Radial sign-changing solutions to biharmonic nonlinear Schrodinger equations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Lions-type compactness and Rubik actions on the Heisenberg group

In this paper we prove a Lions-type compactness embedding result for symmetric unbounded domains of the Heisenberg group. The natural group action on the Heisenberg group TeX is provided by the unitary group U(n) × {1} and its appropriate subgroups, which will be used to construct subspaces with specific symmetry and compactness properties in the Folland-Stein’s horizontal Sobolev space TeX. As an application, we study the multiplicity of solutions for a singular subelliptic problem by exploiting a technique of solving the Rubik-cube applied to subgroups of U(n) × {1}. In our approach we employ concentration compactness, group-theoretical arguments, and variational methods.

A Semi-Lagrangian Particle Level Set Finite Element Method for Interface Problems

We present a quasi-monotone semi-Lagrangian particle level set (QMSL-PLS) method for moving interfaces. The QMSL method is a blend of first order monotone and second order semi-Lagrangian methods. The QMSL-PLS method is easy to implement, efficient, and well adapted for unstructured, either simplicial or hexahedral, meshes. We prove that it is unconditionally stable in the maximum discrete norm, � · �h,∞, and the error analysis shows that when the level set solution u(t) is in the Sobolev space Wr+1,∞(D), r ≥ 0, the convergence in the maximum norm is of the form (KT/Δt)min(1,Δt � v �h,∞ /h)((1 − α)hp + hq), p = min(2, r + 1), and q = min(3, r + 1),where v is a velocity. This means that at high CFL numbers, that is, when Δt > h, the error is O( (1−α)hp+hq) Δt ), whereas at CFL numbers less than 1, the error is O((1 − α)hp−1 + hq−1)). We have tested our method with satisfactory results in benchmark problems such as the Zalesak’s slotted disk, the single vortex flow, and the rising bubble.

A Finite Element Numerical Algorithm for Modelling and Data Fitting in Complex Systems

Numerical modelling methodologies are important by their application to engineering and scientific problems, because there are processes where analytical mathematical expressions cannot be obtained to model them. When the only available information is a set of experimental values for the variables that determine the state of the system, the modelling problem is equivalent to determining the hyper-surface that best fits the data. This paper presents a methodology based on the Galerkin formulation of the finite elements method to obtain representations of relationships that are defined a priori, between a set of variables: y = z(x1, x2,...., xd). These representations are generated from the values of the variables in the experimental data. The approximation, piecewise, is an element of a Sobolev space and has derivatives defined in a general sense into this space. The using of this approach results in the need of inverting a linear system with a structure that allows a fast solver algorithm. The algorithm can be used in a variety of fields, being a multidisciplinary tool. The validity of the methodology is studied considering two real applications: a problem in hydrodynamics and a problem of engineering related to fluids, heat and transport in an energy generation plant. Also a test of the predictive capacity of the methodology is performed using a cross-validation method.

Ornstein-Uhlenbeck operator in convex domains of Banach spaces

Studiamo l'operatore di Ornstein-Uhlenbeck e il semigruppo di Ornstein-Uhlenbeck in un sottoinsieme aperto convesso $\Omega$ di uno spazio di Banach separabile $X$ dotato di una misura Gaussiana centrata non degnere $\gamma$. In particolare dimostriamo la disuguaglianza di Sobolev logaritmica e la disuguaglianza di Poincaré, e grazie a queste disuguaglianze deduciamo le proprietà spettrali dell'operatore di Ornstein-Uhlenbeck. Inoltre studiamo l'equazione ellittica $\lambdau+L^{\Omega}u=f$ in $\Omega$, dove $L^\Omega$ è l'operatore di Ornstein-Uhlenbeck. Dimostriamo che per $\lambda>0$ e $f\in L^2(\Omega,\gamma)$ la soluzione debole $u$ appartiene allo spazio di Sobolev $W^{2,2}(\Omega,\gamma)$. Inoltre dimostriamo che $u$ soddisfa la condizione di Neumann nel senso di tracce al bordo di $\Omega$. Questo viene fatto finita approssimazione dimensionale.

Determining the temperature from Cauchy data in corner domains

We consider the problem of reconstruction of the temperature from knowledge of the temperature and heat flux on a part of the boundary of a bounded planar domain containing corner points. An iterative method is proposed involving the solution of mixed boundary value problems for the heat equation (with time-dependent conductivity). These mixed problems are shown to be well-posed in a weighted Sobolev space.

Determining the temperature from incomplete boundary data

An iterative procedure for determining temperature fields from Cauchy data given on a part of the boundary is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. A convergence proof of this method in a weighted L2-space is included, as well as a stopping criteria for the case of noisy data. Moreover, a solvability result in a weighted Sobolev space for a parabolic initial boundary value problem of second order with mixed boundary conditions is presented. Regularity of the solution is proved. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Representations of Inverse Functions by the Integral Transform with the Sign Kernel

Mathematics Subject Classification: Primary 30C40

Existence of Solutions for a Class of Quasi-Linear Singular Integro-Differential Equations

2000 Mathematics Subject Classification: 45F15, 45G10, 46B38.

A Note on Div-Curl Lemma

2000 Mathematics Subject Classification: 42B30, 46E35, 35B65.

On the suboptimality of the p-version interior penalty discontinuous Galerkin method

We address the question of the rates of convergence of the p-version interior penalty discontinuous Galerkin method (p-IPDG) for second order elliptic problems with non-homogeneous Dirichlet boundary conditions. It is known that the p-IPDG method admits slightly suboptimal a-priori bounds with respect to the polynomial degree (in the Hilbertian Sobolev space setting). An example for which the suboptimal rate of convergence with respect to the polynomial degree is both proven theoretically and validated in practice through numerical experiments is presented. Moreover, the performance of p- IPDG on the related problem of p-approximation of corner singularities is assessed both theoretically and numerically, witnessing an almost doubling of the convergence rate of the p-IPDG method.

Sobolev Type Decomposition of Paley-Wiener-Schwartz Space with Application to Sampling Theory

2000 Mathematics Subject Classification: 94A12, 94A20, 30D20, 41A05.

Holomorphic Sobolev spaces, Hermite and special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform

The images of Hermite and Laguerre-Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterized. These are used to characterize the image of Schwartz class of rapidly decreasing functions f on R-n and C-n under these semigroups. The image of the space of tempered distributions is also considered and a Paley-Wiener theorem for the windowed (short-time) Fourier transform is proved.

Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.

Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces Hs(Ω) and tildeHs(Ω), for s in R and an open Ω in R^n. We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.