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Let BL:H01,2(Ω)×H01,2(Ω)→R be the bilinear form associated with the elliptic operator
Lu:=−i,j=1∑n∂i(aij∂ju)+cua) Show that there exists a constant μ>0 depending only on bounds for (aij) and Ω such that BL satisfies the conditions of the Lax-Milgram theorem whenever c≥−μ.
b) Deduce that for any f∈L2(Ω), there exists a unique weak solution u∈H01,2(Ω) to the equation Lu=f whenever c≥−μ.
(6 marks)
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Let η∈C01(Ω) and u∈W1,2(Ω).
a) Show that u⋅η2∈H01,2(Ω) and
∂i(u⋅η2)=∂iu⋅η2+2uη∂iη.b) Suppose X∈Ω and R>0 is such that B(X,R)⋐Ω. Define the mapping ηR:Ω→R by
ηR(x)=((1−R2∣x−X∣2)+)2,where for t∈R, t+=t if t≥0 and t+=0 otherwise. Show that ηR∈C01(Ω) and the following inequalities hold:
- ηR≥169 on B(X,2R);
- 0≤ηR≤1; and
- ∣DηR∣≤R4.
(6 marks)
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Suppose that u∈H01,2(Ω) is a weak solution to the boundary-value problem
Lu:=−i,j=1∑n∂i(aij∂ju)+i=1∑nbi∂iu+cuu∣∂Ω=f on Ω=0,(*)and let η∈C01(Ω).
a) Show that there exists a constant C>0 depending only on L (i.e. the various bounds on aij, b and c) such that
∫Ω∣Du∣2η2≤C⋅(∫Ωu2⋅(η2+∣Dη∣2)+∫Ωf2η2)b) Deduce using ηR as defined in the preceding exercise that
∫B(X,2R)∣Du∣2≤92162C((1+R216)∫B(X,R)u2+∫B(X,R)f2).(8 marks)
To be submitted by 10 a.m. on Friday December 20.