Research On Minkowski-type Problems Of Geometric Optics

The L_p Brunn-Minkowski theory and the L_p Minkowski problem have rapidly de-veloped into a significant branch of theory and a topical issue in convex geometric anal-ysis.In this thesis,we mainly apply the basic properties of the L_p Brunn-Minkowski theory and the methods for solving the L_p Minkowski problem to study several classes of geometric measures in geometric optics-the reflector measure、the L_q -reflector mea-sure（all real numbers q）、theΣ-reflector measure with non-Euclidean norm sphere、the L_q Σ-reflector measure（all real numbers q）.And we formulate some Minkowski-type problems in geometric optics,which concern the characterization of these geometric measures.Firstly,we study the existence of solutions to the L_q reflector problem for all real numbers q.The reflector measure was proposed by Caffarelli-Oliker in 1994,after which Caffarelli-Oliker have shown that a weak solution to the reflector problem in the Far-field can be reached as a limit of a sequence of solutions to discrete problems.In this thesis,we introduce the L_q -reflector measure as a class of geometric optics measures which is a L_q extension of the reflector measure.A problem for characterizing the L_q -reflector measure is called the L_q reflector problem,given a non-negative Borel measureμit is natural to look for the necessary and sufficient conditions forμto be the L_q -reflector measure of a reflector.When q=0,the L_q -reflector measure is the reflector measure,and the L_q reflector problem is the Far-Field reflector problem in geometric optics.In order to discuss the existence part of the reflector problem（q=0）and its extension-the L_q reflector problem（all real numbers q）,we not only collect notions of the reflector and properties of the reflector measure,but also introduce fundamental concepts and properties such as the L_q -reflector combination,the L_q -reflector measure,the reflector shape,and the convex hull in Chapter 2.Following that,we establish a number of useful variational formulas for the L_q -reflector measure by the Aleksandrov’s variational method in Chapter 3.In addition,some dual variational formulas are proved due to the duality of the reflector shape and the convex hull.Based on these preparations,the specific approach is to transform the L_q reflector problem into an entropy maximization problem,and then solve the existence by finding the solution to the Euler-Lagrange equations for the entropy maximization problem.Secondly,we consider the L_q Σ-reflector problem（all real numbers q）,which is a generalization of the L_q reflector problem from the Euclidean unit sphere S^n-1to the general unit norm sphereΣ（non-Euclidean unit sphere）.Caffarelli-Huang have set down theΣ-reflector measure.This presentation is given in the more general setting of reflector measures,which are introduced in Chapter 2 along with the fact thatΣmust be a strictly convex unit norm sphere.For this reason research of reflectors in L_q space becomes more difficult.Combined with the knowledge of convex analysis and Fermat’s principle in optics,this rule leads to several geometric notions and properties,such as theΣ-reflector,theΣ-reflector measure,the L_q Σ-reflector combination and the L_q Σ-reflector measure.We describe the so-called L_q Σ-reflector problem as follows:Given a non-negative Borel measure on a strictly convex unit norm sphereΣ,under what sufficient and necessary conditions can a reflector be found such that the given measure is the same as the L_q Σ-reflector measure.In the case of q=0,the L_q Σ-reflector measure is theΣ-reflector measure,and the L_q Σ-reflector problem is the Far-FieldΣ-reflector problem in geometric optics.In Chapter 4,we still present a series of variational formulas of the L_q Σ-reflector measure and following a similar route we deduce the necessary and sufficient conditions of solutions to the L_q Σ-reflector problem（all real numbers q）.It should be emphasized that,due to technical limitations,the L_q Σ-reflector combination and the associated variational formulas in this thesis rely on theΣ-convex hull.Finally,we discuss a class of parabolic flows toward reflectors.Apart from the Aleksandrov’s variational method,the flow equation is also an essential approach to solving Minkowski-type problems.This is motivated by the fact that the solvability of the Minkowski-type problems generally amounts to dealing with one fully nonlin-ear partial differential equation,i.e.,a Monge-Amp`ere type equation on the unit sphere S<sup>n-1</sup>.If we can find a suitable parabolic flow equation,then we prove that it eventually converges to a smooth solution of the Monge-Amp`ere type equation by studying the con-vergence of the parabolic flow.In conclusion,we obtain the existence of a solution to the Minkowski-type problems.In 2003,Schn¨urer solved the reflector problem by designing the parabolic flow equation towards reflectors.In Chapter 5,we derive the existence of solutions to the L_q reflector problem according to the convergence of the parabolic flow towards reflectors in L_q space,together with the approximation theory.