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1.
In this article we construct parametrices and obtain dispersive estimates for a large class of principally normal pseudodifferential operators. The main motivation for this comes from unique continuation problems. Such estimates can be used to prove Lq Carleman inequalities, which in turn yield unique continuation results for various partial differential operators with rough potentials. © 2004 Wiley Periodicals, Inc.  相似文献   

2.
We prove a new unique continuation result for solutions to partial differential equations, “interpolating” between Holmgren's Theorem and Hörmander's Theorem. More precisely, under some partial analyticity assumptions on the coefficients we obtain an intermediate unique continuation result which is in general weaker than Holmgren's Theorem (which applies to problems with analytic coefficients) but stronger than Hörmander's Theorem (which applies to problems with C1 coefficients). Some applications to the wave and the Schroedinger equation are considered next. In particular we obtain a result conjectured by Hörmander, namely that for the wave equation with C1 but time independent coefficients one has unique continutaion across any noncharacteristic surface.  相似文献   

3.
We prove an analog of the Berry-Esseen estimate for the heat kernel of second order elliptic differential operators with quasiperiodic coefficients. As an application of this result, we prove the Lp boundedness of the associated Riesz transform operators.  相似文献   

4.
We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequality is derived, is proved in an elementary and direct way based on standard integration by parts.  相似文献   

5.
For canonical Dirac systems of differential equations with locally integrable coefficients, we prove the existence of transformation operators and estimate the kernels of these operators. We also give estimates for these kernels for the case in which the coefficients belong to the space L loc 2 . We establish a relationship between the kernel of the transformation operators and the potential matrix.  相似文献   

6.
In this paper we prove a unique continuation theorem for elliptic operators of the formP(D)+V(x), whereP(D) has orderm≥2 and simple complex characteristics, andV(x)∈L n/m (R n ). To prove our main theorem we use a restriction theorem for the Fourier transform to manifolds of codimension 2.  相似文献   

7.
We address the strong unique continuation problem for higher order elliptic partial differential equations in 2D with Gevrey coefficients. We provide a quantitative estimate of unique continuation (observability estimate) and prove that the solutions satisfy the strong unique continuation property for ranges of the Gevrey exponent strictly including non-analytic Gevrey classes. As an application, we obtain a new upper bound on the Hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients.  相似文献   

8.
In this paper, we prove Gårding’s weighted inequality for degenerate elliptic operators in an arbitrary (bounded or unbounded) domain of n-dimensional Euclidean space ? n and use this inequality to study the unique solvability of a specific variational problem. It is assumed that the lower coefficients of the operators under consideration belong to some weighted L p -spaces.  相似文献   

9.
In this paper we prove the unique continuation property of the solution for the elastic transversely isotropic dynamical systems with smooth coefficients satisfying some conditions and apply it to extending the Dirichlet to Neumann map. The proof is based on the localized Fourier-Gauss transformation and Carleman type estimate.  相似文献   

10.
In this paper we prove L p -boundedness properties of spectral multipliers associated with multidimensional Bessel operators. In order to do this we estimate the L p -norm of the imaginary powers of Bessel operators. We also prove that the Hankel multipliers of Laplace transform type on (0,∞) n are principal value integral operators of weak type (1,1).  相似文献   

11.
We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large-scale Ck, 1 estimate scale exponentially in k , just as for the classical estimate for harmonic functions, and the minimal scale grows at most linearly in k . As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations that exhibit growth like O(exp(δ| x| )) for small δ > 0 . The large-scale analyticity also implies quantitative unique continuation results, namely a three-ball theorem with an optimal error term as well as a proof of the nonexistence of L2 eigenfunctions at the bottom of the spectrum. © 2020 Wiley Periodicals LLC.  相似文献   

12.
In this paper we prove the following kind of unique continuation property. That is, the zero on each geodesic of the solution in a real analytic hypersurface for second order anisotropic hyperbolic systems with real analytic coefficients can be continued along this curve.

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13.
This paper is devoted to the approximate solution of one-dimensional pseudodifferential equations on a closed curve via spline collocation methods with variable collocation points and represents a continuation of [11]. We give necessary and sufficient conditions ensuring the L2-convergence for operators with smooth and piecewise continuous coefficients.  相似文献   

14.
We consider second-order parabolic equations with time independent coefficients. Under reasonable assumptions, it is known that the fundamental solution satisfies certain Gaussian bounds related to the associated geodesic distance. In this article we prove a sharp unique continuation property at the initial time which matches exactly the above-mentioned kernel bounds.

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15.
In this paper, we prove convergence rates for spherical spline Hermite interpolation on the sphere Sd−1 via an error estimate given in a technical report by Luo and Levesley. The functionals in the Hermite interpolation are either point evaluations of pseudodifferential operators or rotational differential operators, the desirable feature of these operators being that they map polynomials to polynomials. Convergence rates for certain derivatives are given in terms of maximum point separation.  相似文献   

16.
This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L p-norms of the operators of this semigroup. Some spectral properties of such operators in L pspaces are also studied.  相似文献   

17.
We prove the approximate controllability for the heat equation with potential with a cost of order ec/ε when the target is in with a precision in L2(Ω) norm. Also a quantification estimate of the unique continuation for initial data in L2(Ω) of the heat equation with potential is established.  相似文献   

18.
We prove Lp and smoothing estimates for the resolvent of magnetic Schrödinger operators. We allow electromagnetic potentials that are small perturbations of a smooth, but possibly unbounded background potential. As an application, we prove an estimate on the location of eigenvalues of magnetic Schrödinger and Pauli operators with complex electromagnetic potentials.  相似文献   

19.
The unique continuation property has been intensively studied for a long time due to the important role that plays in the applications. The validity of the unique continuation property for symmetric regularized long wave equation is showed in this paper. The result is established by using an appropriate Carleman‐type estimate for a partial differential operators closely related to our problem. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

20.
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