Mixed volumes and anisotropic potentials

Hou, Shaoxiong (2017) Mixed volumes and anisotropic potentials. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

This PhD-thesis, comprizing five chapters deals with some topics combining potential theory and convex geometry through studying the mixed volumes and the anisotropic potentials, whence their applications in information theory and elliptic PDEs. Chapter 1 is the introduction and overview for the whole dissertation. Chapter 2 studies a mixed volume induced by the anisotropic Riesz-potential including its reverse Minkowski-type inequality. It turns out that such a mixed volume is equal to the anisotropic Cordes-Nirenberg-capacity. Two restrictions on the Lorentz spaces are characterized. Besides, we also prove a Minkowski-type inequality and a log- Minkowski type inequality as well as its reverse form. Chapter 3 investigates a mixed volume from the anisotropic potential with natural logarithm as a better complement to the end point case of the mixed volumes from the anisotropic Riesz-potential. An optimal polynomial log-inequality is not only discovered but also applicable to produce a polynomial dual for the conjectured fundamental log-Minkowski inequality in convex geometry analysis. Moreover, the star body with respect to the origin is characterized in terms of anisotropic potentials over the Euclidean spaces. Chapter 4 establishes an interpretation of a functional type of mixed volume, the f-divergence via the Orlicz addition of measures. Fundamental inequalities, such as a dual functional Orlicz-Brunn-Minkowski inequality, are established. We also investigate an optimization problem for the f-divergence. Chapter 5 characterizes the embeddings of associate Morrey spaces to Cordes- Nirenberg spaces and Cordes-Nirenberg spaces to Morrey spaces and hence produces the embedding chain. The trace of Riesz-Cordes-Nirenberg potentials, i.e., the boundedness of the Riesz operator mapping Cordes-Nirenberg spaces to the Radon measure based Campanato space, is also established with both sufficient and necessary conditions. Consequently, the regularity of an elliptic equation living on the Cordes- Nirenberg spaces can be characterized by means of the Campanato spaces.

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