Mean Value Formulae for Laplace and Heat Equation

Transcription

1 Mean Vale Formlae for Laplace and Heat Eqation Abhinav Parihar December 7, 03 Abstract Here I discss a method to constrct the mean vale theorem for the heat eqation. To constrct sch a formla ab initio, I first generalize the method sed in the mean-vale theorem for the Laplace eqation which is discssed. Most of the generalization comes from sing the co-area formla and its modified version. I fond ot that a mean vale formla on the srface is also possible for the heat eqation as in the Laplace Eqation. Both these forms and many other mean vale formlae are possible sing a general mean vale formla for each of the Laplace and Heat eqations. Co-area Formla I will be sing the Coarea Formla given in the book (C- Theorem 5 and a modified version of it as below. Following the book, the coarea formla is: n dx ( v r ds dr Let s define U(r and as: U(r : {x n v(x r} : {x n v(x r} I assme that v a fnction sch that the set U(r always decreases in size with r. emember that r is not the radis bt the vale of the fnction v on the level set. Using U(r, I propose

2 dx dx dx dx ( ( ( ds dr ds ds dr ds dr (.a (.b (.c (.d. A note I will enconter sch eqation bt with D x v instead of Dv. For that, dx T( T ( T( ( T ( T( dx U x(,t ( T ( U x(r,t U x(r,t dt dx dr dt dx dt dr Note that the integral T ( T ( can be replaced by T (r T (r as in or case, T (r r 0 and T(r r 0, de to the kinds of v we are considering. This gives s, dx T(r T (r ( dx dt dr U x(r,t ds dr (. This is strange as (. is same as (.c with only replaced by. I don t have a good nderstanding why this is so, bt I tried it for calclating the volme of a sphere and it works! I am not sre bt this has something to do with the integral U x(r,t D dx being 0 at the ends, i.e. at T xv (r and T (r. The differential form of eqation (. will be, and ( dx ( dx ds ds dr (.3 dr (.4

3 Laplace Eqation. General case of v I will try to generalize the the mean vale formla sing the co-area formla discssed above. The srface normal n Using Green s Formla, 0 U(r D n Dv. Now sing.d 0 r U(r D Dv 0 v + r U(r r v + r U(r r D Dv Dv Dv (. This gives a hint of sing v which satisfies the laplace eqation and makes the first integral 0. In case of the fndamental soltion φ or φ(x x 0 ; t t 0, φ is a δ-distribtion, which gives, Hence, r r or U(r x φ r (x constant : k(v (. k(v can be calclated by finding the limiting vale as r, k(v lim Eqation (. holds for any v that satisfies the heat eqation, or like φ acts as a δ distribtion. 3

4 . Special case of φ Let s proceed with v φ(x x 0. I ll se x for x x 0 and φ for φ(x x 0. Hence, k(φ lim Dφ (x 0 (x 0 Dφ Dφ nα(n x x 0 n (x 0 (.3 which is the mean vale theorem for the Laplace eqation. The other mean vale theorem can be derived from. as follows. I coldn t generalize this part more and had to se a φspecific method. Using the fact that Dφ is constant on a level set, mltiply both sides of.3 by Dφ. Dφ (x 0 Dφ Now, integrating from to, we get Dφ U(r (x 0 Dφ (α(n x x 0 n (x 0 (x 0 (.4 U(r Eqations.4 and.3 are the mean vale formlae pair for the Laplace Eqation. Both of these and other mean vale formlas are possible from a general mean vale formla on the srface of a level set v given by.. An example of another mean vale formla possible is as follows..3 Example of another mean vale formla I will start with eqation.3 and instead of mltiplying by Dφ, it is mltiplied by φ (which is same as r on the srface and then integrate. 4

5 (x 0 (x 0 r (x 0 Dφ (n (x 0 (n Dφ φ x x Here, is the vale of φ at the srface. replacing it in terms of r (radis of the ball, we get (x 0 3 Heat Eqation 3. General case of v (n r n B r(x 0 x I try to follow the same procedre as above for the Heat Eqation. n t t x t x U U n t D x n U The srface normals are characterized by n vt. Hence, U Dv, n x D xv D xv and n t Using (.3, v t D x D x v Using Green s Formla, v t r U(r r v t ( T r U(r r T ( x v r U(r U x(r,t U(r D x D x v x v + + r ( U x(r,t D x v D x v 5

6 This gives the following eqation which is similar to (.. 0 (v t + x v + (3. r U(r r Similar argments hold as before. I choose v that satisfies the heat eqation and makes the first integral 0. In case of the fndamental soltion φ or φ(x x 0 ; t t 0, φ t + x φ is a δ distribtion which gives, Hence, r U(r r or (φ t + x φ r (x 0; t constant : k v (3. k(v can be calclated by finding the limiting vale as r, k v lim Eqation (3. holds for any v that satisfies the heat eqation, or like φ acts as a δ distribtion. 3. Special case of φ Let s proceed with v φ(x x 0 ; t t 0. I ll se x for x x 0, t for t t 0 and φ for φ(x x 0. Now, D x φ x t φ which gives, k φ lim D x φ (x 0 ; t 0 (3.3 k φ (x 0 ; t 0 r D x φ φ t x x t r x t (3.4 6

7 This gives s the srface form of the mean vale theorem for the heat eqation like we had in the laplace case. Now again I se a φ specific method. Mltiplying by r on both sides, r Integrating from to, x t r k φ r x t r k φ r Bt from (.4, x t r k φ k φ r x t r (x 0 ; t 0 4 x t r D xφ x t x 4t r (r x t x t (3.5 Eqations 3.5 and 3.4 are the mean vale formlae pair for the Heat Eqation, similar to the mean vale formlae pair for the Laplace Eqation. Both of these and other many mean vale formlas are possible from a general mean vale formla on the srface of a level set v given by 3.. I coldn t prove the limit in 3.3. I got close to it by approximating the integral by taking it on the cylinder enclosing it, bt i think it doesn t converge to the given integral in the limit. My approximation gave me ( ( n n/ e k φ nα(n (x 0 ; t 0 πe 7