Tutorial sheet 5

In all of the following, Ω⊂Rn\Omega\subset\mathbb{R}^{n} is open, bounded and nonempty and all functions are real-valued.

  1. For λ≥0\lambda\geq 0 and f∈L2(Ω)f\in L^{2}(\Omega), let Eλ:H01,2(Ω)→RE_{\lambda}:H_{0}^{1,2}(\Omega)\rightarrow\mathbb{R} be the energy functional

    Eλ(u)=∫Ω12∣Du∣2−12λu2−fu.E_{\lambda}(u)=\int_{\Omega}\frac{1}{2}|\D u|^{2} - \frac{1}{2}\lambda u^{2} - fu.

    a) Show using the Poincaré inequality (i.e. ∫Ωu2≤CPoinc∫Ω∣Du∣2\int_{\Omega}u^{2}\leq C_{\textup{Poinc}}\int_{\Omega}|\D u|^{2}) that there exist constants C0,C1>0C_{0}, C_{1}>0 with the same dependencies as the Poincaré constant such that whenever λ≤C0\lambda \leq C_{0} and u∈H01,2(Ω)u\in H_{0}^{1,2}(\Omega),

    Eλ(u)≥14∫Ω∣Du∣2−C1∫Ωf2.E_{\lambda}(u)\geq \frac{1}{4}\int_{\Omega}|\D u|^{2} - C_{1}\int_{\Omega} f^{2}.

    b) Suppose henceforth that λ≤C0\lambda \leq C_{0} and let {uj}⊂H01,2(Ω)\{u_{j}\}\subset H_{0}^{1,2}(\Omega) be a minimising sequence for EλE_{\lambda}, i.e. Eλ(uj)→j→∞inf⁡u∈H01,2(Ω)Eλ(u)E_{\lambda}(u_{j})\xrightarrow{j\rightarrow\infty}\inf_{u\in H_{0}^{1,2}(\Omega)}E_{\lambda}(u). By the preceding part, this infimum is finite. Show that there is a constant C2>0C_{2}>0 depending only on C1C_{1} and ∣∣f∣∣2||f||_{2} such that for all sufficiently large jj

    ∫Ω∣Duj∣2≤C2.\int_{\Omega}|\D u_{j}|^{2}\leq C_{2}.

    c) Verify that there exists a u∈H01,2(Ω)u\in H_{0}^{1,2}(\Omega) and a subsequence {ujk}k=1∞\{u_{j_{k}}\}_{k=1}^{\infty} of {uj}j=1∞\{u_{j}\}_{j=1}^{\infty} such that

    ujk⇁k→∞u in H01,2(Ω)andujk→k→∞u in L2(Ω).\begin{aligned}&u_{j_{k}}\xrightharpoondown{k\rightarrow\infty} u\ \textup{in }H_{0}^{1,2}(\Omega)\\ \textup{and}\qquad\\& u_{j_{k}}\xrightarrow{k\rightarrow\infty} u\ \textup{in }L^{2}(\Omega).\end{aligned}

    Deduce the following:

    ∫Ω∣Du∣2≤lim inf⁡k→∞∫Ω∣Dujk∣2∫Ωu2=lim⁡k→∞∫Ωujk2∫Ωfu=lim⁡k→∞∫Ωfujk\begin{aligned}\int_{\Omega}|\D u|^{2}&\leq \liminf_{k\rightarrow\infty}\int_{\Omega}|\D u_{j_{k}}|^{2}\\ \int_{\Omega}u^{2} &=\lim_{k\rightarrow\infty}\int_{\Omega}u_{j_{k}}^{2}\\ \int_{\Omega} f u &= \lim_{k\rightarrow\infty}\int_{\Omega}f u_{j_{k}} \end{aligned}

    d) Conclude that Eλ(u)=inf⁡v∈H01,2(Ω)Eλ(v)E_{\lambda}(u)=\inf_{v\in H_{0}^{1,2}(\Omega)}E_{\lambda}(v).

    (12 marks)

  2. Suppose that aij∈C1(Ω‾)a_{ij}\in C^{1}(\overline{\Omega}) for all i,j∈{1,…,n}i,j\in\{1,\dots,n\} and the following conditions are satisfied:

    ∑i,j=1naij(x)vivj≥λ0∣v∣2.\sum_{i,j=1}^{n} a_{ij}(x)v^{i}v^{j}\geq \lambda_{0}|v|^{2}.

    Write LL for the elliptic operator

    Lu:=−∑i,j=1naij⋅∂i∂ju.Lu:=-\sum_{i,j=1}^{n}a_{ij}\cdot\partial_{i}\partial_{j}u.

    a) Show that if u1,u2∈C2(Ω)u_{1},u_{2}\in C^{2}(\Omega), then the product rule

    L(u1⋅u2)=Lu1⋅u2+u1⋅Lu2−2∑i,j=1naij⋅∂iu1⋅∂ju2L(u_{1}\cdot u_{2}) = L u_{1}\cdot u_{2} + u_{1}\cdot Lu_{2} - 2\sum_{i,j=1}^{n}a_{ij}\cdot\partial_{i}u_{1}\cdot \partial_{j}u_{2}

    holds.

    b) Let u∈C3(Ω)∩C1(Ω‾)u\in C^{3}(\Omega)\cap C^{1}(\overline{\Omega}) satisfy the equation Lu=0Lu=0 on Ω\Omega and set v:=∣Du∣2+λu2v:=|\D u|^{2}+\lambda u^{2}. Show that there is a constant C>0C>0 depending only on LL such that if λ≥C\lambda\geq C, then Lv≤0Lv\leq 0 on Ω\Omega.

    c) Deduce that there is a constant C′>0C'>0 with the same dependencies as CC such that

    sup⁡Ω∣Du∣≤C′(sup⁡∂Ω∣u∣+sup⁡∂Ω∣Du∣).\sup_{\Omega}|\D u|\leq C'\left(\sup_{\partial\Omega}|u| + \sup_{\partial\Omega}|\D u| \right).

    (8 marks)

To be submitted by 10 a.m. on Friday January 17.