Discussion
Let \(u_{1}\) and \(u_{2}\) be solution of the equation :
\(\Delta u = 1\) in D
\(u = 0\) on \(\partial D\)
where \(D = \{(x, y) \in \mathbf{R}^{2}/(x - 2)^{2} + (y - 1)^{2} \le 1/4\}\) and \(\partial D = \{(x, y) \in \mathbf{R}^{2}/(x - 2)^{2} + (y - 1)^{2} = 1/4\}\) is the boundary of D.
If \(u_{1}(2, 1) = u_{2}(2, 1)\); then :
Answer: D
The answer is \(u_{1} \equiv u_{2}\) in D.
The Dirichlet problem for Poisson's equation on a bounded domain has at most one solution: \(w = u_{1} - u_{2}\) satisfies \(\Delta w = 0\) in D and \(w = 0\) on \(\partial D\), so by the maximum principle \(w\) attains its maximum and minimum on the boundary, where it is 0. Hence \(w \equiv 0\). The condition \(u_{1}(2, 1) = u_{2}(2, 1)\) is automatically true.
Neither solution is constant: a constant has \(\Delta u = 0\), not 1. (Explicitly, \(u = \tfrac{1}{4}((x - 2)^{2} + (y - 1)^{2} - \tfrac{1}{4})\).)
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development