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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

1 vote
0 answers
80 views

Compact imbedding for weight space

We begin with some definitions. Let $\gamma \geqslant 1,\,p \in \left[ {1,\infty } \right)$, we define $$L_\gamma ^p\left( {0,1} \right) = \left\{ {v:\left( {0,1} \right) \to \mathbb{R}:{{\left\| v \ …
0 votes
1 answer
221 views

Existence of subsequences convergence with weak topology

Let $\left\{ {{\varphi _n}} \right\}$ is the sequence bounded in ${L^\infty }\left( {0,\infty ;H_0^1\left( {0,1} \right)} \right)$. Is there exists $\varphi \in {L^\infty }\left( {0,\infty ;H_0^1\lef …
0 votes
0 answers
85 views

Prove that the solution belong to ${L^2}\left( {0,T;{L^2}\left( \Omega \right)} \right)$

Let $k \in {L^\infty }\left( {0,T} \right)$ and we assume that $$\phi :t \mapsto u\left( t \right) + \int_0^t {k\left( s \right)u\left( s \right)ds} \in {L^\infty }\left( {0,T;{L^2}\left( \Omega \ri …
1 vote
0 answers
155 views

orthonormal basis of ${H^2} \cap H_0^1$

we consider the following eigenvalue problem for the Laplacian $$ - \Delta w\left( x \right) = \lambda w\left( x \right),\,x \in \left( {0,1} \right),\,w\left( 0 \right) = w\left( 1 \right) = 0.$$ By …