(Oral Ens Mp/Mpi)
| Question a) Soit {f\in\mathcal{C}^2(\mathbb{R}^2,\mathbb{R})} admettant un maximum local en {(x_0,y_0)}. Que dire de : {\dfrac{\partial^2f}{\partial x^2}(x_0,y_0)+\dfrac{\partial^2f}{\partial y^2}(x_0,y_0)\,?} |
| Question b) Soit {D=\left\{(x,y)\in\mathbb{R}^2,\,x^2+y^2\lt 1\right\}}. Soient {\varepsilon,T>0} et {f:[0,T]\times\mathbb{R}^2\rightarrow\mathbb{R}} de classe {\mathcal{C}^2}, vérifiant : {\begin{array}{ll}\bullet&f(0,x,y)\geqslant\varepsilon\quad\text{sur }\overline D,\\[8pt]\bullet&x\partial_xf(t,x,y)+y\partial_yf(t,x,y)=0\\[4pt]&\text{sur }[0,T]\times\partial D,\\[8pt]\bullet&\partial_tf-\partial_{xx}f-\partial_{yy}f\geqslant\varepsilon\\[4pt]&\text{sur }[0,T]\times\overline D.\end{array}}Montrer que {f>0} sur {[0,T]\times\overline D}. |