Partial Differential Equations

Let $T$ be a distribution on $\mathbb {R}$ and suppose that $T' = 0$ on $\mathbb {R}$. Show that $T = \operatorname {const}$; i.e., show that there is a number $a$ such that

Let $u \in H^s(\mathbb {R}^{n})$ with $s > n/2$. Show that $\lim_{\left|x\right| \to \infty } u(x) = 0$.

Let $u$ be defined for $f \in \mathcal{D}((0,1))$ by

Determine whether $u$ is a distribution on $(0,1)$, and support your answer.

Let

For which $p \geq 1$ does the function

belong to $W^{1,p}(B)$?

Let $\Sigma$ be a smooth hypersurface dividing $\mathbb {R}^{n}$ into two disjoint open regions $\Omega_1$ and $\Omega_2$. Denote by $\nu = \nu (x)$ the unit normal to $\Sigma$ pointing into $\Omega_2$. Suppose $v : \mathbb {R}^{n} \to \mathbb {R}^{n}$ solves the equation

in the classical ($C^1$) sense where $b$ is a continuous function. Assume furthermore that $v$ lies in both $C^0(\overline{\Omega_1})$ and $C^0(\overline{\Omega_2})$ (but not necessarily $C^0(\mathbb {R}^{n})$!!) with

for each $x_0 \in \Sigma$. Derive a necessary and sufficient condition in terms of $v^+$, $v^-$ and $\nu$ for $v$ to be a distribution solution of (*) on all of $\mathbb {R}^{n}$.

Recall the definition of the curl operator in two dimensions acting on a vector field $U(x,y) = (u(x,y), v(x,y))$:

Let $p$ be a polynomial in $n$ variables, and assume that the set $\left\{ x : p(ix) = 0\right\}$ is bounded. Prove that there exists a tempered distribution $E$ on $\mathbb {R}^{n}$ such that $p(D)E - \delta \in S$. Here $S$ is the Schwartz class of rapidly decreasing functions, and $\delta$ is the Dirac distribution defined by $\delta \cdot \phi = \phi (0)$.

Let $P(D)$ be an $m$-th order linear constant coefficient partial differential operator on $\mathbb {R}^{n}$; i.e.,

where $\alpha$ denotes a multi-index $\alpha = (\alpha_1, \alpha_2, \ldots , \alpha_n)$ and the $C_{\alpha }$ are constants. Suppose

If $u \in S'$ satisfies $P(D)u = 0$, prove that $u$ must be a polynomial. (Here $S'$ denotes the space of tempered distributions on $\mathbb {R}^{n}$.)

Let $u$ be the solution of the initial problem

u_tt = -u_x_j x_j x_k x_k, x R^3, t > 0, u(x,0) = 0, u_t(x,0) = (x),

where summation is understood in the pde, and $\phi$ is a $C^{\infty }$ function on $\mathbb {R}^{3}$ with compact support. Assume that there is a constant $C$ such that

and prove that

Given $\phi \in S$ (rapidly decreasing functions) over $\mathbb {R}^{1}$, consider the solution $u$ of the Schrodinger equation

iu_t + u_xx = 0 (x R^1, t > 0), u(x,0) = (x).

Show that

Here $\hat{\phi }$ denotes the Fourier Transform of $\phi$.

Suppose $u$ is a smooth solution of the following problem

u_xxt + u_xx - u^3 = 0, in [0,1] [0,), u(0,t) = u(1,t) = 0 for t [0,)

with initial data $u(x,0) = x(x-1)$. Derive a differential inequality for $w := \int_0^1 \left|u_x(x,t)\right|^2\, dx$, and show that $u(x,t)$ uniformly tends to zero as $t \to \infty$.

Assume that $g \in L^2(\mathbb {R})$ and consider the Cauchy problem

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 $\varepsilon \to +0$.

Consider the Cauchy problem for the Heat Operator in $\mathbb {R}^{1}$:

u_t = u_xx (-< x < , t > 0) u(x,0) = f(x) (-< x < ),

where $f$ is bounded, continuous, and satisfies

Show that there exists a constant $C$ such that

for all $-\infty < x < \infty$, $t > 0$.

Let $\Omega \subset \mathbb {R}^{n}$ be an open set with smooth boundary and suppose $u \in C^{\infty }(\Omega \times [0,\infty ))$ is a solution of the equation

with $u = 0$ on $\partial \Omega \times [0,\infty )$. Assume that

Let

Prove that $\psi (t) \to 0$ as $t \to +\infty$.

Consider the Cauchy problem

u_t = u_xx (-< x < , t > 0), u(x,0) = f(x),

where $f$ is bounded and continuous.

Consider the problem

u_t = u_xx (x R, t > 0) u(x,0) = u_0(x) L^2(R).

Find a bound on $\int_{-\infty }^{\infty } \left|u_x(x,t)\right|^2\, dx$ in $t > 0$.

For the indicated domain $\Omega$ (interior of the parabola) let $\Omega_T$ denote the points $(x,t)$ with $x \in \Omega$ and $t < T$. Let $B_1$ denote the open line-segment forming the top of $\partial \Omega_T$, and let $B_2 = \partial \Omega_T \setminus B_1$. Let $u(x,t) \in C^0(\overline{\Omega_T})$ with $u_{xx}, u_t \in C^0(\Omega_T \cup B_1)$ be a solution of

with $a, f \in C^0(\overline{\Omega })$. Show that if $a < 0$, $f \leq 0$ in $\Omega_T$, then every non-constant solution $u$ assumes its negative minimum (if one exists) on $B_2$. What can you say if $a < 0$, $f \geq 0$ in $\Omega_T$?

Consider the boundary-value problem

u_xx + u_t = 0 (0 x and t > 0), B.C. u(0,t) = 0, u_x(,t) = 0, I.C. u(x,0) = f(x).

Use the sequence $\left\{ f_n(x) = \dfrac {2}{n+1} \sin \left(\dfrac {n+1}{2}x\right)\right\}$ and an appropriate space of continuous functions to decide whether this problem is well-posed.

In Probability Theory one encounters the Ornstein-Uhlenbeck process, in which the particles of Brownian motion are subjected to an elastic force. One is required to solve

u_t = u_xx - xu_x (x R, t > 0) u(x,0) = u_0(x).

Assume $u_0 \in S$ and find $u$. (You should be able to find $u$ explicitly; if not, leave your result in the form of a one-dimensional integral.)

Let $L$ be the usual heat operator

let $\phi \in D(\mathbb {R}^{n})$, and let $h$ be a fixed point of $\mathbb {R}^{n}$. Consider the problem

(Lu)(x,t) = u(x-h,t), x R^n, t > 0, u(x,0) = (x), x R^n

for an unknown function $u(x,t)$.

Let $\Omega \subset \mathbb {R}^{n}$ be a bounded domain and assume $u(x,t) \geq 0$ is a function with $u \in C^2(\overline{\Omega } \times [0,\infty ))$, which solves the equation

(heat conduction with heat loss due to radiation) with the boundary condition $u|_{\partial \Omega } = 0$. Prove that we can find a constant $C$ such that

regardless of the initial value $u(x,0)$.

Let

satisfy

with $g,h$ continuous and $g,h < 1$. Prove that $\max u < 1$ on $\overline{B(0,r)} \times [0,T]$.

Let $u \in C^2(\Omega \times (0,T)) \cap C^0(\overline{\Omega } \times [0,T])$ be a solution to the problem

where $\Omega$ is a smooth bounded domain in $\mathbb {R}^{n}$.