On The Regularity And Existences Of Free Discontinuity Problems

Free discontinuity problems are introduced and investigated from about 1990.These variational problems that deal with discontinuous sets are united and called freediscontinuity problems by E. De Giorgi. Some phenomena in image analysis, liquidcrystals, fracture medchanics can be modelled as free discontinuity problems.Theoretically,free discontinuity problems are new problems that concern with modern partial differ-ential equations,variational methods and Geometric Measure Theory. This thesis dealswith problems in three settings:one is to discuss partical regularity of free discontinuityproblems in the SBV framework;the second,we discuss free discontinuity problems inthe SBD framework;finally, we discuss free discontinuity problems in the SBH frame-work.Chapter 1 deals with the background and survey of the study on the free dis-continuity problems, and the main topic, some main results obtained and employedmethods in this thesis. Chapter 2 is some preliminaries, notation and some lemmas.In Chapter 3, we first discuss a regularity property for minimizers of the Mumford-Shah functional. By Using the excision method,we obtain a Lipschitz property forthe minimizers of the Mumford-Shah functional;the second,we discuss the Hausdorffdimension estimate on the singular part of discontinuous sets for minimizers of freediscontinuity problems, give some expressions on the singular partsΣ(u) of singularsets for minimizers of the Mumford-Shah functional. We don't assume the higherintegrability on the gradient,namely, |â–½u |∈L_loc^p(Ω),and prove that H- dim(Σ)≤N-2,and give a partial answer on the De Giorgi conjecture.In Chapter 4,we discuss the existence for the minmizers of free discontinuity prob-lems in the SBH space. First,we show a compactness theorem in SBH(Ω),then byusing this compactness theorem,we discuss variational problems for two different func-tionals.In Chapter 5, we discuss the lower semicontinuity and relaxation result of integralfunctionals in the BD space. First,we consider the lower semicontinuity property fora functional with linear growth in LD(Ω);the second, in the SBD space,we discussthe lower semicontinuity of an integral functional that the integrand is a Carathéodoryfunction and that satisfies a symmetric quasi-convex assumption, by the compactnesstheorem of the SBD space,blow-up method and Morrey theorem,prove that integral functional is lower semicontinuous with respect to L¹- convergence;then by using theone-dimentional sections method and the structure theorem of the BD functions, dis-cuss the lower semicontinuity of the integral functional in the whole BD space. Fi-nally, we discuss a relaxation result of an integral functional whose density functiondepends on x,u andεu.By using a condition on the integrand f(x, u(x),εu(x)) similarto a condition for the integrand f(x, u(x),â–½u(x))in the SBV case,symmetric quasi-convexity, Lusin's theorem,global method,obtain a integral expression of the integralfunctional.In Chapter 6, we show some special results on the chain rule of the BD func-tions.First,we prove a result that may apply to the homogenization theory;then givesome chain rule results with divergence form.In Chapter 7,we discuss the existences for the minimizers of free discontinuityproblems in the SBD space. First,we prove existence for the minimizers of generalfree discontinuity problems:namely, we consider variational problems for two differentfunctionals.By using the compactness theorem of SBD space, Poincaréinequality onthe BD functions,we prove the existence for variational problems. Here, we don'tsuppose the constraint that u is bounded.At last, we give the existence for the minmizerof a special free discontinuity problem (namely, piecewise rigid body case).It is provedby using the compactness for Borel partitions,the properties of the SBD functions andsets with finite perimeter, and the direct method in the Calculus of Variations.