Math 7731 { Mathematical Problems in Industry Assignment 2 (2017) 1. In lectures we defined the Sobolev space H0 1(V ) with the norm jjfjjH1 0 = ZV f;if;i dV 1=2 : However, since H0 1(V ) ⊂ H1(V ), it will also have the norm associated with the Sobolev space H1(V ), jjfjjH1 = ZV (f 2 + f;if;i) dV 1=2 : We want to show that these are equivalent norms. That is, there are constants k; K > 0 such that for any f 2 H0 1(V ), kjjfjjH1 0 ≤ jjfjjH1 ≤ KjjfjjH1 0 : (1) (a) Show that if (1) is true, then there also exists k0; K0 > 0 such that k0jjfjjH1 ≤ jjfjjH1 0 ≤ K0jjfjjH1: (2) So equivalence is symmetric, as we would expect. (b) We will show this equivalence for the one-dimensional case where V is the interval (a; b), though it is true for n = 2; 3 dimensions also. The left hand inequality of (1) is obvious (with k = 1). Let f be any bounded and continuous function with bounded and continuous first derivatives, and which is zero on the boundary of V , that is at x = a and x = b. Then for any x 2 (a; b), f(x) = Zax f0(x) dx; where f0 = df=dx. Why? (c) Apply the Cauchy-Schwarz inequality to the right hand side of this to obtain jf(x)j ≤ Zax jf0j2 dx1=2 Zax 1 dx1=2 : (d) Show that jf(x)j2 ≤ jjf0jj2 L2jx − aj where jj:jjL2 is the L2 norm defined in lectures. (e) By integrating both sides with respect to x over the entire interval (a; b), conclude that jjfjj2 L2 ≤ Cjjf0jj2 L2 for some C > 0 which does not depend on f. What does C depend upon? (f) Use the result (e) to prove the right hand inequality of (1). (g) We have now proved (1) for the special case of f as described in (b). How do you think we would prove it for any f 2 H0 1(a; b)? (Describe in words.) 12. We want to prove the result that there exists M > 0 such that for any f 2 H1(V ), ZV f 2 dV ≤ M Z@V f 2 dS + ZV f;if;i dV  : (3) Intuitively, this result is saying that the size of f on V is \controlled" by the size of f on the boundary and the size of the first partial derivatives of f on V , where size is measured by the L2 norm. Again we will only consider the case where V is the interval (a; b), though the result is true for n = 2; 3 dimensions also. In the one-dimensional case the boundary integral over @V on the right hand side takes the simple form f(a)2 + f(b)2. (a) Let f be any bounded and continuous function with bounded and continuous first derivatives. Using a similar argument as in Question 1(b), show that or any x 2 V , jf(x)j ≤ f(a) + Zax f0 dx ≤ jf(a)j + Zax f0 dx ; and so jf(x)j2 ≤ 2 jf(a)j2 + Zax f0 dx 2! ≤ 2 jf(a)j2 + jjf0jj2 L2jx − aj
(c) Integrate both sides with respect to x over the entire interval (a; b) to conclude that jjfjjL2 ≤ C f(a)2 + jjf0jjL2
where C does not depend on f. What does C depend upon? (d) Deduce from (3) that there exists M0 > 0 such that for any f 2 H1(V ), jjfjj2 H1 ≤ M0 Z@V f 2 dS + ZV f;if;i dV  : (4) 3. We next want to prove the result that there exists M > 0 such that for any f 2 H1(V ), the boundary integral Z@V f 2 dS ≤ Mjjfjj2 H1: (5) Intuitively, this is saying that the size of f on the boundary is controlled by the H1 norm of f on V . Again we will only consider the case where V is the interval (a; b), though the result is true for n = 2; 3 dimensions also. In this one-dimensional case the boundary integral on the left hand side takes the simple form f(a)2 + f(b)2. (a) Let f be any bounded and continuous function with bounded and continuous first derivatives. Using a similar argument as in Question 2(a), show that or any x 2 V , jf(a)j ≤ jf(x)j + Zax f0 dx ; and jf(a)j2 ≤ 2 jf(x)j2 + Zax f0 dx 2! ≤ 2 jf(x)j2 + jjf0jj2 L2jx − aj
(c) Integrate both sides with respect to x over the entire interval (a; b) to conclude that jf(a)j2 ≤ Cjjfjj2 H1 where C does not depend on f. What does C depend upon? 2(d) Prove a similar result for jf(b)j2. 4. Consider the weak formulation of the steady state Dirichlet problem in the form: Find v 2 H0 1(V ) such that a(v; φ) = f(φ) 8φ 2 H0 1(V ); where the bilinear form a(:; :) and the linear functional f(:) are defined by a(v; φ) = ZV k(x)v;iφ;i dV and f(φ) = − ZV gφ dV − ZV k(x)U ~;iφ;i dV: Here k(x) is the non-constant conductivity, g(x) is a source term and U ~ is a H1(V ) extension of the boundary value U(x) to all of V . (a) Suppose that there are constants m; M > 0 such that for any x 2 V , m ≤ k(x) ≤ M. Show that the energy norm (a(:; :))1=2 derived from the bilinear form a(:; :) is equivalent to the H0 1 norm. (Hint: Use the property of integrals that for any functions f(x) ≥ h(x) then ZV f dV ≥ ZV h dV:) (b) Assume that the source term g 2 L2(V ). Show that the linear functional f(:) is bounded on H0 1(V ): (Hint: Use the result (1) from Question 1 above to bound the first integral in f(:).) We have therefore shown that the assumptions of the Lax-Milgram Theorem are true for this problem. 5. Consider the weak formulation of the cooling problem described in Question 3(b) of Assignment 1: Find u 2 H1(V ) such that a(u; φ) = f(φ) 8φ 2 H1(V ); where the bilinear form a(:; :) and linear functional f(:) are defined by a(u; φ) = ZV k(x)u;iφ;i dV + Z@V huφ dS and f(φ) = − ZV gφ dV + Z@V hu0φ dS: Here k(x) is the non-constant conductivity, g(x) is a source term, h(x) is the heat transfer coefficient on the boundary and u0(x) is the temperature of the external environment. (a) With the same conditions on k as in Question 4(a) above, and suppose that there are constants hmin; hmax > 0 such that hmax ≥ h(x) ≥ hmin, show that the energy norm (a(:; :))1=2 derived from the bilinear form a(:; :) is equivalent to the H1 norm. (Hint: Use the general results (4) above (5) to handle the surface integral term in the bilinear form.) (b) Assume that the source term g 2 L2(V ). Show that the linear functional f(:) is bounded on H1(V ): We have therefore shown that the assumptions of the Lax-Milgram Theorem are also true for this problem. 3