Tutorial sheet 3

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

  1. Suppose that (X,<⋅,⋅>)(X,\left<\cdot,\cdot\right>) is a Hilbert space, {xm}m=1∞⊂X\{x_{m}\}_{m=1}^{\infty}\subset X and x∈Xx\in X.

    a) Suppose that xm⇁m→∞xx_{m}\xrightharpoondown{m\rightarrow\infty}x. Show that ∣∣x∣∣≤lim inf⁡m→∞∣∣xm∣∣||x||\leq \liminf_{m\rightarrow\infty}||x_{m}||.

    b) Show that xm→m→∞xx_{m}\xrightarrow{m\rightarrow\infty}x in XX iff ∣∣xm∣∣→m→∞∣∣x∣∣||x_{m}||\xrightarrow{m\rightarrow\infty}||x|| and xm⇁m→∞xx_{m}\xrightharpoondown{m\rightarrow\infty}x.

  2. Suppose that u∈W1,2(Ω)u\in W^{1,2}(\Omega) and f∈L2(Ω)f\in L^{2}(\Omega). We call uu a weak solution to the equation −Δu=f-\Delta u = f on Ω\Omega if for all φ∈C0∞(Ω)\varphi\in \cs(\Omega),

    ∫ΩDu⋅Dφ=∫Ωf⋅φ\int_{\Omega}\D u\cdot \D\varphi=\int_{\Omega}f\cdot \varphi

    For each ε>0\eps>0, let uε:=Jεuu_{\eps}:=J_{\eps}u and fε:=Jεff_{\eps}:=J_{\eps}f be the mollified versions of uu and ff respectively. Show that the equation

    −Δuε=fε-\Delta u_{\eps}=f_{\eps}

    holds on Ωε:={x∈Ω:dist(x,∂Ω)>ε}\Omega_{\eps}:=\{x\in\Omega:\dist(x,\partial\Omega)>\eps\}.

  3. Suppose that u∈W1,2(Ω)u\in W^{1,2}(\Omega) is a weak solution to the equation −Δu=f-\Delta u = f on Ω\Omega.

    a) Verify that the equation

    ∫ΩDu⋅Dv=∫Ωf⋅v\int_{\Omega}\D u\cdot \D v = \int_{\Omega}f\cdot v

    holds for all v∈H01,2(Ω)v\in H_{0}^{1,2}(\Omega).

    b) Suppose that Ω\Omega is bounded. Show that if u∈H01,2(Ω)u\in H_{0}^{1,2}(\Omega), there exists a constant C>0C>0 depending only on Ω\Omega such that the estimate

    ∣∣u∣∣1,2≤C∣∣f∣∣2||u||_{1,2}\leq C||f||_{2}

    holds.

  4. Suppose that Ω\Omega is bounded. Recall that the Poincaré inequality for u∈H01,2(Ω)u\in H_{0}^{1,2}(\Omega) states that there is a constant C(Ω)>0C(\Omega)>0 such that

    ∫Ω∣u∣2≤C⋅∫Ω∣Du∣2,\int_{\Omega}|u|^{2}\leq C\cdot\int_{\Omega}|\D u|^{2},

    which implies that λ:=inf⁡{∫Ω∣Du∣2∫Ω∣u∣2:u∈H01,2(Ω)\{0}}>−∞\lambda:=\inf\{\frac{\int_{\Omega}|\D u|^{2}}{\int_{\Omega}|u|^{2}}: u\in H_{0}^{1,2}(\Omega)\backslash\{0\}\}>-\infty. Suppose that there exists a v∈C2(Ω‾)∩H01,2(Ω)v\in C^{2}(\overline\Omega)\cap H_{0}^{1,2}(\Omega) such that

    ∫Ω∣Dv∣2∫Ω∣v∣2=λ.\frac{\int_{\Omega}|\D v|^{2}}{\int_{\Omega}|v|^{2}}=\lambda.

    By considering the function

    t↦∫Ω∣D(v+tφ)∣2∫Ω∣v+tφ∣2t\mapsto \frac{\int_{\Omega}|\D (v+t\varphi)|^{2}}{\int_{\Omega}|v+t\varphi|^{2}}

    for φ∈C0∞(Ω)\varphi\in\cs(\Omega), show that vv solves the equation −Δv=λv-\Delta v = \lambda v on Ω\Omega.

To be submitted by 10 a.m. on Friday December 6.