1
vote
0answers
23 views

A projection error estimation in Sobolev space of Fractional order (1D)

Let $\Pi_0$ be the $L_2$ projector that maps $u \in L_2(I)$ to a constant function, where $I$ is taken as the interval $(0,1)$. I would like to find an error estimation for $u \in H^{1/2}(I)$, such ...
4
votes
1answer
147 views

What can be said about the eigenvalues of the Laplace operator in $H^k(\mathbb{T}^2)$

Consider the Laplace operator $$\Delta: H^{k+2}(\mathbb{T}^2) \to H^k(\mathbb{T}^2)$$ where $\mathbb{T^2}$ is the two-dimensional torus (which is a compact manifold without boundary), so that $$ ...
1
vote
0answers
84 views

Solving a Sturm-Liouville differential equation variationally

This is a problem from Haim Brezis' functional analysis book (Exercise 8.41). I solved parts of it, but am stuck on some parts/want confirmation on the method. The problem is as follows: Let $q ...
1
vote
0answers
94 views

Eigenfunction associated to the smallest eigenvalue of an elliptic operator

Let $(T_n)$ be a sequence of elliptic operators defined in $H^2(\Omega)\cap H_0^1(\Omega)$ to $L^2(\Omega)$, with $\Omega$ being a bounded domain with smooth boundary. All of them have a smallest ...