Oscillatory integral operators with homogeneous phase functions

The purpose of this paper is to show well-posedness results for Dirichlet problems for the Stokes and Navier–Stokes systems with $$L^{\infty }$$ -variable coefficients in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. First, we refer to the Dirichlet problem for the nonsmooth coefficient Stokes system on Lipschitz domains in compact Riemannian manifolds and show its well-posedness by employing a variational approach that reduces the boundary value problem of Dirichlet type to a variational problem defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. We show also the equivalence between some transmission problems for the nonsmooth coefficient Stokes system in complementary Lipschitz domains on compact Riemannian manifolds and their mixed variational counterparts, and then their well-posedness in $$L^2$$ -based Sobolev spaces by using the remarkable Necas–Babuska–Brezzi technique (see Babuska in Numer Math 20:179–192, 1973; Brezzi in RAIRO Anal Numer R2:129–151, 1974; Necas in Rev Roum Math Pures Appl 9:47–69, 1964). As a consequence of these well-posedness results we define the layer potential operators for the nonsmooth coefficient Stokes system on Lipschitz surfaces in compact Riemannian manifolds, and provide their main mapping properties. These properties are used to construct explicitly the solution of the Dirichlet problem for the Stokes system. Further, we combine the well-posedness of the Dirichlet problem for the nonsmooth coefficient Stokes system with a fixed point theorem to show the existence of a weak solution to the Dirichlet problem for the nonsmooth variable coefficient Navier–Stokes system in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. The well developed potential theory for the smooth coefficient Stokes system on compact Riemannian manifolds (cf. Dindos and Mitrea in Arch Ration Mech Anal 174:1–47, 2004; Mitrea and Taylor in Math Ann 321:955–987, 2001) is also discussed in the context of the potential theory developed in this paper.

This workshop in real and harmonic analysis surveyed recent investigations in areas including geometric measure theory and restriction theory, multiple ergodic averages, local smoothing estimates, Schrödinger and Hörmander-type oscillatory integral operators, multilinear estimates and analysis on the Hamming cube. In particular, the workshop emphasized the recent solutions of two longstanding problems: the Kakeya phenomenon in dimension three, and almost everywhere convergence of long time averages associated to multiple commuting measure-preserving transformations. The methods presented during the workshop have yielded applications in ergodic theory, number theory, and computer science.

We prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, G\'erard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schr\"odinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d=2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.

We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 2017, 157–192] concerning the spectral cluster bounds for orthonormal systems at p = ∞ {p=\infty} , on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [ λ 2 , ( λ + 1 ) 2 ) {[\lambda^{2},(\lambda+1)^{2})} to [ λ 2 , ( λ + ϵ ⁢ ( λ ) ) 2 ) {[\lambda^{2},(\lambda+\epsilon(\lambda))^{2})} , where ϵ ⁢ ( λ ) {\epsilon(\lambda)} is a function of λ that goes to 0 as λ goes to ∞ {\infty} . In achieving this, we invoke the method developed in [J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p L^{p} -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 2015, 3, 1483–1527].

Boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach

Best constants in Sobolev trace inequalities

We study the wave equation on a bounded domain $M$ in $\\mathbb{R}^m$ or on a compact Riemannian manifold with boundary. Assume that we do not know the coefficients of the wave equation but are given only the hyperbolic Robin-to-Dirichlet map that corresponds to physical measurements on a part of the boundary. In this paper we show that at a fixed time $t_0$ a wave can be cut off outside a suitable set. That is, if $N\\subset M$ is a union of balls in the travel time metric having centers at the boundary, then we can modify a given Robin boundary value of a wave such that at time $t_0$ the modified wave is arbitrarily close to the original wave inside $N$ and arbitrarily small outside $N$. Also, at time $t_0$ the time derivative of the modified wave is arbitrarily small in all of $M$. We apply this result to construct a sequence of Robin boundary values so that at a time $t_0$ the corresponding waves converge to a delta distribution $\\delta_{\\widehat{x}}$ while the time derivative of the waves converge to zero. Such boundary values are generated by an iterative sequence of measurements. In each iteration step we apply time reversal and other simple operators to measured data and compute boundary values for the next iteration step. A key feature of this result is that it does not require knowledge of the coefficients in the wave equation, that is, of the material parameters inside the media. However, we assume that the point $\\widehat{x}$ where the wave focuses is known in travel time coordinates, and $\\widehat{x}$ satisfies a certain geometrical condition.

Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold

Uniform stabilization of the wave equation on compact manifolds and locally distributed damping – a sharp result

In this paper, we discuss two well-known open problems in the regularity theory for nonlinear, conformally invariant elliptic systems in dimensions n ≥3 , with a critical nonlinearity: H -systems (equations of hypersurfaces of prescribed mean curvature) and n-harmonic maps into compact Riemannian manifolds. For n = 2 several solutions of these problems are known but they all break down in higher dimensions (unless one considers special cases, e.g. hypersurfaces of constant mean curvature or manifolds with symmetries). We discuss some of the known proofs and hint at the main difficulties. We also state a few new results (such as positive answers for all solutions of class W^{n/2,2} for even n , instead of W^{1,n} and list some open questions of independent interest — including specific endpoint variants of the Coifman–Rochberg–Weiss theorem, addressing the boundedness of commutators of fractional and singular integrals with multiplication by bounded functions of class W^{1,n} — that would lead to solutions of these two problems.

On the wave equation on a compact Riemannian manifold without conjugate points

We consider a wave equation on a compact Riemannian manifold in the Hamiltonian form. The measure obtained as the product of Wiener measure on path space and the one of the (flat) white noise is infinitesimally invariant under the action of the corresponding flow.KeywordsBrownian MotionWave EquationRiemannian ManifoldInvariant MeasureHamiltonian FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

We determine lower and exact estimates of Kolmogorov, Gelfand and linear n n -widths of unit balls in Sobolev norms in L p L_{p} -spaces on compact Riemannian manifolds. As it was shown by us previously these lower estimates are exact asymptotically in the case of compact homogeneous manifolds. The proofs rely on two-sides estimates for the near-diagonal localization of kernels of functions of elliptic operators.

This paper will be concerned with some estimates on the lower bound of the dimension of the Euclidean space in which a compact Riemann manifold can be imbedded isometrically, if its curvatures satisfy certain conditions. Our basic geometrical idea is a very simple one. Denote by M a compact Riemann manifold of dimension n in an Euclidean space E of dimension n + N, the Riemann metric on M being induced by the imbedding. Let be a fixed point of E. The distance OP, P e M, is a continuous function in M and attains a maximum at a point P0 e M, since M is compact. It is intuitively clear that M will be concave toward 0 at Po, so that there will be some restrictions on the Riemann curvature of M at P0. If M is given abstractly, the imbedding will not be possible, if these restrictions are not fulfilled by the given Riemann metric at any of the points of M. Actually, however, if the difference N of the dimensions of M and E is greater than one, the implication of this geometrical fact on the Riemann curvature of M is not very simple. The question leads to algebraic problems which probably do not have simple answers. We propose to give in this paper a few conclusions which can be drawn. It should be mentioned that the above geometrical idea has been used by Tompkins1 to prove that a locally flat compact Riemann manifold of dimension n cannot be isometrically imbedded in an Euclidean space of dimension 2n 1. Among other things this theorem will be generalized and the invariants entering in the problem geometrically interpreted. As for the differentiability assumptions we suppose our manifold and the imbedding to be of class > 4.