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Register a new userLimited data problems for the generalized Radon transform in $mathbb{R}^n$
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We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in $mathbb{R}^n$. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in $mathbb{R}^n$ and define modifi
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In this article, we consider the limited data problem for spherical mean transform. We characterize the generation and strength of the artifacts in a reconstruction formula. In contrast to the thirds author work [Ngu15b], the observation surface considered in this article is not flat. Our results are comparable to those obtained in [Ngu15b] for flat observation surface. For the two dimensional problem, we show that the artifacts are $k$ orders smoother than the original singularities, where $k$ is vanishing order of the smoothing function. Moreover, if the original singularity is conormal, then the artifacts are $k+frac{1}{2}$ order smoother than the original singularity. We provide some numerical examples and discuss how the smoothing effects the artifacts visually. For three dimensional case, although the result is similar to that [Ngu15b], the proof is significantly different. We introduce a new idea of lifting the space.
We study the generalized eigenvalue problem in $mathbb{R}^N$ for a general convex nonlinear elliptic operator which is locally elliptic and positively $1$-homogeneous. Generalizing article of Berestycki and Rossi in [Comm. Pure Appl. Math. 68 (2015), no. 6, 1014-1065] we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.
In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the $L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data.
Consider a nonlinear Kirchhoff type equation as follows begin{equation*} left{ begin{array}{ll} -left( aint_{mathbb{R}^{N}}| abla u|^{2}dx+bright) Delta u+u=f(x)leftvert urightvert ^{p-2}u & text{ in }mathbb{R}^{N}, uin H^{1}(mathbb{R}^{N}), & end{array}% right. end{equation*}% where $Ngeq 1,a,b>0,2<p<min left{ 4,2^{ast }right}$($2^{ast }=infty $ for $N=1,2$ and $2^{ast }=2N/(N-2)$ for $Ngeq 3)$ and the function $fin C(mathbb{R}^{N})cap L^{infty }(mathbb{R}^{N})$. Distinguishing from the existing results in the literature, we are more interested in the geometric properties of the energy functional related to the above problem. Furthermore, the nonexistence, existence, unique and multiplicity of positive solutions are proved dependent on the parameter $a$ and the dimension $N.$ In particular, we conclude that a unique positive solution exists for $1leq Nleq4$ while at least two positive solutions are permitted for $Ngeq5$.
The transform considered in the paper averages a function supported in a ball in $RR^n$ over all spheres centered at the boundary of the ball. This Radon type transform arises in several contemporary applications, e.g. in thermoacoustic tomography and sonar and radar imaging. Range descriptions for such transforms are important in all these areas, for instance when dealing with incomplete data, error correction, and other issues. Four different types of complete range descriptions are provided, some of which also suggest inversion procedures. Necessity of three of these (appropriately formulated) conditions holds also in general domains, while the complete discussion of the case of general domains would require another publication.
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