

This paper considers the equation of the type − + + = , (,) ∈ ℝ × (0,); which is the Black-Scholes option pricing model that includes the presence of transaction cost. The existence, uniqueness and continuous dependence of the weak solution of the Black-Scholes model with transaction cost are established.The continuity of weak solution of the parameters was discussed and similar solution as in literature obtained.

We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on the notion of autoregressive Hilbert processes that represent a generalization of the classical autoregressive processes to random variables with values in a Hilbert space. A careful analysis reveals, in particular, that this approach is related to the theory of function estimation in linear ill-posed inverse problems. In the deterministic literature, such problems are usually solved by suitable regularization techniques. We describe some recent approaches from the deterministic literature that can be adapted to obtain fast and feasible predictions. For large sample sizes, however, these approaches are not computationally efficient.

We prove that if Ω ⊆ R 2 is bounded and R 2 \ Ω satisfies suitable structural assumptions (for example it has a countable number of connected components), then W 1,2 (Ω) is dense in W 1,p (Ω) for every 1 ≤ p < 2. The main application of this density result is the study of stability under boundary variations for nonlinear Neumann problems of the form −div A(x, ∇u) + B(x, u) = 0 in Ω, A(x, ∇u) • ν = 0 on ∂Ω, where A : R 2 × R 2 → R 2 and B : R 2 × R → R are Carathéodory functions which satisfy standard monotonicity and growth conditions of order p.

We are motivated by the problem of control for a non-homogeneous elastic string with memory. We reduce the problem of controllability to a non-standard moment problem. The solution of the latter problem is based on an auxiliary Riesz basis property result for a family of functions quadratically close to the nonharmonic exponentials. This result requires the detailed analysis of an integro-differential equation and is of interest in itself for Function Theory. Controllability of the string implies observability of a dual system.

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 a recent paper of 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).

Communicated by T. Hishida Kato, Ponce, Beale and Majda prove the existence and uniqueness of maximal solution of Euler and Navier-Stokes equations and some blow-up criterion. In the periodic case, we establish that if the maximum time T is finite, then the growth of ku.t/k H m is at least of the order of .T t/ 2m=5 .

We study different Sobolev spaces associated with multidimensional Laguerre expansions. To do this we establish an analogue of P.A. Meyer's multiplier theorem, prove some transference results between higher order Riesz-Hermite and Riesz-Laguerre transforms, and introduce fractional derivatives and integrals corresponding to the Laguerre setting. Hypercontractivity of the Laguerre semigroups and Calderón's reproduction formula are also discussed.  2004 Elsevier SAS. All rights reserved.

Motivated by recent developments on calculus in metric measure spaces (X, d, m), we prove a general duality principle between Fuglede's notion of p-modulus for families of finite Borel measures in (X, d) and probability measures with barycenter in L q (X, m), with q dual exponent of p ∈ (1, ∞). We apply this general duality principle to study null sets for families of parametric and non-parametric curves in X. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on p-Modulus ([21], [23]) and suitable probability measures in the space of curves ([6], [7]).

We prove a regularity result for a Fredholm integral equation with weakly singular kernel, arising in connection with the neutron transport equation in an infinite cylindrical domain. The theorem states that the solution has almost two derivatives in L1, and is proved using Besov space techniques. This result is applied in the error analysis of the discrete ordinates method for the numerical solution of the neutron transport equation. We derive an error estimate in the L1-norm for the scalar flux, and as a consequence, we obtain an error bound for the critical eigenvalue.

An approach to analysis on path spaces of Riemannian manifolds is described. The spaces are furnished with 'Brownian motion' measure which lies on continuous paths, though differentiation is restricted to directions given by tangent paths of finite energy. An introduction describes the background for paths on R m and Malliavin calculus. For manifold valued paths the approach is to use 'Itô' maps of suitable stochastic differential equations as charts. 'Suitability' involves the connection determined by the stochastic differential equation. Some fundamental open problems concerning the calculus and the resulting 'Laplacian' are described. A theory for more general diffusion measures is also briefly indicated. The same method is applied as an approach to getting over the fundamental difficulty of defining exterior differentiation as a closed operator, with success for one and two forms leading to a Hodge-Kodaira operator and decomposition for such forms. Finally there is a brief description of some related results for loop spaces.

This paper presents a direct boundary integral equation method for solving the exterior Neumann problem of the Helmholtz equation which is a mathematical formulation of the acoustic scattering problem at a perfectly hard body. It is proved that the integral equation obtained from the Helmholtz representation is equivalent to the original boundary value problem and it has a unique solution in a suitable Sobolev space in the framework of pseuo-differential operators. Moreover, the numerical treatment and error estimate are given by using a Galerkin approximation.

Absolutely continuous functions of two variables Embeddings Strictly singular embeddings Superstrictly singular embeddings Beppo Levi spaces Sobolev spaces Rearrangement invariant spaces A negative solution of Problem 188 posed by Max Eidelheit in the Scottish Book concerning superpositions of separately absolutely continuous functions is presented. We discuss here this and some related problems which have also negative solutions. Finally, we give an explanation of such negative answers from the "embeddings of Banach spaces" point of view.

We consider elliptic equations of the form L * µ = ν for measures on R n . The membership of solutions in the Sobolev classes W p,1 (R n ) is established under appropriate conditions on the coefficients of L. Bounds of the form (x) ≤ CΦ(x) −1 for the corresponding densities are obtained. 1991 Mathematics Subject Classification. 35J15, 35Q80, 60J60, 35R05.

Recently, multiple input, single output, single hidden layer, feedforward neural networks have been shown to be capable of approximating a nonlinear map and its partial derivatives. Specifically, neural nets have been shown to be dense in various Sobolev spaces (Hornik, Stinchcombe and White, 1989). Building upon this result, we show that a net can be trained so that the map and its derivatives are learned. Specifically, we use a result of Gallant (1987b) to show that least squares and similar estimates are strongly consistent in Sobolev norm provided the number of hidden units and the size of the training set increase together. We illustrate these results by an applic~tion to the inverse problem of chaotic dynamics: recovery of a nonlinear map from a time series of iterates. These results extend automatically to nets that embed the single hidden layer, feedforward network as a special case.

In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ andφ in L 2 (R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψ jk := 2 j/2 ψ(2 j • − k) (j, k ∈ Z) form a Riesz basis for L 2 (R). If, in addition, φ lies in the Sobolev space H m (R), then the derivatives 2 j/2 ψ (m) (2 j • − k) (j, k ∈ Z) also form a Riesz basis for L 2 (R). Consequently, {ψ jk : j, k ∈ Z} is a stable wavelet basis for the Sobolev space H m (R). The pair of φ andφ are not required to be biorthogonal or semi-orthogonal. In particular, φ andφ can be a pair of B-splines. The added flexibility on φ andφ allows us to construct wavelets with relatively small supports.

In this note we give a proof of a result on immersions of domains of fractional powers of certain sectorial operators associated to strongly elliptic operators in Sobolev spaces; such immersions preserve information on fractional derivatives. We also briefly comment on the application of this result to a problem of optimal control of mosquito populations.

The functional statistical framework is considered to address the problem of least-squares estimation of the realizations of fractal and long-range dependence Gaussian random signals, from the observation of the corresponding response surface. The statistical methodology applied is based on the functional regression model. The geometrical properties of the separable Hilbert spaces of functions, where the response surface and the signal of interest lie, are considered for removing the ill-posed nature of the estimation problem, due to the non-locality of the integro-pseudodifferential operators involved. Specifically, the local and asymptotic properties of the spectra of fractal and long-range dependence random fields in the Linnik-type, Dagum-type and auxiliary families are analyzed to derive a stable solution to the associated functional estimation problem. Their pseudodifferential representation and Reproducing Kernel Hilbert Space (RKHS) characterization are also derived for describing the geometrical properties of the spaces where the functional random variables involved in the corresponding regression problem can be found.

Probability models are estimated by use of penalized log-likelihood criteria related to AIC and MDL. The accuracies of the density estimators are shown to be related to the tradeoff between three terms: the accuracy of approximation, the model dimension, and the descriptive complexity of the model classes. The asymptotic risk is determined under conditions on the penalty term, and is shown to be minimax optimal for some cases. As an application, we show that the optimal rate of convergence is simultaneously achieved for log-densities in Sobolev spaces W s 2 (U ) without knowing the smoothness parameter s and norm parameter U in advance. Applications to neural network models and sparse density function estimation are also provided.

Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is trivial for small p but non trivial for large p. This answers an open question posed by several authors. We also give a full description of the p-weak gradient for any locally finite Borel measure on the real line.

Evolution operators and wavelets are very interesting and attractive, not only by their extremely wide range of applications, but also by their theories of great importance. It is very difficult to show the relations between evolution operators, wavelets and other subjects in pure and applied mathematics. However, perhaps taking into account the obvious relations between the microlocal and wavelet analysis and the white noise; the Brownian motion and the index formulae for de Rham complex; even only on archimedean fields, we can partially illustrate such interesting relations. In our brief talk, let us present some recent results on a semilinear non-classical evolution problem and a Galerkin-wavelet method to solve a very complicated linear evolution one. Then some general results on stationary problems, from which we can continue to study the respective non-stationary cases, will be introduced.

We study the structure of the spectrum of differen- tial operators which arise in the problems modelling the inner oscilla- tions of viscous compressible barotropic exponentially stratified three- dimensional fluid.For Dirichlet problem, we prove that the essential spectrum consists of three real points.We find the sector of the com- plex plane to which all the eigenvalues belong. 1. Viscous Stratified Compressible Fluid Let us consider a bounded domain Ω ⊂ R 3 with the boundary ∂ Ωo f the

We study different Sobolev spaces associated with multidimensional Laguerre expansions. To do this we establish an analogue of P.A. Meyer's multiplier theorem, prove some transference results between higher order Riesz-Hermite and Riesz-Laguerre transforms, and introduce fractional derivatives and integrals corresponding to the Laguerre setting. Hypercontractivity of the Laguerre semigroups and Calderón's reproduction formula are also discussed.

We study the exact controllability problem for a ring under stretching tension that varies in time. We are looking for a couple of forces, which drive the state solution to rest. We show that applying two forces is necessary for controllability and the ring is controllable in the time interval greater than the optical length of the string. We also explain why one force would not be enough to control the ring. We use the method of moments to reduce the controllability problem to a moment problem for the controlling forces. The solution of that problem is based on an auxiliary basis property result. Both method of moments and proof of the basis property are developed for the model with time-dependent parameters.

We develop multicomponent AM-FM models for multidimensional signals. The analysis is cast in a general -dimensional framework where the component modulating functions are assumed to lie in certain Sobolev spaces. For both continuous and discrete LSI systems with AM-FM inputs, powerful new approximations are introduced that provide closed form expressions for the responses in terms of the input modulations. The approximation errors are bounded by generalized energy variances quantifying the localization of the filter impulse response and by Sobolev norms quantifying the smoothness of the modulations. The approximations are then used to develop novel spatially localized demodulation algorithms that estimate the AM and FM functions for multiple signal components simultaneously from the channel responses of a multiband linear filterbank used to isolate components. Two discrete computational paradigms are presented. Dominant component analysis estimates the locally dominant modulations in a signal, which are useful in a variety of machine vision applications, while channelized components analysis delivers a true multidimensional multicomponent signal representation. We demonstrate the techniques on several images of general interest in practical applications, and obtain reconstructions that establish the validity of characterizing images of this type as sums of locally narrowband modulated components.