Elliptic Equations

Let $u \not\equiv 0$ satisfy $u \in C^2(\mathbb {R}^{n})$, $\Delta u = 0$ on $\mathbb {R}^{n}$. Show that

does not exist.

Let $u \in C^0(\Omega )$ be weakly harmonic in an open set $\Omega$, i.e., the relation

holds for all $\phi \in C_0^2(\Omega )$. Show that $u$ is then harmonic in $\Omega$.

Let $f(x,y)$ be a locally bounded function, harmonic in $x \in \mathbb {R}^{2}$ and continuous in $y \in \mathbb {R}^{2}$. Show that $f$ is continuous in $(x,y)$, i.e., as a function on $\mathbb {R}^{4}$.

Suppose $u$ is a function defined on $\mathbb {R}^{n}$ such that

Let $u(x)$ be $C^2$ on the half-space $\mathbb {R}^{n}_+ = \left\{ x : x_n > 0\right\}$ and continuous on the boundary $\partial \mathbb {R}^{n}_+ = \left\{ x : x_n = 0\right\}$, and let $g(x)$ be a compactly supported, $C^1$ function on $\partial \mathbb {R}^{n}_+ = \left\{ x : x_n = 0\right\}$. If $u$, bounded, satisfies

show that its ``tangential'' derivatives $\dfrac {\partial u}{\partial x_j}$, $j = 1, \ldots , n-1$, are bounded in magnitude by $\max \left|\nabla_g\right|$ on all of $\mathbb {R}^{n}_+ = \left\{ x : x_n > 0\right\}$. (Warning. You must justify any assumptions of regularity you make on the solution $u$.)

Show that the problem

u = -1 for x < 1, y < 1, u = 0 for x = 1, - = 0 for y = 1

has at most one solution. (Here $u = u(x,y)$ is a function of two variables $x$ and $y$, and is a classical solution.)

Let $\Omega$ be a bounded normal domain in $\mathbb {R}^{3}$, and suppose $\alpha$ is such that a non-trivial solution exists to

Let $\Omega \subset \mathbb {R}^{n}$ be an open set. Let $u$ be a continuous function on $\Omega$. Prove the equivalence of these two notions of ``subharmonic''.

Let $\Omega \subseteq \mathbb {R}^{n}$ be an open set, and let $u$ be harmonic in $\Omega$ and continuous in $\overline{\Omega }$.

Let $\Omega \subseteq \mathbb {R}^{3}$ be bounded and open with smooth boundary. Let $u \in C^1(\overline{\Omega }) \cap C^2(\Omega )$ solve

Derive an appropriate mean-value property for the solution.

Let $u \in C^2(\mathbb {R}^{2})$ satisfy

Prove that its spherical mean, defined by

and $v(0) = u(0)$, also satisfies (1), for $x \neq 0$. Here, $B(0,r)$ denotes the ball of radius $r$, $dS_y$ the element of arc length.

Let $B = \left\{ (x,y) \mid x^2 + y^2 < 1\right\}$. Prove that the problem

has exactly one weak solution $u$ in $H_0^1(B)$.

Let $\Omega$ be a unit ball in $\mathbb {R}^{n}$ centered at zero. Consider the following problem.

Find $u \in H_0^1(\Omega )$ such that

where $\mathbb {R}^{n} \ni x = (x_1, \ldots , x_{n-1}, x_n) = (x', x_n)$. Prove the existence of a unique solution of that problem.

Denote $u^+ = u|_{\Omega \cap \left\{ x_n>0\right\} }$ and $u^- = u|_{\Omega \cap \left\{ x_n<0\right\} }$, and assume that $u^{\pm }$ are regular enough. Show that $\Delta u^{\pm } = 0$ and that

Consider functions $u_n, v_n, w \in W^{2,2}(\Omega ) \cap W_0^{1,2}(\Omega )$, such that

-v_n + u_n = w, -u_n - n v_n = 0,

in $\Omega$. Prove that $u_n \to w$ strongly in $L^2(\Omega )$, as $n \to +\infty$.

Assume $\Omega \subset \mathbb {R}^{n}$ is a bounded open set with smooth boundary and let $f \in C_c^{\infty }(\Omega )$. Suppose $u_k \in C^{\infty }(\Omega )$ satisfies $\Delta u_k + u_k = f$ and has the property that for some positive number $M$,

for $k = 1, 2, \ldots$. Prove that there exist a function $u \in C^{\infty }(\Omega )$ satisfying $\Delta u + u = f$ and a subsequence $\left\{ u_{k_j}\right\}$ such that $u_{k_j}$ converges uniformly to $u$ on each compact subset of $\Omega$.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb {R}^{n}$, and

where $h_1, h_2 : \Omega \to \mathbb {R}$ are given smooth functions.

Let $\Omega \subset \mathbb {R}^{n}$, $n > 1$, be a bounded domain with smooth boundary. Let $u \in C^1(\overline{\Omega })$ be harmonic in $\Omega$.

Let $\Omega$ be an open subset of $\mathbb {R}^{n}$. Suppose $u \in C^2(\overline{\Omega })$ is a solution of the equation $\Delta u = u^3$ with the property that $\left|\nabla u(x)\right| \leq 1$ for each $x \in \partial \Omega$. Prove that $\left|\nabla u(x)\right| \leq 1$ for all $x \in \Omega$.

Consider (a nonlinear) equation

-u = u^2(1-u) in , u = 0 on , 1

where $\lambda > 0$ is a parameter, and $\partial \Omega$ sufficiently smooth.

Let $\Omega$ be a bounded normal domain in $\mathbb {R}^{n}$ with smooth boundary $\partial \Omega$. Let $\overrightarrow {n}$ be the exterior unit normal on $\partial \Omega$. Show that the only solution of the boundary value problem for the biharmonic equation:

(u) = 0 in u = 0 on , = 0 on ,

is the trivial one $u \equiv 0$. Assume $u \in C^4(\overline{\Omega })$.

Let $\Omega$ be a bounded domain in $\mathbb {R}^{n}$ with smooth boundary $\partial \Omega$ and assume $u \in C^2(\overline{\Omega })$ satisfies the following

where $f \in C^0(\overline{\Omega })$ and $\phi \in C^0(\partial \overline{\Omega })$. Let $B_R(O)$ be the smallest ball centered at the origin $O$ such that $\Omega \subset B_R(O)$. Show that

Also give an example to demonstrate that the estimate is sharp, i.e., the equality sign can be attained in your example.

Inside unit disc $\Omega = \left\{ x \in \mathbb {R}^{2}, \left|x\right| \leq 1\right\}$, construct two solutions $u_1, u_2$ of the Hamilton-Jacobi equation

u^2 - 1 = 0 for a.e. x u = 0 for x = 1,

with $u_1 \geq 0$ while $u_2 \leq 0$. Prove that the negative solution $u_2$ can not be obtained as a limit of parabolic (elliptic) approximation. Indeed, if $u^{\varepsilon }$ denotes a solution to

u^^2 - 1 = u^ for x u^ = 0 for x = 1,

then $u^{\varepsilon }$ cannot converge to $u_2$ as $\varepsilon \to +0$.