grandes-ecoles 2012 QVII.A

grandes-ecoles · France · centrale-maths1__psi Differential equations Eigenvalue Problems and Operator-Based DEs

We assume that $f$ is positive and that $E$ is neither empty nor equal to $\mathbb{R}$. We denote by $\alpha$ its infimum.
VII.A.1) Show that if $Lf$ is bounded on $E$, then $\alpha \in E$.
VII.A.2) If $\alpha \notin E$, what can we say about $Lf(x)$ when $x$ tends to $\alpha^+$?

We assume that $f$ is positive and that $E$ is neither empty nor equal to $\mathbb{R}$. We denote by $\alpha$ its infimum.

VII.A.1) Show that if $Lf$ is bounded on $E$, then $\alpha \in E$.

VII.A.2) If $\alpha \notin E$, what can we say about $Lf(x)$ when $x$ tends to $\alpha^+$?

Paper Questions

QI.A QI.B QI.C QII.A QII.B QII.C QIII.A QIII.B QIII.C QIII.D QIV.A QIV.B QIV.C QIV.D QV.A QV.B QV.C QV.D QV.E QV.F QV.G QVI.A QVI.B QVI.C QVII.A QVII.B QVIII.A QVIII.B QVIII.C QVIII.D QVIII.E QVIII.F QVIII.G