The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

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1 Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply the equation by any v C ( ) and integrate, we have As a consequence, we have = ( u) v = u v. ( ) ( u + v) = + v. ( 3) In other words, u is the minimizer of the function D( u). ( 4) Conversely, if u is a minimizer, then for any v C, and t >, we have which implies ( u + t v) by taking t and consequently when u C. t v t u v ( 5) ( u) v = u v = ( 6) From the above discussion we conclude the following Dirichlet principle. Dirichlet principle. u solves if and only if u minimizes u = ( 7) u = in, u = g on. ( 8) D( u). ( 9) A moment s inspection reveals that the principle cannot be automatically true without specifying the class of functions u should belong to: D( u) is well-defined when u C 1 but u needs to be in C to satisfy the Laplace s equation. And furthermore, it is not clear yet why D( u) should have a minimizer. We first establish Theorem 1. D( u) has a minimizer u satisfying <, that is u L. Proof. Let u n be a minimizing sequence, that is lim D( u n) = inf D( u). ( 1 ) n

2 Then one calculates Since we see that D( u n u m ) = u n u m = u n u n u m + u m = u n + u m ( un + u = D( u n ) + D( u m ) 4 D m u n + u m ). ( 1 1 ) ( ) un + u 4 D m 4 inf D( u) = lim [ D( u n ) + D( u m ) ], ( 1 ) D( u n u m ) n, m ( 1 3) or equivalently { u n } is a Cauchy sequence in the space L of all square integrable functions. Thus there is a limit function w = lim u n which is square integrable. It turns out that 1. w = u for some function u in the sense of distributions.. D( u) lim D( u n ) which means u is a minimizer. From the above theorem we see that only the existence of u ( as a square integrable function) is guaranteed. Therefore the Dirichlet principle only makes sense when we re-define the Laplace equation to its weak formulation: u v = v C ( ), u = g on. ( 1 4). The Sobolev space W 1, ( ). Definition. The Sobolev space W 1, ( ) is defined as the space of those u L ( ) whose distributional derivatives x i u also belong to L ( ). Proposition 3. i. W 1, ( ) becomes a Hilbert space after we define the inner product ( u, v) W 1, u v + n ( ) x i u x i v. ( 1 5) i= 1 The induced norm is ii. C ( ) is dense in W 1, ( ). u W 1, ( ( u, u) 1 /. ( 1 6) ) W 1, ( ) Proof. See J. Jost Partial Differential Equations, 7.. Example u( x) = x W 1, (, 1 ) ; { 1 x < 1. u( x) = < x < W 1, (, 1 ). Definition 5. The closure of C ( ) in W 1, ( ) is denoted W 1, ( ).

3 Example u( x) = 1 x W 1, (, 1 ).. u( x) 1 W 1, (, 1 ). 1 Remark 7. Intuitively, W 1, ( ) are those functions in W 1, ( ) which are on the boundary. The following properties are important for studying PDEs. We will omit their proofs, details can be found in J. Jost Partial Differential Equations, 7.. Lemma 8. For u W 1, ( ), f C 1 ( ), suppose Then f u W 1, ( ) and D( f u) = f ( u) D u. sup f ( y) <. ( 1 9) y R The above is still true when f Lip( ). In particular, if u W 1, ( ), so is u, and D u = ( sign u) D u. ( ) Now recall the minimization problem in the Dirichlet principle: min, u = g on. ( 1 ) We would like to rigorously specify over which set the minimization is taking place. This set is exactly the space W 1,. Thus we would like to minimize over all functions in W 1, with boundary value g. Recall that if u n is a minimizing sequence, then u n is a Cauchy sequence in L. If furthermore u n is a Cauchy sequence in L too, we know that the sequence is a Cauchy sequence in W 1,. The following Poincaré inequality guarantees that u n is a Cauchy sequence in L. Lemma 9. There is a constant C, depending on the bounded set only, such that for all u W 1, ( ), we have Proof. We prove by contradiction. Assume that there are u k such that u L ( ) C u L ( ). ( ) u k L ( ) > k u k L ( ). ( 3) Rescaling, we can set u k L = 1. Thus u k in L. By the compactness theorem of Rellich 3, there is a subsequence u k j which convergens in L. Thus { } u k j converges in 1, W to some limit function u satisfying 1, u L = 1, u =, u W ( 4) where the contradiction is obvious. 1. To see this, we assume the contrary, that is there are u n C (, 1 ) such that u n u 1 in W 1,. This means 1 ( u n ) 1 ( ), un. ( 1 7) B ut the latter implies x 1 u n( x) u n u ( ) 1 ( ) n un ( 1 8) for any x (, 1 ) and furthermore the convergence is uniform in x. Contradiction.. The idea is to approximate f b y C 1 functions, and use Lebesgue s dominated convergence Theorem. 3. Theorem. Let R n be open and bounded, and let { u n } be a bounded sequence in W 1, ( ), then there is a sub sequence which converges strongly in L ( ).

4 Remark 1. The following argument yields an explicit C when is contained in a box of side R ( denote the box by C R ) : Extend u by outside we obtain a W 1, function, still denoted u, defined on the box. Without loss of generality we assume the box is x i R. Integrating from x n = we have x n u( x 1,, x n ) = x n u( x 1,, x n, t) dt. ( 5) Now we have u = ( x n ) u( x 1,, x n ) dt 1 ( ) R u( x 1,, x n ) dt 1 n n R u( x 1,, x n ) u( x 1,, t) dt 1 n ( ) 1 / R ( u( x 1,, x n ) u( x 1,, t) dt 1 R 1 / ( ) 1 / ( u R 1 / n ) 1 / ) 1 /. ( 6) and thus obtaining u L ( ) R u L ( ). ( 7) A more refined ( and more general, as it can be applied to unbounded regions) estimate has the constant ( ) 1 / n C = ( 8) α( n) where is the volume of and α( n) is the volume of the n-dimensional unit ball. See the proof of Theorem 7.. in J. Jost Partial Differential Equations. 3. Weak formulation. From the above discussion we see that the minimizer of the Dirichlet functional is in W 1, ( ). Now we are ready to give the definition of a solution u W 1, ( ) : Definition 1 1. u W 1, ( ) is a weak solution of the Laplace equation if u =, in, u = g, on ( 9) u v = v W 1, ( ) ; u g W 1, ( ). ( 3) Remark This definition requires that the boundary value g can be extended to a function in W 1, ( ). This can indeed be done. See e. g. R. A. Adams Sobolev Spaces.. Since C 1, is dense in W ( ) ( by definition! ), we can also use v C in the definition. The current definition is however more convenient. For example, when the solution u exists, the nonsmooth function max {, u k } W ( ) ( if k is bigger than the boundary values) can be used as 1, test functions This choice of test functions is used in the so-called De G iorgi method, which obtains L bound from energy bound.

5 4. Poisson equation. The above discussions can be applied to the Poisson equation u = f, in, u = g, on ( 31 ) with little modification. In this case, the definition for weak solutions is Definition 1 3. u W 1, ( ) is a weak solution of the Poisson equation if u v + u = f, in, u = g, on ( 3) f v = v W 1, ( ) ; u g W 1, ( ). ( 33) The weak formulation is advantageous in getting quick estimates. For example, when g =, we have u W 1, C f L ( 34) for some constant C. 1, To see this, note that when g =, u W can be used as a test function, which gives = f u f L u L. ( 35) Applying Poincaré inequality gives the desired estimate. 5. Introduction to the direct method. The direct method shows the existence/ uniqueness of the solution of PDEs by studying its variational formulation. We sketch this approach by studying the Poisson equation with zero boundary condition: u = f, u W 1, ( ). ( 36) We know that any weak solution to this problem is a minimizer of the functional D( u) = + f u. ( 37) We would like to show that the minimizer exists and is unique. An outline of the argument is the following. Assume f L. 1. Writing D( u) ε u f ( 38) 4 ε and recalling the Poincaré s inequality, we see that D( u) has finite infimum.. Let u n be such that D( u n ) inf u 1, W D( u). We claim that there is a subsequence converging to 1, some limit u W. 5 To see this, note that a uniform bound on D( u n ) implies a uniform bound on u n, since D( u) u L u L f L u L by Hölder s inequality and Poincaré s inequality. C u L = ( u L C ) u L. ( 39) 3. Uniform boundedness of u n L implies that u n is uniformly bounded in W 1, and thus has a weakly 6 converging subsequence, still denoted by u n. We denote the limit by u. 5. This actually cannot be guaranteed now. 6. The weak convergence is in W 1,. Recall that a sequence {u n } in a Hilbert space H is weakly convergent with weak limit u H if ( u n, v) ( u, v) for any v H. Furthermore, using Rellich s theorem, we see that when u n converges to u weakly in W 1,, we can find a subsequence, still denoted u n, converging to u strongly in L, at the same time u n converges to u weakly in L.

6 4. The convexity of the functional D( u) then guarantees that which means u is a minimizer. D( u ) liminf D( u n ) = inf D( u) ( 4) n 1, u W 5. The convexity of D( u) also guarantees the uniqueness of the minimizer. The last few steps in general involve much technicality. 7 Interested readers can refer to the book by B. Dacorogna for details. Remark 1 4. This approach easily generalizes to certain nonlinear equations of the form: ( ) F ( x, u, u) + F ( x, u, u) =. ( 43) p u where F( x, u, p) is smooth and convex in p, with certain growth condition at infinity. The key observation is that this equation is the condition for minimizers of the functional D( u) = F( x, u, u). ( 44) The books by B. Dacorogna and L. C. Evans are good texts for direct methods in variational problems. Further readings. J. Jost, Partial Differential Equations, Chap. 7. B. Dacorogna, Direct Methods in the C alculus of Variations. L. C. Evans, Weak C onvergence Methods for Nonlinear Partial Differential Equations. Exercises. Exercise 1. Consider the functional D p( u) where 1 < p <. What equation should its minimizer satisfy? 8 p, u = g on. ( 45) 7. In our case much technicality is not involved. The only thing required is some familiarity with weak convergence in Hilbert spaces. For example, note that when u n u weakly, ( u n, u n) ( u, u ) = lim [ ( u n u, u n u ) ]. ( 41 ) One can even show similarly that the convergence is in fact strong. Furthermore if u and v are both minimizers of the norm, we have = ( u, u ) ( v, v ) = ( u v, u v ) ( 4) since ( v, u v ) must vanish due to the fact that v is a minimizer ( local minimizer is enough). 8. The resulting operator is called the p-laplacian.