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In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at hand, we can determine the diffusion, cavity and coefficients simultaneously by knowing the corresponding localized Dirichlet-Neumann operators.
The purpose of the paper is to initiate the development of the theory of Newton Okounkov bodies of curve classes. Our denition is based on making a fundamental property of NewtonOkounkov bodies hold also in the curve case: the volume of the NewtonOkounkov body of a curve is a volume-type function of the original curve. This construction allows us to conjecture a new relation between NewtonOkounkov bodies, we prove it in certain cases.
Although everyone is familiar with using algorithms on a daily basis, formulating, understanding and analysing them rigorously has been (and will remain) a challenging task for decades. Therefore, one way of making steps towards their understanding is the formulation of models that are portraying reality, but also remain easy to analyse. In this thesis we take a step towards this way by analyzing one particular problem, the so-called group testing problem. R. Dorfman introduced the problem in 1943. We assume a large population and in this population we find a infected group of individuals. Instead of testing everybody individually, we can test group (for instance by mixing blood samples). In this thesis we look for the minimum number of tests needed such that we can say something meaningful about the infection status. Furthermore we assume various versions of this problem to analyze at what point and why this problem is hard, easy or impossible to solve.
We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian (−Δ)s in bounded open Lipschitz sets in the small order limit s→0+. While it is easy to see that all eigenvalues converge to 1 as s→0+, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with Fourier symbol 2log|ξ|. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that L2-normalized Dirichlet eigenfunctions of (−Δ)s corresponding to the k-th eigenvalue are uniformly bounded and converge to the set of L2-normalized eigenfunctions of the logarithmic Laplacian. In order to derive these spectral asymptotics, we establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.