Regularity for the near field parallel refractor and reflector

C.GUTIERREZ Y TOURNIER F.

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS

SPRINGER

Lugar: Berlin; Año: 2015

0944-2669

Suppose we have a domain Rn and a domain contained in an n dimensionalsurface inRn+1; is referred as the target domain or screen to be illuminated.Let n1 and n2 be the indexes of refraction of two homogeneous and isotropic mediaI and II, respectively, and suppose that from the region surrounded by mediumI, radiation emanates in the vertical direction en+1 with intensity f (x) for x 2 , and is surrounded by media II. That is, all emanating rays from are collimated.A parallel refractor is an optical surface R interface between media I and II, suchMarch 18, 2014.The first author was partially supported by NSF grant DMS?1201401.12 C. E. GUTI´ERREZ AND F. TOURNIERthat all rays refracted by R into medium II are received at the surface , and theprescribed radiation intensity received at each point p 2 is (p). AssumingRno loss of energy in this process, we have the conservation of energy equation f (x) dx =R (p) dp. Under general conditions on and , and when is aRadon measure in D, the existence of parallel refractors is proved in [?].The purpose of this paper is to study regularity of parallel refractors and reflectors.Indeed, under suitable conditions on the target and the measure , we proveinterior C1; estimates. We prove that if u is a parallel refractor in , the target satisfies the local condition (3.6) from (x?; u(x?)), and the measure satisfies alocal condition at that point, condition (5.1), then u 2 C1; in a neighborhood ofthe point x?.