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[solution] » HOMEWORK PROBLEMS SET 2 1. Assime 1 = p = 8 and U is bounded. (a) Prove that if u ? W1,p(U), then

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HOMEWORK PROBLEMS SET 2 1. Assime 1 = p = 8 and U is bounded. (a) Prove that if u ? W1,p(U), then
More:Document Preview: HOMEW ORK PROBLEMS SET 21. Assime 1 =p = 8 and U is bo unded. (a) Pro v e that if u? W 1,p (U ), then u+ , u- ? W 1,p (U ), a nd (0.1) ?u+ = ( ? u a.e. on {u > 0 } 0a.e. on {u = 0 } (hin t: u+ = limo?0 Fo (u ) for (0.2) ? Fo (z ) = ( (z2 + o2 ) 1/2 -o on {z > 0 } 0 on {z = 0 } )(b) Prov e that if u? W 1,p (U ), then ?u= 0 a.e. on {u= 0 }. 2. W e sa y that the uniformly elliptic operatorLu = - nXi,j =1 aij ux i x j + nXi=1 bi ux i + cusatis?es the w eak maximum principle if for all u? C2 (U )nC ( ¯ U), (0.3) ( Lu = 0 in U u=0 on ? U implies u= 0 in U . Supp ose that there is a v? C2 (U )nC ( ¯ U), suc h that Lv = 0 in U and v > 0 on ¯ U. Sho w that L satis?es the w eak maxim um principle. (hin t: ?nd an elliptic op erator M with no zeroth order term suc h that w := u/v satis?es M w = 0 in the region { u > 0} . T o do this, ?rst compute (v2 wx i )x j . ) 3. (Courant minim ax principle) Let Lu =- P n i,j =1 (a ij u x i ) x j , where a ij = aj i , L is elliptic. Assume the op erato r L, with zero b oundary conditions, has eigen v alues 0 < ?1 < ?2 = ... . Sho w that ?k = max S?Sk-1 minu? S? ,k ukL2 =1 B[u, u ], k = 1, 2,... Here Sk-1 denotes the collection o f (k- 1)- dimensional subspaces of H 1 0(U). 1 2 HOMEWORK PR OBLEMS SET 24. Assume(0.4) ( u k ? u in L2 (0 , T ; H 1 0(U)) u' k? v in L 2 (0, T ; H -1(U)) Pro v e that v =u' . (Hint: let f? C 1 c(0, T ), w? H 1 0(U). Then ZT 0hu' k, fw i dt =-Z T 0huk , f' w idt )5. Suppose H is a Hilb ert space and uk ? u in L2 (0 , T ;H ). Assume further that there is C > 0, indep enden t of k , suc h that ess sup0=t =T kuk ( t ) k = C, k = 1, 2,... Pro v eess sup0=t =T ku ( t ) k = C. 6. Assume that ? =? ( t )= 0, ?? C1 and there is a p o lynomial P , with nonnegativ e co e?cien ts, suc h that?' ( t)= P( ?( t)) sho w that there are T > 0, M > 0, dep ending only on ? (0), suc h that ? ( t )= M for all t? [0, T ]. 7. Assume that f :Rn × R? R satis?es (a) there is a constan t M > 0, suc h that |f ( x, t )-f ( y , t ) | = M |x -y | , for an y x, y ?Rn ,t? [0, T ]; (b) f (0, t )? L1 ([0, T ]). Sho w that the initial v alue problem(0.5) ( x ' =f( x, t) x(0) = x0 has a unique solution x = x ( t ) for t? [0, T ], moreov er, x = x ( t ) is absolutely con tin uous on [0 , T ].

 







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