MATH P.D.E. II

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1 Homework # 1 (due Thursday, 4/16/215) 1. Let IR n be a bounded open set. Consider the hyperbolic equation utt u = c(x)u+g(u) x, u = x. (1) Construct a function φ(x,u) such that, for every smooth solution of (1), the total energy E = ( ut u 2 2 ) +φ(x,u) dx is constant in time. (Hint: write an identity that should be satisfied by the partial derivative φ u, in order that de(t) dt =. ) 2. On the space L 1 (IR), consider the semigroup S t ; t } generated by the operator Au = u xx. (i) Show that this is a contractive semigroup on the space L 1 (IR). (ii) Using Duhamel s formula, write out the mild solution to the Cauchy problem u t = u xx + u(,x) = e x. t 1+x 2, (2) (iii) Write an integral equation satisfied by the mild solution of the semilinear Cauchy problem u t = u xx +sinu, (3) u(,x) = f(x). Explain why this equation has a unique solution, defined for all t. 3. Decide which of the following maps is Lipschitz continuous from L 1 (IR) into itself. (Λ 1 u)(x) = sinu(x), (Λ 2 u)(x) = cosu(x), (Λ 3 u)(x) = u(y)dy 1+x 2, (Λ 4 u)(x) = u 2 (x). In the positive case, determine the Lipschitz constant. In other words, find C such that Λu Λv L 1 C u v L 1. 1

2 Homework # 9 (due Thursday, 4/9/215) 1. On the open interval =],3[, consider the boundary value problem uxx = 1 < x < 3, u() = u(3) =. (1) Consider the two linearly independent functions ϕ 1,ϕ 2 H 1 (],3[), defined by x if x [,1], if x [,1], ϕ 1 (x) = 2 x if x [1,2], ϕ 2 (x) = x 1 if x [1,2], if x [2,3], 3 x if x [2,3]. Explicitly compute the Galerkin approximation U(x) = c 1 ϕ 1 (x)+c 2 ϕ 2 (x) such that B[U,ϕ i ] = 3 U x ϕ i,x dx = Compare U with the exact solution of (1). 3 1 ϕ i dx = (1,ϕ i ) L 2 i = 1,2. 2. On the open interval =],3[, consider the parabolic problem u t = u xx < x < 3, u(t,) = u(t,3) =, u(,x) = 1. (2) Explicitly compute the Galerkin approximation U(t,x) = c 1 (t)ϕ 1 (x) + c 2 (t)ϕ 2 (x), choosing the functions c 1,c 2 so that ( U(, ) 1, ϕi ) L 2 = i = 1,2, t =, ( Ut, ϕ i ) L 2 +B[U, ϕ i ] = i = 1,2, t >. Compare U with the exact solution of (2). 3. Let = (x,y); x 2 1 +x2 2 < 1} be the open unit disc in IR2, and let u be a smooth solution to the equation u tt = 2u x1 x 1 +x 2 u x1 x 2 +3u x2 x u x 1 on [,T], (3) u = on [,T]. (i) Write the equation (3) in the form u tt + Lu =, showing that the operator L is symmetric (i.e., a 12 = a 21 ) and uniformly elliptic on the domain. (ii) Define a suitable energy e(t) = [kinetic energy] + [elastic potential energy], and check that it is constant in time. 2

3 Homework # 8 (due Thursday, 4/2/215) 1. Consider the space of square summable sequences of real numbers: } X =. x = (x 1,x 2,x 3,...); x =. ( x k 2) 1/2 < k (i) Show that the linear operator Ax = ( x 1, 2x 2, 3x 3,...) is not bounded on X. (ii) Explicitly write out the semigroup generated by A. In other words, for every x X, compute S t x. Prove that this semigroup is contractive. (iii) Prove that, for every x X and t > one has S t x Dom(A), even if x / Dom(A). Show that AS t sup λ> λe tλ <. 2. Let IR n be a bounded open set. A function u H 2 () is a weak solution of the biharmonic equation 2 u = f x, u = u ν = x, (1) if u vdx = f vdx for all v H 2 (). (2) Given f L 2 (), prove that the boundary value problem (1) has a unique weak solution. Hint: show that the bilinear form B[u,v] on H 2 () defined by the left hand side of (2) is strictly positive definite on H(). 2 Notice that, if u Cc (), integrating by parts one obtains u xi x j u xi x j dx = u xi x i u xi x j dx 1 2 (u 2 x i x i +u 2 x i x j )dx. 3

4 Homework # 7 (due Thursday, 3/26/215) 1. Let A be the generator of a contractive semigroup, on a Banach space X. Show that, for every λ >, the bounded linear operator A λ. = λa(λi A) 1 also generates a contractive semigroup. 2. On the space L 1 (IR), consider the linear operators S t defined by (S t f)(x) = e 2t f(x+t). (i) Prove that the family of linear operators S t ; t } is a strongly continuous, contractive semigroup on L 1 (IR). Find the generator A of this semigroup. What is Dom(A)? (ii) Show that the family of operators S t ; t } is NOT a strongly continuous semigroup on L (IR). 3. Let S t ; t } be a strongly continuous semigroup of linear operators on IR n. Prove that there exists an n n matrix A such that S t = e ta for every t. 4. On the space L 1 (IR), consider the operator Au = xu with domain } Dom(A) = u L 1 (IR); u is absolutely continuous, u x L 1 (IR). (i)] Describe the semigroup S t ; t } generated by A. (ii) For any u L 1 (IR) and h >, construct the backward Euler approximation E h u. 5 (extra credit) Fix a time T >. On the space X = L 1 ([,1]), construct a strongly continuous, contractive semigroup of linear operators S t ; t } such that S t = 1 for t < T but S t = for t T. 4

5 Homework # 6 (due Thursday, 3/5/215) 1. On the space L 2( [, [ ), consider the linear operator Lu(x). = u(x+1). (i) Find the adjoint operator L, such that (u,l v) L 2 = (Lu,v) L 2 for every u,v L 2 ([, [). (ii) Find the kernel and the range of L and L. Do the two kernels have the same dimension? 2. Assume f L 2( [,π] ). Show that the problem u +9u = f, u() = u(π) = has a solution if and only if Is the solution unique? π f(x) sin3xdx =. 3. Let IR n be a bounded open set, and consider the operator Lu. = ij (a ij (x)u xi ) xj + i b i u xi +c(x)u, where the coefficients b i are constant. Assume that a ij, c L (), with c(x), and that L is uniformly elliptic, so that ij aij ξ i ξ j θ i ξ2 i, for some θ >. (i) Write the corresponding bilinear form B[u, v]. Prove that it is strictly positive definite on H 1 (). (Hint: compute the divergence of the vector (b1,...,b n ) u2 2 ) (ii) Prove that, for every f L 2 (), the problem Lu = f x, u = x, has a unique weak solution. 5

6 Homework # 5 (due Thursday, 2/26/215) 1. Let = (x 1,x 2 ); x 2 1 +x 2 2 < 1 } be the unit disc in IR 2. Consider the operator ( ) Lu = u x1 x 1 +u x2 x 2 +x 1 u x1 x 2 +3x 2 2 u x1 (i) Check whether the operator L is elliptic on. (ii) Given f L 2 (), explain what it means for a function u to be a weak solution of the equation Lu = f x, u = x, 2. Let IR n be a bounded open set, and let a,c : [1,2] be smooth functions. Define H to be the Hilbert space H 1 () with the equivalent inner product. (u,v) H = a(x) u(x) v(x)dx+ c(x)u(x)v(x) dx. Let ι : H L 2 () be the identity map, and let ι : L 2 () H be the adjoint operator. Prove that, for every f L 2 (), the function ι f H provides the weak solution to an elliptic boundary value problem. Which equation does ι f solve? 3. Let IR n be a bounded open set, and let u be a weak solution to u+u = f x, u = x, (i) Prove that u L 2 f L 2. (ii) Prove that u L 2 f L 2. Hint: in both cases, use the definition of weak solution taking v = u as test function. 6

7 Homework # 4 (due Thursday, 2/19/215) 1. Let be an open ball in IR n. Show that there exists a constant C such that u 2 (x)dx C u(x) 2 dx+c u(x) dx Let IR n be a bounded open set and let c( ) be a continuous function on. (i) Show that there exists µ > such that, if c(x) > µ for all x, then the bilinear operator B[u,v] = u vdx+ c(x)u(x)v(x) dx satisfies the assumption of the Lax-Milgram theorem on the space H 1 () = W 1,2 (). (ii) By (i), for every f L 2 (), there exists u H) 1 () such that B[u,v] = (f,v) L 2 for every v H 1 (). What PDE + boundary conditions does u solve, in a weak sense? 3. Consider the open interval =] 1,1[, and the Hilbert-Sobolev space H 1 (). (i) Show that the Dirac measure δ : v v() (= a unit mass concentrated at the origin) is a continuous linear functional on H 1 (). Is the Dirac measure an element of the space H 1 ()? (ii) Explicitly determine a function u H 1 () such that (u,v) H 1 = v() for every v H 1 (). Hint: solve the boundary value problem u u = x ] 1,[ ],1[, u( 1) = u(1) =, u ( ) u (+) = 1. 7

8 Homework # 3 (due Thursday, 2/12/215) 1. Prove that, for any 1 p <, one has W 1,p (IR n ) = W 1,p (IR n ). Hint: Let u W 1,p (IR n ). Given ε >, find ũ C (IR n ) such that ũ u W 1,p < ε. Then take a C function ψ : IR : [,1], with ψ(r) = 1 if x, and ψ(r) = if r > 1. Show that the functions u k (x) = ũ(x) ψ( x k) are C with compact support. Prove that u k ũ W 1,p as k. 2. Consider the sequence of functions u k (x) = sinkx For a fixed ε >, define the regularization u ε k (x) = 1 2ε x+ε x ε u k (y)dy. (i) Prove that, for a fixed ε >, all functions u ε k, k 1 are uniformly Lipschitz continuous. (ii) By (i), on the compact interval [, 2π], by Ascoli s theorem (possibly taking a subsequence), one has the uniform convergence lim k u ε k = ūε. Describe the function ū ε. What is ū = lim ε ū ε? (iii) Estimate the limit 2π lim u ε k (x) u k (x) dx. k Does this approach zero as ε? Is it true that u k ū L1 ([,π])? 3. (i) Let IR 2 be an open ball. Assume that f W 1.p (). For which values of p,q can one conclude that the product f D x1 f lies in L q ()? (ii) Answer the same question in the case where is a ball in IR 3. 8

9 Homework # 2 (due Thursday, 2/5/215) 1. Let ξ : IR IR be a C 1, strictly increasing function (with ξ (x) > for every x), mapping =]a,b[ onto =]c,d[. Let f W 1,p () and define the composite function g(x) = f ( ξ(x) ). Prove that g W 1,p ( ), and compute the weak derivative of g. 2. Verify that if n > 1 the unbounded function ( u = loglog 1+ 1 ) x is in the space W 1,n (). Here IR n is the unit ball centered at the origin. This shows that functions u W 1,n (IR n ) need not be continuous, or bounded. 3. Let IR n be the unit ball centered at the origin. Consider the problem of minimizing the integral u p dx among all continuous functions that vanish on and take the value 1 at the origin. Assuming that u(x) = w( x ) is radially symmetric, this leads to the standard problem in the Calculus of Variations: minimize 1 r n 1 w (r) p dr subject to: w() = 1, w(1) =. (1) Study whether this problem has solutions, depending on n, p. Hint: consider the Euler-Lagrange equations L w = d L dr w, L(r,w,w ) = r n 1 w p. (2) Prove that (i) If p > n, the second order ODE (2) has a solution with boundary data (1) (ii) If p < n, no solution exists. In this case, for any ε > one can find a function w satisfying (1), such that 1 r n 1 w (r) p dr < ε. As a consequence, the norm u C () = sup x u(x) cannot be controlled by the norm u W 1,p (). 9

10 Homework # 1 (due Thursday, 1/29/215) 1. Which of the following spaces ( X, ) is a Banach space? Motivate your answers. (i) X= set of all continuous functions on [,1], u = 1 u(x) dx. (ii) X= set of all polynomials of degree 3, u = 1 u(x) dx. (iii) X= set of all convex functions on [,1], u = sup u(x). x [,1] (iv) X= set of all absolutely continuous functions on [, 1], u = 1 u() + u (x) dx. 2. On the open interval ] 1, 1[, consider the function u(x) = x 2 sin 1 x 4 x, u() =. - Does this function have a weak derivative on the open set = ], [? - Does this function have a weak derivative on the open set = ] 1,1[? (If yes, compute it. If no, explain why). 3. Let =]a,b[ be an open interval. Prove that (after a possible modification on a set of measure zero): (i) W 1, (]a,b[) is the space of all functions which are Lipschitz continuous on ]a,b[ (ii) W 1,p (]a,b[) is the space of absolutely continuous functions f : ]a,b[ IR with Df L p (]a,b[), (iii) For 1 p < one has W,p (]a,b[) = W,p (]a,b[) (iv) (extra credit) What is W, (]a,b[)? lim f(x) = lim f(x) = x a+ x b. = L p (]a,b[). 1

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