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4.0

Jun 30, 2018
06/18

Jun 30, 2018
by
Matthew D. Blair; Christopher D. Sogge

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We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to the geodesic flow.

Topics: Mathematics, Analysis of PDEs, Classical Analysis and ODEs, Differential Geometry

Source: http://arxiv.org/abs/1409.1286

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6.0

Jun 30, 2018
06/18

Jun 30, 2018
by
Shanlin Huang; Christopher D. Sogge

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We prove families of uniform $(L^r,L^s)$ resolvent estimates for simply connected manifolds of constant curvature (negative or positive) that imply the earlier ones for Euclidean space of Kenig, Ruiz and the second author \cite{KRS}. In the case of the sphere we take advantage of the fact that the half-wave group of the natural shifted Laplacian is periodic. In the case of hyperbolic space, the key ingredient is a natural variant of the Stein-Tomas restriction theorem.

Topics: Mathematics, Analysis of PDEs, Classical Analysis and ODEs, Differential Geometry

Source: http://arxiv.org/abs/1406.1940

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7.0

Jun 30, 2018
06/18

Jun 30, 2018
by
Christopher D. Sogge; Steve Zelditch

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In this note we show how improved $L^p$-estimates for certain types of quasi-modes are naturally equaivalent to improved operator norms of spectral projection operators associated to shrinking spectral intervals of the appropriate scale. Using this, one can see that recent estimates that were stated for eigenfunctions also hold for the appropriate types of quasi-modes.

Topics: Mathematics, Analysis of PDEs, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1401.0345

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6.0

Jun 29, 2018
06/18

Jun 29, 2018
by
Christopher D. Sogge

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If $(M,g)$ is a compact Riemannian manifold of dimension $n\ge 2$ we give necessary and sufficient conditions for improved $L^p(M)$-norms of eigenfunctions for all $2

Topics: Differential Geometry, Spectral Theory, Classical Analysis and ODEs, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1610.06639

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3.0

Jun 29, 2018
06/18

Jun 29, 2018
by
Christopher D. Sogge; Yakun Xi; Cheng Zhang

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We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_\lambda$ over geodesics go to zero at the rate of $O((\log\lambda)^{-1/2})$ if $\lambda$ are their frequencies. As discussed in \cite{CSPer}, no such result is possible in the constant curvature case if the curvature is $\ge0$. Notwithstanding, we also show that these bounds for period integrals are valid...

Topics: Differential Geometry, Spectral Theory, Classical Analysis and ODEs, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1604.03189

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3.0

Jun 29, 2018
06/18

Jun 29, 2018
by
Changxing Miao; Christopher D. Sogge; Yakun Xi; Jianwei Yang

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We obtain an improvement of the bilinear estimates of Burq, G\'erard and Tzvetkov in the spirit of the refined Kakeya-Nikodym estimates of Blair and the second author. We do this by using microlocal techniques and a bilinear version of H\"ormander's oscillatory integral theorem.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1602.06936

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Jun 28, 2018
06/18

Jun 28, 2018
by
Christopher D. Sogge

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We prove new improved endpoint, $L^{p_c}$, $p_c=\tfrac{2(n+1)}{n-1}$, estimates (the "kink point") for eigenfunctions on manifolds of nonpositive curvature. We do this by using energy and dispersive estimates for the wave equation as well as new improved $L^p$, $2

Topics: Differential Geometry, Analysis of PDEs, Classical Analysis and ODEs, Mathematics

Source: http://arxiv.org/abs/1512.03725

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Jun 28, 2018
06/18

Jun 28, 2018
by
Christopher D. Sogge

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We survey recent results related to the concentration of eigenfunctions. We also prove some new results concerning ball-concentration, as well as showing that eigenfunctions saturating lower bounds for $L^1$-norms must also, in a measure theoretical sense, have extreme concentration near a geodesic.

Topics: Differential Geometry, Analysis of PDEs, Mathematics, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1510.07723

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5.0

Jun 28, 2018
06/18

Jun 28, 2018
by
Matthew D. Blair; Christopher D. Sogge

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We use Toponogov's triangle comparison theorem from Riemannian geometry along with quantitative scale oriented variants of classical propagation of singularities arguments to obtain logarithmic improvements of the Kakeya-Nikodym norms introduced in \cite{SKN} for manifolds of nonpositive sectional curvature. Using these and results from our paper \cite{BS15} we are able to obtain log-improvements of $L^p(M)$ estimates for such manifolds when $2

Topics: Differential Geometry, Analysis of PDEs, Mathematics, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1510.07726

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Jun 27, 2018
06/18

Jun 27, 2018
by
Christopher D. Sogge

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We use a straightforward variation on a recent argument of Hezari and Rivi\`ere~\cite{HR} to obtain localized $L^p$-estimates for all exponents larger than or equal to the critical exponent $p_c=\tfrac{2(n+1)}{n-1}$. We are able to this directly by just using the $L^{p}$-bounds for spectral projection operators from our much earlier work \cite{Seig}. The localized bounds we obtain here imply, for instance, that, for a density one sequence of eigenvalues on a manifold whose geodesic flow is...

Topics: Differential Geometry, Analysis of PDEs, Mathematics, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1503.07238

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Jun 27, 2018
06/18

Jun 27, 2018
by
Christopher D. Sogge; Xing Wang; Jiuyi Zhu

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We study the interior nodal sets, $Z_\lambda$ of Steklov eigenfunctions in an $n$-dimensional relatively compact manifolds $M$ with boundary and show that one has the lower bounds $|Z_\lambda|\ge c\lambda^{\frac{2-n}2}$ for the size of its $(n-1)$-dimensional Hausdorff measure. The proof is based on a Dong-type identity and estimates for the gradient of Steklov eigenfunctions, similar to those in \cite{SZ1} and \cite{SZ2}, respectively.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1503.01091

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Sep 24, 2013
09/13

Sep 24, 2013
by
Christopher D. Sogge; Steve Zelditch

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On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not...

Source: http://arxiv.org/abs/math/0103172v2

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Sep 23, 2013
09/13

Sep 23, 2013
by
Yi Du; Jason Metcalfe; Christopher D. Sogge; Yi Zhou

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We show the obstacle version of the Strauss conjecture holds when the spatial dimension is equal to 4. We also show that an almost global existence theorem of H\"ormander for (4+1)-dimensional Minkowski space holds in the obstacle setting. We use weighed space-time variants of the energy inequality and a variant of the classical Hardy inequality.

Source: http://arxiv.org/abs/0710.2026v2

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Sep 23, 2013
09/13

Sep 23, 2013
by
Hart F. Smith; Christopher D. Sogge; Chengbo Wang

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We establish the Strauss conjecture for nontrapping obstacles when the spatial dimension $n$ is two. As pointed out in \cite{HMSSZ} this case is more subtle than $n=3$ or 4 due to the fact that the arguments of the first two authors \cite{SmSo00}, Burq \cite{B} and Metcalfe \cite{M} showing that local Strichartz estimates for obstactles imply global ones require that the Sobolev index, $\gamma$, equal 1/2 when $n=2$. We overcome this difficulty by interpolating between energy estimates ($\gamma...

Source: http://arxiv.org/abs/1012.3183v2

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Sep 23, 2013
09/13

Sep 23, 2013
by
Jason Metcalfe; Christopher D. Sogge; Ann Stewart

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In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neumann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results...

Source: http://arxiv.org/abs/math/0411513v2

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Sep 22, 2013
09/13

Sep 22, 2013
by
Jason Metcalfe; Makoto Nakamura; Christopher D. Sogge

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We prove global existence of solutions to multiple speed, Dirichlet-wave equations with quadratic nonlinearities satisfying the null condition in the exterior of compact obstacles. This extends the result of our previous paper by allowing general higher order terms. In the currect setting, these terms are much more difficult to handle than for the free wave equation, and we do so using an analog of a pointwise estimate due to Kubota and Yokoyama.

Source: http://arxiv.org/abs/math/0409363v1

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Sep 22, 2013
09/13

Sep 22, 2013
by
Jason Metcalfe; Makoto Nakamura; Christopher D. Sogge

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In this paper we prove global existence for certain multispeed Dirichlet-wave equations with quadratic nonlinearities outside of obstacles. We assume the natural null condition for systems of quasilinear wave equations with multiple speeds. The null condition only puts restrictions on the self-interactions of each wave family. We use the method of commuting vector fields and weighted space-time $L^2$ estimates.

Source: http://arxiv.org/abs/math/0409362v1

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Sep 22, 2013
09/13

Sep 22, 2013
by
Matthew D. Blair; Hart F. Smith; Christopher D. Sogge

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We establish Strichartz estimates for the Schr\"odinger equation on Riemannian manifolds $(\Omega,\g)$ with boundary, for both the compact case and the case that $\Omega$ is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents $(p,q)$ for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key $L^4_tL^\infty_x$ estimate, which we use to...

Source: http://arxiv.org/abs/1004.3976v2

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Sep 22, 2013
09/13

Sep 22, 2013
by
Kunio Hidano; Jason Metcalfe; Hart F. Smith; Christopher D. Sogge; Yi Zhou

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The purpose of this paper is to show how local energy decay estimates for certain linear wave equations involving compact perturbations of the standard Laplacian lead to optimal global existence theorems for the corresponding small amplitude nonlinear wave equations with power nonlinearities. To achieve this goal, at least for spatial dimensions $n=3$ and 4, we shall show how the aforementioned linear decay estimates can be combined with "abstract Strichartz" estimates for the free...

Source: http://arxiv.org/abs/0805.1673v2

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Sep 22, 2013
09/13

Sep 22, 2013
by
Matthew D. Blair; Hart F. Smith; Christopher D. Sogge

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We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcricital case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in...

Source: http://arxiv.org/abs/0805.4733v3

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Sep 22, 2013
09/13

Sep 22, 2013
by
Xuehua Chen; Christopher D. Sogge

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We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $\R$, we are able to improve the corresponding $L^2$-restriction bounds of Burq, G\'erard and Tzvetkov and Hu. Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved $L^4$-estimates for restrictions to geodesics, which, by...

Source: http://arxiv.org/abs/1210.7520v4

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Sep 22, 2013
09/13

Sep 22, 2013
by
Hans Lindblad; Makoto Nakamura; Christopher D. Sogge

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A combination of some weighted energy estimates is applied for the Cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions. Alternative proofs for global solutions are shown including the exterior domain problems.

Source: http://arxiv.org/abs/1210.8237v1

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Sep 22, 2013
09/13

Sep 22, 2013
by
Xuehua Chen; Christopher D. Sogge

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If $(M,g)$ is a compact Riemannian surface then the integrals of $L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length are uniformly bounded. Also, if $(M,g)$ has negative curvature and $\gamma(t)$ is a geodesic parameterized by arc length, the measures $e_j(\gamma(t))\, dt$ on $\R$ tend to zero in the sense of distributions as the eigenvalue $\la_j\to \infty$, and so integrals of eigenfunctions over periodic geodesics tend to zero as $\la_j\to \infty$. The assumption...

Source: http://arxiv.org/abs/1302.5597v3

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Sep 22, 2013
09/13

Sep 22, 2013
by
Christopher D. Sogge; John A. Toth; Steve Zelditch

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On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${\phi_{\lambda}}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not...

Source: http://arxiv.org/abs/0908.0688v1

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Sep 21, 2013
09/13

Sep 21, 2013
by
Christopher D. Sogge; Chengbo Wang

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We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\R^d, \mathfrak{g})$, $d \geq 3$, when metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $ x ^{- \rho}$ with $\rho>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $\rho> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $\rho>0$ for $d= 3$.

Source: http://arxiv.org/abs/0901.0022v4

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Sep 21, 2013
09/13

Sep 21, 2013
by
Jean Bourgain; Peng Shao; Christopher D. Sogge; Xiaohua Yao

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We address an interesting question raised by Dos Santos Ferreira, Kenig and Salo about regions ${\mathcal R}_g\subset {\mathbb C}$ for which there can be uniform $L^{\frac{2n}{n+2}}\to L^{\frac{2n}{n-2}}$ resolvent estimates for $\Delta_g+\zeta$, $\zeta \in {\mathcal R}_g$, where $\Delta_g$ is the Laplace-Beltrami operator with metric $g$ on a given compact boundaryless Riemannian manifold of dimension $n\ge3$. This is related to earlier work of Kenig, Ruiz and the third author for the...

Source: http://arxiv.org/abs/1204.3927v2

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Sep 21, 2013
09/13

Sep 21, 2013
by
Christopher D. Sogge

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In this article we shall go over recent work in proving dispersive and Strichartz estimates for the Dirichlet-wave equation. We shall discuss applications to existence questions outside of obstacles and discuss open problems.

Source: http://arxiv.org/abs/math/0311150v1

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Sep 21, 2013
09/13

Sep 21, 2013
by
Christopher D. Sogge; Steve Zelditch

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Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the general results of Burq, G\'erard and Tzvetkov \cite{burq}. By earlier results of Bourgain \cite{bourgainef} and the first author \cite{Sokakeya}, they are equivalent to improvements of the general $L^p$-estimates in \cite{soggeest} for $n=2$ and $2

Source: http://arxiv.org/abs/1108.2726v2

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Sep 21, 2013
09/13

Sep 21, 2013
by
Matthew D. Blair; Christopher D. Sogge

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We extend a result of the second author \cite[Theorem 1.1]{soggekaknik} to dimensions $d \geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2

Source: http://arxiv.org/abs/1301.7468v1

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Sep 21, 2013
09/13

Sep 21, 2013
by
Christopher D. Sogge; Steve Zelditch

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Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with Laplacian, $\Delta_g$. If $e_\lambda$ are the associated eigenfunctions of $\sqrt{-\Delta_g}$ so that $-\Delta_g e_\lambda = \lambda^2 e_\lambda$, then it has been known for some time \cite{soggeest} that $\|e_\lambda\|_{L^4(M)}\lesssim \lambda^{1/8}$, assuming that $e_\lambda$ is normalized to have $L^2$-norm one. This result is sharp in the sense that it cannot be improved on the standard sphere because of highest...

Source: http://arxiv.org/abs/1011.0215v1

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Sep 20, 2013
09/13

Sep 20, 2013
by
Jason Metcalfe; Christopher D. Sogge

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We prove global existence for quasilinear wave equations outside of a wide class of obstacles. The obstacles may contain trapped hyperbolic rays as long as there is local exponential energy decay for the associated linear wave equation. Thus, we can handle all non-trapping obstacles. We are also able to handle non-diagonal systems satisfying the appropriate null condition.

Source: http://arxiv.org/abs/math/0301290v3

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Sep 20, 2013
09/13

Sep 20, 2013
by
Jason Metcalfe; Christopher D. Sogge

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We prove global existence of solutions to quasilinear wave equations with quadratic nonlinearities exterior to nontrapping obstacles in spatial dimensions four and higher. This generalizes a result of Shibata and Tsutsumi in spatial dimensions greater than or equal to six. The technique of proof would allow for more complicated geometries provided that an appropriate local energy decay exists for the associated linear wave equation.

Source: http://arxiv.org/abs/math/0404420v2

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Sep 19, 2013
09/13

Sep 19, 2013
by
Jason Metcalfe; Christopher D. Sogge

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We study long time existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. In particular, we obtain exterior domain analogs of the four dimensional results of H\"ormander where the nonlinearity is permitted to depend on the solution not just its first and second derivatives. Previous proofs in exterior domains omitted this dependence as it did not mesh well with the energy methods in use.

Source: http://arxiv.org/abs/0910.0433v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
Daniel Oberlin; Hart Smith; Christopher D. Sogge

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We establish $L^p$ Sobolev mapping properties for averages over certain curves in $\R^3$, which improve upon the estimates obtained by $L^2-L^\infty$ interpolation.

Source: http://arxiv.org/abs/math/9912205v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
Hart Smith; Christopher D. Sogge

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The authors prove global Strichartz estimates for compact perturbations of the wave operator in odd dimensions when a non-trapping assumption is satisfied.

Source: http://arxiv.org/abs/math/9912204v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
Marcus Keel; Hart Smith; Christopher D. Sogge

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We prove global existence for semilinear hyperbolic equations that satisfy the null condition of Christodoulou and Klainerman in the exterior of convex domains. We use a combination of the conformal method of Christodoulou and the direct method of Klainerman.

Source: http://arxiv.org/abs/math/9912201v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
V. Georgiev; Hans Lindblad; Christopher D. Sogge

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We prove certain weighted Strichartz estimates and use these to prove a sharp theorem for global existence of small amplitude solutions of $\square u= |u|^p$, thus verifying the so-called "Strauss conjecture".

Source: http://arxiv.org/abs/math/9912206v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
William Minicozzi II; Christopher D. Sogge

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We expand on counterxamples of Bourgain showing how Nikodym maximal estimates and oscillatory integral estimates can break down in the non-Euclidean case. Our examples show the role that the non-existence of totally geodesic submanifolds can play in these problems.

Source: http://arxiv.org/abs/math/9912202v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
Hart Smith; Christopher D. Sogge

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The authors show that bilinear estimates for null forms hold for Dirichlet-wave equations outside of convex obstacle. This generalizes results for the Euclidean case of Klainerman and Machedon, and of Sogge for the variable coefficient boundaryless case. The estimates are used to prove a local existence theorem for semilinear wave equations satisfying the null condition.

Source: http://arxiv.org/abs/math/9912207v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
Christopher D. Sogge

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We obtain estimates for maximal functions that arise when one studies Nikodym-type sets. We also formulate a curvature condition that allows favorable estimates for these maximal functions.

Source: http://arxiv.org/abs/math/9912203v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
Christopher D. Sogge; Steve Zelditch

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Let $\ncal_{\phi_{\lambda}}$ be the nodal hypersurface of a $\Delta$-eigenfunction $\phi_{\lambda}$ of eigenvalue $\lambda^2$ on a smooth Riemannian manifold. We prove the following lower bound for its surface measure: $\hcal^{n-1}(\ncal_{\phi_{\lambda}}) \geq C \lambda^{\frac74-\frac{3n}4} $. The best prior lower bound appears to be $e^{- C \lambda}$.

Source: http://arxiv.org/abs/1009.3573v3

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Sep 18, 2013
09/13

Sep 18, 2013
by
Christopher D. Sogge

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We shall be concerned with the Cauchy problem for quasilinear systems in three space dimensions of the form \label{i.1} \partial^2_tu^I-c^2_I\Delta u^I = C^{IJK}_{abc}\partial_c u^J\partial_a\partial_b u^K + B^{IJK}_{ab}\partial_a u^J\partial_b u^K, \quad I=1,..., D. Here we are using the convention of summing repeated indices, and $\partial u$ denotes the space-time gradient, $\partial u=(\partial_0 u, \partial_1 u, \partial_2 u, \partial_3u)$, with $\partial_0=\partial_t$, and...

Source: http://arxiv.org/abs/math/0202031v3

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Sep 18, 2013
09/13

Sep 18, 2013
by
Christopher D. Sogge

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The purpose of this paper is to give a simple proof of sharp $L^\infty$ estimates for the eigenfunctions of the Dirichlet Laplacian on smooth compact Riemannian manifolds $(M,g)$ of dimension $n\ge 2$ with boundary $\partial M$ and then to use these estimates to prove new estimates for Bochner Riesz means in this setting. Our eigenfunction estimates involve estimating the $L^2\to L^\infty$ mapping properties of the operators $\chi_\lambda$ which project onto unit bands of the spectrum of the...

Source: http://arxiv.org/abs/math/0202032v4

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Sep 18, 2013
09/13

Sep 18, 2013
by
Matthew D. Blair; Hart F. Smith; Christopher D. Sogge

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We prove sharp bilinear estimates for Dirichlet or Neumann eigenfunctions in domains in the plane. These are the natural analog of earlier estimates for the boundaryless case of Burq, G\'erard, and Tzvetkov.

Source: http://arxiv.org/abs/math/0611443v1

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Sep 18, 2013
09/13

Sep 18, 2013
by
Jason Metcalfe; Christopher D. Sogge

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We provide a proof of global existence of solutions to quasilinear wave equations satisfying the null condition in certain exterior domains. In particular, our proof does not require estimation of the fundamental solution for the free wave equation. We instead rely upon a class of Keel-Smith-Sogge estimates for the perturbed wave equation. Using this, a notable simplification is made as compared to previous works concerning wave equations in exterior domains: one no longer needs to distinguish...

Source: http://arxiv.org/abs/math/0611489v1

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Sep 18, 2013
09/13

Sep 18, 2013
by
Hart F. Smith; Christopher D. Sogge

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We use microlocal and paradifferential techniques to obtain $L^8$ norm bounds for spectral clusters associated to elliptic second order operators on two-dimensional manifolds with boundary. The result leads to optimal $L^q$ bounds, in the range $2\le q\le\infty$, for $L^2$-normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp $L^q$ estimates in higher dimensions for a range of exponents...

Source: http://arxiv.org/abs/math/0605682v1

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140

Sep 17, 2013
09/13

Sep 17, 2013
by
Markus Keel; Hart Smith; Christopher D. Sogge

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We prove almost global existence for semilinear wave equations outside of nontrapping obstacles. We use the vector field method, but only use the generators of translations and Euclidean rotations. Our method exploits 1/r decay of wave equations, as opposed to the much harder to prove 1/t decay.

Source: http://arxiv.org/abs/math/0108016v1

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51

Jul 20, 2013
07/13

Jul 20, 2013
by
Matthew D. Blair; Hart F. Smith; Christopher D. Sogge

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We prove local Strichartz estimates on compact manifolds with boundary. Our results also apply more generally to compact manifolds with Lipschitz metrics.

Source: http://arxiv.org/abs/math/0609455v1

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96

Jul 20, 2013
07/13

Jul 20, 2013
by
Hans Lindblad; Jason Metcalfe; Christopher D. Sogge; Mihai Tohaneanu; Chengbo Wang

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We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the Strauss conjecture on the Schwarzschild and Kerr, with small angular momentum, black hole backgrounds. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and a localized energy estimate on the black hole background, which handles the behavior in the remaining...

Source: http://arxiv.org/abs/1304.4145v2

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52

Jul 20, 2013
07/13

Jul 20, 2013
by
Hamid Hezari; Christopher D. Sogge

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We prove a natural inequality which implies the known lower bounds for the $(n-1)$-dimensional Hausdorff measure of nodal sets for smooth compact manifolds.

Source: http://arxiv.org/abs/1107.3440v3