Numerical Solution of Partial Differential Equations, SS 2014 Exercise Sheet 4
Prof. Peter Bastian Deadline 21. May 2014
IWR, Universit¨at Heidelberg
EXERCISE1 OPERATORS ONHILBERT SPACE
LetHbe a Hilbert space andY a closed subspace ofH. Define the mapP :HÑY for eachvPHas
@yPY :pPpvq, yq pv, yq.
Let us prove that:
1. OperatorP is linear and continuous.
2. ForvPH it holds
}Ppvq v} min
yPY }yv} (applyLax-Milgram TheoremandCharacterization Theorem).
5 points
EXERCISE2 PROJECTIONS
LetY be a subspace of a normed vector spaceX. An operatorP :XÑXis said to be a projection onY if
P2 P and RangepPq Y.
Show the following:
1. P is a projection if and only ifP :X ÑY andP I onY.
2. IfP is a projection, thenXKerpPq `RangepPq, where`denotes a direct sum.
3. OperatorP defined in exercise 1 is a projection.
5 points EXERCISE3 UNBOUNDED LINEAR OPERATORS
The real trigonometrical polynomials have the form
tpxq a0
¸8 n1
ancosnx bnsinnx
wherean, bn PR. LetX be the space of all real trigonometrical polynomials onΩ pπ, πqwith a finite norm
}t}
»π
π
|tpxq|dx.
1. Prove, that the derivative BBx is a linear operator fromXtoX.
2. Show, that this operator is not bounded and therefore not continuous.
4 points
EXERCISE4 H1-NORM ANDH ¨OLDER-NORM
LetΩ ra, bs R. The H ¨older-Norm of a real functionf : ΩÑR,mPN, αP p0,1sis defined as }f}Cm,α : ¸
|s|¤m
}Bsf}8 ¸
|s|m
supt|fpxq fpyq|
|xy|α ;x, yPΩ, xyu
Moreover, let1 p¤ 8andα:11p.
Prove: Tthere exists a constantCPRandx0PΩ, that forf PC1pΩqit holds:
}f}C0,α ¤ |fpx0q| C}f1}Lp
Use H ¨older-inequality:
Letf PLppΩq, gPLqpΩqand 1p 1q 1, then it holdsf gPL1pΩqand }f g}L1 ¤ }f}Lp}g}Lq.
3 points
EXERCISE5 WEAK DIFFERENTIABILITY
Continuous, piecewise-smooth functions in 1D are weakly differentiable (see example 5.31 in the lecture notes). The continuity of the function is crucial.
Consider the function
fpxq
"
1 xP p1,0s 1 xP p0,1q
and show that the weak derivative off does not exist. 2 points
