Tutorial sheet 4

Throughout this sheet, assume that Ω⊂Rn\Omega\subset\mathbb{R}^{n} is open and bounded and aij,bi,c∈L∞(Ω)a_{ij},b^{i},c\in L^{\infty}(\Omega) and f∈L2(Ω)f\in L^{2}(\Omega) are as introduced in Week 8.

  1. Let BL:H01,2(Ω)×H01,2(Ω)→RB_{L}:H_{0}^{1,2}(\Omega)\times H_{0}^{1,2}(\Omega)\rightarrow\mathbb{R} be the bilinear form associated with the elliptic operator

    Lu:=−∑i,j=1n∂i(aij∂ju)+cuLu:=-\sum_{i,j=1}^{n}\partial_{i}(a_{ij}\partial_{j}u) + cu

    a) Show that there exists a constant μ>0\mu>0 depending only on bounds for (aij)(a_{ij}) and Ω\Omega such that BLB_{L} satisfies the conditions of the Lax-Milgram theorem whenever c≥−μc\geq -\mu.

    b) Deduce that for any f∈L2(Ω)f\in L^{2}(\Omega), there exists a unique weak solution u∈H01,2(Ω)u\in H_{0}^{1,2}(\Omega) to the equation Lu=fLu=f whenever c≥−μc\geq -\mu.

    (6 marks)

  2. Let η∈C01(Ω)\eta\in C_{0}^{1}(\Omega) and u∈W1,2(Ω)u\in W^{1,2}(\Omega).

    a) Show that u⋅η2∈H01,2(Ω)u\cdot\eta^{2} \in H_{0}^{1,2}(\Omega) and

    ∂i(u⋅η2)=∂iu⋅η2+2uη∂iη.\partial_{i}(u\cdot\eta^{2})=\partial_{i}u\cdot\eta^{2} + 2u\eta\partial_{i}\eta.

    b) Suppose X∈ΩX\in\Omega and R>0R>0 is such that B(X,R)⋐ΩB(X,R)\Subset\Omega. Define the mapping ηR:Ω→R\eta_{R}:\Omega\rightarrow\mathbb{R} by

    ηR(x)=((1−∣x−X∣2R2)+)2,\eta_{R}(x)=\left(\left(1-\frac{|x-X|^{2}}{R^{2}} \right)_{+} \right)^{2},

    where for t∈Rt\in\mathbb{R}, t+=tt_{+}=t if t≥0t\geq 0 and t+=0t_{+}=0 otherwise. Show that ηR∈C01(Ω)\eta_{R}\in C_{0}^{1}(\Omega) and the following inequalities hold:

    (6 marks)

  3. Suppose that u∈H01,2(Ω)u\in H_{0}^{1,2}(\Omega) is a weak solution to the boundary-value problem

    Lu:=−∑i,j=1n∂i(aij∂ju)+∑i=1nbi∂iu+cu=f on Ωu∣∂Ω=0,(*)\begin{aligned}Lu:=-\sum_{i,j=1}^{n}\partial_{i}(a_{ij}\partial_{j}u) + \sum_{i=1}^{n}b^{i}\partial_{i}u + cu&=f\ \textup{on }\Omega\\ \left.u\right|_{\partial\Omega}&=0,\end{aligned}\tag{*}

    and let η∈C01(Ω)\eta\in C_{0}^{1}(\Omega).

    a) Show that there exists a constant C>0C>0 depending only on LL (i.e. the various bounds on aija_{ij}, bb and cc) such that

    ∫Ω∣Du∣2η2≤C⋅(∫Ωu2⋅(η2+∣Dη∣2)+∫Ωf2η2)\int_{\Omega}|\D u|^{2}\eta^{2} \leq C\cdot\left(\int_{\Omega}u^{2}\cdot(\eta^{2}+|\D \eta|^{2}) + \int_{\Omega}f^{2}\eta^{2} \right)

    b) Deduce using ηR\eta_{R} as defined in the preceding exercise that

    ∫B(X,R2)∣Du∣2≤162C92((1+16R2)∫B(X,R)u2+∫B(X,R)f2).\int_{B(X,\frac{R}{2})}|\D u|^{2}\leq \frac{16^{2}C}{9^{2}}\left((1+\frac{16}{R^{2}})\int_{B(X,R)}u^{2}+\int_{B(X,R)}f^{2} \right).

    (8 marks)

To be submitted by 10 a.m. on Friday December 20.