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Consider the following problem on $[0,1]$ with mixed boundary condition

a. Identify an appropriate Sobolev space on $[0,1]$ and derive the variational formulation of this problem using this space.

b. Show that the bilinear form of the variational formulation is coercive and bounded.

Consider the equation

where $𝑎:\mathrm{\Omega }\to {ℝ}^{𝑛\times 𝑛}$ is ${𝐶}^{\infty }$ and satisfies ${\sum }_{𝑖,𝑗=0}^{\infty }{𝑎}_{𝑖𝑗}(𝑥){𝜉}_{𝑖}{𝜉}_{𝑗}\ge 𝜆{|𝜉|}^2$ for all $𝑥\in \mathrm{\Omega }$ and $𝜉\in {ℝ}^{𝑛}$, with some $𝜆>0$. We define the energy as $𝐸[𝑢](𝑡)=\frac{1}{2}{\int }_{\mathrm{\Omega }}{𝑢}^2(𝑡,𝑥)d𝑥$.

a. Let $\mathrm{\Omega }\subset {ℝ}^{𝑛}$ be a bounded open set with smooth boundary and let $𝑢(𝑡,𝑥)$ be a ${𝐶}^2$ solution to this equation with the boundary condition $𝑢{|}_{𝜕\mathrm{\Omega }}=0$, and an initial condition . Prove that there exists ${𝑐}_1>0$ (depending only on $\mathrm{\Omega }$ and $𝜆$) such that

b. Let $\mathrm{\Omega }={ℝ}^{𝑛}$ and $𝑢(𝑡,𝑥)$ be a ${𝐶}^{\infty }({ℝ}^{𝑛})$ solution in ${ℝ}^{𝑛}$. Show that there is a constant ${𝐶}_{𝑛}(𝜆)$ that depends only on $𝜆$ and the dimension $𝑛$ so that

Hint: part (a) of problem 2 might be useful for part b.

Let $𝐵(𝑥;𝑟)$ be the closed ball of radius $𝑟$ and center $𝑥$ in ${ℝ}^{𝑑}$. Fix $𝜀\in (0,1/2]$ and let ${\{{𝑥}_{𝑘}\}}_{𝑘\ge 1}$ be a sequence of points independently identically distributed according to the uniform distribution in the ball $𝐵(0;1+𝜀)$. Consider the random set ${𝑆}_{𝑛}(𝜀)≔𝐵(0;1)\setminus ({\bigcup }_{𝑖\le 𝑛}𝐵({𝑥}_{𝑖};𝜀))$, and let $𝑁≔max\{𝑛:{𝑆}_{𝑛}(𝜀)\ne \varnothing \}$.

a. Show that $𝑁<\infty$ almost surely.

b. Define ${𝑛}_{\ast }(𝜃)=\frac{𝑑log(1/𝜀)}{{(𝜃𝜀)}^{𝑑}}$. Show that, for any $𝛿>0$, there is such that for all $𝑑\ge 1$, $𝜀\in (0,1/2]$.

c. Show that, for any $𝛿>0$, there exists such that for all $𝑑\ge 1$, $𝜀\in (0,1/2]$.

Find the line integral

where $𝛾$ is the curve from $-1$ to $1$ taken along a semicircle.

A domain $\mathrm{\Omega }$ in $ℂ$ is said to be holomorphically simply connected if for any holomorphic function $𝑓$ on $\mathrm{\Omega }$ and any simple closed piecewise ${𝐶}^1$ curve $𝛾$ in $\mathrm{\Omega }$, one has

(a) Prove that

is holomorphically simply connected.

(b) Prove that $\mathrm{\Omega }=𝐷(0,1)\backslash \{0\}$ is not holomorphically simply connected.

(a) Prove that the series

converges to a holomorphic function in a neighborhood of the point ${𝑧}_0=𝑖$.

(b) If

is represented as a power series

what is its radius of convergence?

Find an explicit conformal mapping of the domain

onto the unit disk.

If $𝑓:ℂ\to ℂ$ is an entire function, define a sequence of functions ${𝑓}^{(𝑛)}$ by

Prove or disprove: for any entire function $𝑓:ℂ\to ℂ$, the sequence of restrictions

forms a normal family.

(a) Suppose that $𝑢:ℂ\to ℝ$ is a harmonic function such that $𝑢{|}_{ℝ}=0$. Does it imply that $𝑢\cong 0$?

(b) Suppose that $𝑢:ℂ\to ℝ$ is a harmonic function such that . Does it imply that $𝑢\cong 0$?

Let $𝑓$ be entire holomorphic such that

where ${𝑝}_{𝑛}(𝑥,𝑦)$ is a polynomial in $𝑥$ and $𝑦$ of degree $𝑛$. Prove that $𝑓$ is a polynomial in $𝑧$.