Questions: Well-Posedness and Hadamard's Conditions

Question 1

Which of Hadamard's three conditions for well-posedness addresses numerical stability?

• A. Existence
• B. Uniqueness
• C. Continuous dependence on data
• D. All three equally

Answer: (C)

Continuous dependence on data ensures that small perturbations in the input (from measurement errors or numerical rounding) produce only small changes in the solution. Without this property, numerical computation of the solution is unreliable because tiny floating-point errors can produce wildly different outputs.

Question 2

The backward heat equation u_t = -kΔu is well-posed as an initial value problem.

• T. True
• F. False

Answer: False

The backward heat equation is the classic example of ill-posedness. While solutions may exist and be unique, they do not depend continuously on the data: Fourier modes e^(kn²t) grow exponentially, amplifying any small perturbation in the initial data without bound. This makes the problem catastrophically unstable.

Question 3

Why is the Cauchy problem for Laplace's equation (specifying u and u_n on part of the boundary) ill-posed?

Model answer: It violates continuous dependence: Hadamard showed that arbitrarily small oscillatory boundary data can produce arbitrarily large solutions in the interior

Hadamard's famous counterexample shows that the Cauchy data u = 0, u_y = sin(nx)/n on y = 0 for Laplace's equation produces the solution sinh(ny)sin(nx)/n², which grows exponentially with n despite the data shrinking to zero. This demonstrates catastrophic instability.

Question 4

Prescribing Dirichlet boundary conditions for Laplace's equation on a bounded domain is well-posed.

• T. True
• F. False

Answer: True

The Dirichlet problem for Laplace's equation satisfies all three Hadamard conditions: existence follows from Perron's method or variational arguments, uniqueness from the maximum principle, and continuous dependence from the maximum principle applied to the difference of two solutions.