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Let $\Omega \subset \mathbf R^n$ be a smooth domain and define $U_s=\{x\in\Omega \mid d(x,\partial \Omega)<s\}$; let $f\in W^{1,p}(\Omega)∩W_{\mathrm{loc}} ^{2,p}(\Omega)$; let $v$ be the unit normal …

Let $Q:=[0,1]^d$ and $g\in L^2(\Omega)$. Consider the PDE : $$ \begin{cases} -\Delta f=g & \text{in $\Omega$} \\ f\equiv 0 & \mbox{on $\partial \Omega$.} \end{cases} $$ I know tha …

I’m interested in using laplacian (−Δ) eigenfunction as a basis for H1(Rn) . I know that in H1(Ω) , Ω bounded this can be done so I was wandering about H1(Rn) . Now let eλ be an eigenfunction correspo …

Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$. Let $u∈L^2 ([0,T];V); u …

I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$ let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and refl …

Proposition 8.14. in Brezis states that:$(W_0^{1,p} (Ω))^*=W^{-1,p^*} (Ω)$ and we have the representation: $∀ F∈(W_0^{1,p} (Ω))^* ∃ f_0...f_n ∈L^{p^*} (Ω)$ such that $∀ u∈W_0^{1,p}(Ω)$ $F(u)=∫_Ω uf …

I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the duali …