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Results tagged with uniqueness-theorems
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user 108637
3
votes
Uniqueness of solution to heat equation when initial condition is a generalized function
(Not sure if I understand the question correctly.)
If $p_{t,x}(y) = p(t, x, y)$ is the fundamental solution (a.k.a. the heat kernel), then $p_{t,x}$ converges as $t \to 0^+$ to the Dirac measure $\del …
- 17.2k
5
votes
Accepted
Uniqueness of minimizers in the Calculus of Variations
There's no uniqueness in general. If $\Omega$ is a unit square, and $$\varphi(x, y) = (1-x-y)^+,$$ then obviously any $u$ which is decreasing with respect to both $x$ and $y$, and which matches the bo …
- 17.2k
3
votes
Accepted
Unique continuation of the Hilbert transform
No. Let $$u(z) = \exp(-(-iz)^{1/2}-(-iz)^{-1/2})$$ for $z$ in the closed upper complex half-plane, with the principal branch of the complex power. Then $u$ is a bounded holomorphic function in the ope …
- 17.2k