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Results tagged with fa.functional-analysis
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user 35874
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
4
votes
References on the obstacle problem for the heat equation
I will get you started, but there are lots of blanks to fill. We are
interested in the PDE
\begin{align*}
\min\left\{ -u_{t}+\Delta u,u-\varphi\right\} & =0 & \text{in }(0,T]\times\mathbb{R}^{n}\\
u( …
- 331
2
votes
Accepted
Equivalence of alternative definitions of 'viscosity solution'
Only the subsolution case is proven, as the supersolution case is identical.
Q1
Suppose $u$ is a subsolution under the definition with strict extremum. Let $\phi$ be a test function such that $u-\ph …
- 331
1
vote
Accepted
On the 'usefulness' of the 'original' definition of viscosity solution
(i). The semijet definition [CEL Theorem 1.1 (i)] is the natural way to derive comparison principles for PDEs using, e.g., the Crandall-Ishii lemma [CIL, Theorem 3.2].
(ii). Here are some various exa …
- 331
1
vote
A result on the convergence of the vanishing viscosity approximation to the viscosity soluti...
More generally, viscosity solutions are stable with respect to limits. The presentation of the following result, which has more general forms, is borrowed from [Tou13].
Theorem: Let $\mathcal{O}$ …
- 331