Feynman
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Let the parabolic problem:
\displaystyle\left\{<br /> \newline\begin{array}{ccc}<br /> \newline\frac{\partial u}{\partial t}-\Delta u= f_{0}& \Omega_{T}=\Omega\times ]0,T[\\ <br /> \newline\frac{\partial{u}}{\partial n}=f_{1}& \Gamma_{T}=\partial\Omega\times ]0,T[ \\ <br /> \newline u(.,0)=u_{0}& \Omega \end{array}\right.
Then the weak formulation of this problem is :<br /> \displaystyle\left\{<br /> \newline\begin{array}{ccc}<br /> \newline Trouver & u\in L^{2}(0,T;H^{1}(\Omega))\cap C(0,T;L^{2}(\Omega))\\ <br /> \newline\int_{0}^{T}[a(u(t),v)\phi(t)-(u(t),v)\phi^{,}(t)]dt=(u_{0},v)\phi(0) +\int_{0}^{T}(f_{0}(t),v)\phi(t) dt\\ <br /> \newline + \int_{0}^{T}<f_{1}(t),v>_{H^{-\frac{1}{2}},H^{\frac{1}{2}}}\phi(t)dt&\forall\phi\inD([0,T[)et\forallv\in H^{1}(\Omega)\end{array}\right.<br /> and \displaystyle (h,g)= \int_{Omega} h(x)g(x) dx and $\displaystyle a(u,v)=(\nabla u,\nabla v).
So how prof that this weak problem have a solution? and u verify .(1)_{1} and (1)_{3}.
u verify .(1)_{3}.?
Merci
\displaystyle\left\{<br /> \newline\begin{array}{ccc}<br /> \newline\frac{\partial u}{\partial t}-\Delta u= f_{0}& \Omega_{T}=\Omega\times ]0,T[\\ <br /> \newline\frac{\partial{u}}{\partial n}=f_{1}& \Gamma_{T}=\partial\Omega\times ]0,T[ \\ <br /> \newline u(.,0)=u_{0}& \Omega \end{array}\right.
Then the weak formulation of this problem is :<br /> \displaystyle\left\{<br /> \newline\begin{array}{ccc}<br /> \newline Trouver & u\in L^{2}(0,T;H^{1}(\Omega))\cap C(0,T;L^{2}(\Omega))\\ <br /> \newline\int_{0}^{T}[a(u(t),v)\phi(t)-(u(t),v)\phi^{,}(t)]dt=(u_{0},v)\phi(0) +\int_{0}^{T}(f_{0}(t),v)\phi(t) dt\\ <br /> \newline + \int_{0}^{T}<f_{1}(t),v>_{H^{-\frac{1}{2}},H^{\frac{1}{2}}}\phi(t)dt&\forall\phi\inD([0,T[)et\forallv\in H^{1}(\Omega)\end{array}\right.<br /> and \displaystyle (h,g)= \int_{Omega} h(x)g(x) dx and $\displaystyle a(u,v)=(\nabla u,\nabla v).
So how prof that this weak problem have a solution? and u verify .(1)_{1} and (1)_{3}.
u verify .(1)_{3}.?
Merci