1. The idea by J.Kigami Before the idea of J.Kigami, a lot of - PDF document

New Development of Fractal PDE Weiyi Su Nanjing University, PRC suqiu@nju.edu.cn Abstract Fractal PDE, as a quite new topic in the area of “Fractal Analysis”, is developing very fast since the end of last century. It is motivated from physics, astronomy, geology, …, for instance, scientists hope mathematicians show the speed of an ant when it moves along a Weierstrass curve, or speed of Brownian motion. Moreover, what are solutions of a drum with a fractal boundary, and so on. Thus, the fractal PDE problems are proposed. In this paper, we will show 4 important ideas to study the fractal PDE with corresponding main methods and main results. Finally, Some open problems are also indicated. 1. The idea is proposed by J. Kigami, developed by R.S. Strichartz, K.-S.Lau, J.X.Hu, et al. 2. The idea is proposed by H. Triebel, developed by M. Zähle, D.C. Yang, et al. 3. The idea is proposed by B.B.Mnadelbrot, developed by F. Tatom, M. Zähle, K.Yao, et al. 4. The diea is proposed by the School of Harmonic-Fractal Analysis in Nanjing University, developed by members in the School. 1. The idea by J.Kigami Before the idea of J.Kigami, a lot of physicists paid their attention to analytic structures of a fractal set, and studied the Brownian motions on fractals, as well as proved the existence of Brownian motions on a Sierpinski gasket, such as, M.Barlow, E.Perkins, T.Lindstrom and R.Bass. Kigami introduces the Dirichelt forms, Laplacians, heat kernels on self-similar sets, …, then show a series theorems and properties, and establish the theory of partial differential equations on fractals. He has published the paper “Harmonic calculus on the Sierpindki spaces” in 1989, then about more than 20 papers are published continuously. The book “ Analysis on Fractals ” [1] has been published in 2001. He has devoted his best to do the research of analysis on fractals, and obtained lots of foundation works in the area. The main contributions of Kigami: (1) Construction of Dirichelt forms and Laplacians on p.c.f. ( post-critical fractal ) self-similar sets For a self-similar set, a topological structure, a harmonic structure and Green’ operator are defined, so that for fractal PDE, the preparations have been established. For example, Derichelt form Let V be a finite set,       be equipped with inner  l V f : V        on   product        u v , , u v , l V ; A symmetric bilinear form l V is u v , u p v p  p V 1

said to be a Dirichlet form on V , if it satisfies          u l V u u , 0 i) ;         u u , 0 u l V V is constant on ii) ;     1, if u p 1                     u l V    u u , u u , 0 with iii)  . u p u p , if 0 u p 1      0, if u p 1 Then, he generalizes the definitions of Dirichlet forms and Laplacian on limits of networks; moreover, on the p.c.f. self-similar structures. (2) Discussion of the spectrum theory Including the eigen-vaules, eigen-functions, the existence and estimates of spectra for the Neumann boundary problems and the Dirichlet boundary problems; as well as the relationships between various fractal dimensions. (3) Construction of heat kernels  Heat kernel Let b represent boundary condition, b N as the Neumann boundary    , , p t x y condition, and b D as the Dirichlet boundary condition. The b -heat kernel b for   t x y       , , 0, K K with K a p.c.f. self-similar structure is defined by          t x y   b  n t b b p , , e x y , b n n  n 1 where         b   , and b   exists for b as the sequence of eigen-values with 0 b 1 n n n 1     K  .  2 b  is a complete orthonormal system in L , n n 1 Then, 1  the parabolic maximum principle is proved; 2  an asymptotic behavior of the heat kernels is reveled; 3  other interesting properties are shown. After the foundation work of Kigami, mathematicians in the world pay their attentions to this topic. Mainly, ① R.S.Strichartz a series research work on fractal analysis is completed, specially, the fractal differential equations since 1998. The summary book “Differential Equations on Fractals” published in 2006 [2] , exhibits abundant new results, such as, 1  electric Network interpretations; 2  normal derivatives; 3  Gauss-Green formula and Green’s function; 4  spectral asymptotic growth; 5  integrals involving eigenfunctions; 6  conformal energy and energy measures on Sierpinski gaskets; 7  spectral decimation on some hierarchical gaskets; 8  resolvent kernel for p.c.f. self-similar fractals; and so on. R.S.Strichartz has excellent jobs for differential equations on fractals, and he has started to study the fractal differential equations on non-self-similar fractals. ② K.-S.Lau a series nice work for the Laplacian on p.c.f. self-similar sets, his main jobs concentrate on the harmonic structures of fractals. Recently, boundary theory on some trees and Martin boundary, as well as exit space on the Sierpinski gaskets are studied. With his colleagues and students, about hundreds papers have been published. 2

③ J.X.Hu he has a series work about fractal analysis, and written a foundational book “Introduction to Fractal Analysis” [3] . In the book, the developments of results of theory and applications of heat kernels are included. For instance, the estimates of various upper and lower bounds of heat kernels, there are accurate methods and nice results in the book. ④ U.Mosco establishes Laplacian on Heisenberg group, and so on. By virtue of developing heat kernel theory on fractals to develop fractal PDE is a good idea . 2. The idea by H.Triebel H.Triebel has an idea to establish a differential equation theory on fractals: by virtue of establishing function spaces on fractals to define fractal pseudo-differential operators over those function spaces, thus a theory of partial differential equations on fractals can be constructed. His book “Fractals and Spectra” [4] is published in 1997. The main contributions of Triebel : (1) Define the function spaces on fractals By virtue of concept d -set to define the function spaces on fractals :    , if for some d with 0 Definition 2.1 ( d -set) Let n   and   n d n ,  satisfies : (i) there exists a Borel measure  on  such that supp    ; (ii) there n     c  c  for all          0 0 d d c r B , r c r exist two constants , , such that 1 2 1 2      is said to be a d -set . n and 0 r 1 . Then     s , n Definition 2.2 ( B-type space on d -set ) The B-type space on a d -set is B p q , defined as                      s , n s n n B f B : f , 0 for with 0 .  p q , p q , (2) Define the distribution dimension of a fractal Definition 2.3 ( Distribution dimension ) For a non empty Borel d -set  with the   Lebesgue measure 0 is defined as              n d , n C dim sup d : is non trivial for a compact . D    . Thus one has an explanation : And he has a very significant result : dim dim D H the distribution dimension is an analytical expression of the Hausdorff dimension. By this result, Triebel proves more properties of the Hausdorff dimensions of fractals. (3) Define fractal pseudo-differential operators on some function spaces 1  Operators on fractals Let    be a compact d -set with 0   n A is d n ; s 3