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Results tagged with ap.analysis-of-pdes
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user 32507
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
Accepted
Bounding $\lVert{u}\rVert_{C^0([0,T];V)} \leq C\left(\lVert{u}\rVert_{L^2(0,T;V)} + \lVert{u...
As mentioned by @TaQ, the embedding $W(V,H) \hookrightarrow C([0,T];V)$ is, in general, not true. However, if your estimate would be true, you can extend the embedding operator from the dense subspace …
- 1,724
1
vote
Seeking references on second-order optimality conditions in $H^1(Ω)$ space
If I understand your comment correctly, you are minimizing over a set
$$
U := \{ u \in H^1(\Omega) \mid a \le u \le b \}
$$
for some $a, b \in L^\infty(\Omega)$.
Such a set is polyhedric in the sense …
- 1,724
6
votes
Accepted
Poincare Inequality for $H^2$ function satisfying homogeneous Robin boundary conditions
This is not true. Take $\Omega = (-1,1)$ and functions $u_M$ like (I hope that I got the constants right)
$$
u_M(x)
=\begin{cases}
-M^2 (|x|-1)(|x|-1+1/M) + M & \text{for } |x| > 1-1/(2M) \\
1/4 + M & …
- 1,724
1
vote
A distributional normal derivative for functions in $H^1(\Omega)$
It is not possible to define a normal derivative for all $u \in H^1(\Omega)$ which depends continuously on $u$.
The reason is that all $C_c^\infty(\Omega)$ is dense in $H^1_0(\Omega)$, but all $u \in …
- 1,724