12. We consider whether the inequality ||u||L² (n) ≤ c||L(u)||L² (2) can hold for open sets that are unbounded. (a) Assume d≥2. Show that for each constant coefficient partial differential operator L, there are unbounded connected open sets for which the above holds for all u € Co (N). (b) Show that ||u|| 1² (Rª) ≤ C||L(u)|| L² (Rª) for all u € C° (Rª) if and only if |P(§)| ≥ c> 0 all , where P is the characteristic polynomial of L.
12. We consider whether the inequality ||u||L² (n) ≤ c||L(u)||L² (2) can hold for open sets that are unbounded. (a) Assume d≥2. Show that for each constant coefficient partial differential operator L, there are unbounded connected open sets for which the above holds for all u € Co (N). (b) Show that ||u|| 1² (Rª) ≤ C||L(u)|| L² (Rª) for all u € C° (Rª) if and only if |P(§)| ≥ c> 0 all , where P is the characteristic polynomial of L.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![12. We consider whether the inequality
||u|| L² (2) ≤c||L(u) || L² (12)
can hold for open sets that are unbounded.
(a) Assume d≥2. Show that for each constant coefficient partial differential
operator L, there are unbounded connected open sets for which the above
holds for all u € Co (N).
(b) Show that ||u|| ₁² (Rd) ≤ C||L(u)||L2 (Rd) for all u € Co (R) if and only if
|P(§)| ≥ c > 0 all , where P is the characteristic polynomial of L.
[Hint: For (a) consider first L = (0/0x₁)" and a strip {x: −1 < x₁ < 1}.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8b8fab0-2681-4f69-af9a-7bc39adf973d%2F3ac4f55a-b0e1-407b-b3c0-6219b823c8ca%2Fy460t7k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:12. We consider whether the inequality
||u|| L² (2) ≤c||L(u) || L² (12)
can hold for open sets that are unbounded.
(a) Assume d≥2. Show that for each constant coefficient partial differential
operator L, there are unbounded connected open sets for which the above
holds for all u € Co (N).
(b) Show that ||u|| ₁² (Rd) ≤ C||L(u)||L2 (Rd) for all u € Co (R) if and only if
|P(§)| ≥ c > 0 all , where P is the characteristic polynomial of L.
[Hint: For (a) consider first L = (0/0x₁)" and a strip {x: −1 < x₁ < 1}.]
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