Question

Solve the Cauchy problem for the two-dimensional Laplace equation subject to the Cauchy data \(u(0, y)=0,(\partial u / \partial x)(0, y)=\epsilon \sin k y\), where \(\epsilon\) and \(k\) are constants. Show that the solution does not vary continuously as the Cauchy data vary. In particular, show that for any \(\epsilon \neq 0\) and any preassigned \(x>0\), the solution \(u(x, y)\) can be made arbitrarily large by choosing \(k\) large enough.

Short answer

The solution to the Cauchy problem for the given Laplace equation is \(u(x,y) = \frac{\epsilon}{2k} (e^{kx} - e^{-kx})( C \cos ky + D \sin ky)\), which can be made arbitrarily large for a preassigned \(x > 0\) by taking \(k\) to be large enough. Therefore, the solution does not vary continuously as the Cauchy data vary.

Step-by-step solution

Write down the Cauchy problem

First step is writing down the given Cauchy problem: \(\nabla^2 u = 0\), with \(u(0,y) = 0\) and \(\partial u / \partial x (0,y) = \epsilon \sin k y\). Here, \(\nabla^2 = \partial^2 / \partial x^2 + \partial^2 / \partial y^2\) denotes the Laplacian.

Solve the two-dimensional Laplace equation

Since the 2-D Laplace equation is separable, propose a solution in the form \(u(x,y) = X(x)Y(y)\). Substitute this into the Laplace equation to get \(X'' Y + X Y'' = 0\), which can be rewritten as \(X''/X = - Y''/Y = k^2\). The negative sign is chosen arbitrarily.

Find the individual solutions for X and Y

Solve the separated equations \(X'' - k^2 X = 0\) and \(Y'' + k^2 Y = 0\) separately, you get \(X(x) = A e^{kx} + B e^{-kx}\) and \(Y(y) = C \cos ky + D \sin ky\). Thus, the general solution is a superposition of eigenfunctions \(u(x,y) = (A e^{kx} + B e^{-kx})( C \cos ky + D \sin ky)\).

Apply the Cauchy conditions

With the Cauchy data, you have \(u(0,y) = 0\) meaning that \(B = -A\) and \(\partial u / \partial x (0,y) = \epsilon \sin ky\), implying \(A=-B=\frac{\epsilon}{2k}\). The solution satisfying the Cauchy conditions is then \(u(x,y) = \frac{\epsilon}{2k} (e^{kx} - e^{-kx})( C \cos ky + D \sin ky)\). Since the Cauchy condition does not involve \(y\), the constants \(C\) and \(D\) are undetermined.

Demonstrate the non-continuity and unboundedness

Now, it should be noted that the solution \(u(x,y)\) can become arbitrarily large for a preassigned \(x>0\) by choosing \(k\) large enough. Therefore, we have shown that the solution does not vary continuously as the Cauchy data vary.

Key concepts

Cauchy Problem

The Cauchy problem is a specific type of problem in mathematics where initial conditions are specified for partial differential equations. These problems are named after the French mathematician Augustin-Louis Cauchy. In the Cauchy problem, the initial conditions generally involve specifying the value of the function and its derivatives at a particular point.
For the Laplace equation discussed here, the Cauchy problem involves specifying the function

• \( u(0, y) \)
• and its derivative \((\partial u / \partial x)(0, y)\)

These conditions are used to find a solution, though it is essential to note that solutions to Cauchy problems can sometimes be unstable or unbounded. This instability is observed in this problem where, even with small variations in Cauchy data, the solution can change unpredictably.

Two-Dimensional Laplace Equation

The two-dimensional Laplace equation is a second-order partial differential equation. It is represented as \( abla^2 u = 0 \), where \( abla^2 = \partial^2 / \partial x^2 + \partial^2 / \partial y^2 \). It describes the behavior of potential fields, such as gravitational, electrostatic, and fluid dynamics.
In simpler terms, the Laplace equation helps model situations where there is no change—or a steady state—throughout the space. Nevertheless, when boundary or initial conditions are applied, finding the solution can become more complex. Such is the case with the Cauchy problem, where initial data at a boundary can lead to unpredictable behaviors or solutions.

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. They are vital in describing various phenomena in physics, engineering, and other sciences, as they model relationships involving rates of change.
PDEs can include several functions and derivatives concerning multiple variables. Notably in this exercise, we are dealing with the Laplace equation, a form of PDE, which provides insights into the potential theory. Solving PDEs often requires specialized techniques such as separation of variables. This is used frequently in problems where functions can be expressed as a product of single-variable functions.

Separation of Variables

Separation of variables is a mathematical method used to solve partial differential equations. It involves the assumption that a multi-variable function can be rewritten as a product of single-variable functions. This breaks down complex equations into simpler, solvable ordinary differential equations.
For the two-dimensional Laplace equation, the separation of variables technique involves assuming a solution of the form \( u(x, y) = X(x)Y(y) \). Substituting this into the original equation helps segregate terms involving \(x\) from those involving \(y\). This method is particularly effective in solving PDEs with homogenous boundary conditions and is a classic strategy in mathematical physics and engineering problems.