1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Weak solutions to PDE with different ICs

  1. Apr 1, 2013 #1
    1. The problem statement, all variables and given/known data
    Let ##U\subset\mathbb{R}^n## be a bounded open set with smooth boundary ##\partial U##. Consider the boundary value problem $$\begin{cases}\bigtriangleup^2u=f&\text{on }U\\u=\frac{\partial u}{\partial n}=0&\text{on }\partial U\end{cases}$$where ##n## is the outward pointing normal on ##\partial U## and ##f\in L^2(U)##.
    a) Define the notion of weak solution and show that the boundary value problem has a unique solution in an appropreately chosen Sobolev space.
    b) Change the boundary conditions to ##u=\bigtriangleup u=0## and prove the existence of a weak solution in the appropreate Sobolev space.

    2. Relevant equations
    The book I am using describes the procedure for second-order partial differential operator ##L=-\sum_{i,j=1}^na_{i,j}u_{x_ix_j}+\sum_{i=1}^nb_iu_{x_i}+cu## with boundary condition ##u=0## as follows:
    Multiply ##Lu=f## by a test function ##v\in C_0^\infty## and integrate over ##U##. After some partial integration one has$$\sum_{i,j=1}^n\int_Ua_{ij}u_{x_i}v_{x_j}+\sum_{i=1}\int_Ub_iu_{x_i}v+\int_Ucuv=\int_Ufv.$$By approximation (property) the equality above can be made to hold for any ##v\in W_0^{1,2}(U)## so they define ##u\in W_0^{1,2}(U)## as the weak solution of the problem if it satisfies the formula above for any ##v\in W_0^{1,2}(U)##.

    3. The attempt at a solution
    Multiplying the equation by a test function ##v\in C_0^\infty(U)## and integrating over ##U## I get (after partial integration, using that all derivatives of ##v## are zero on the boundary):$$\sum_{i,j=1}^n\int_Uu_{x_ix_i}v_{x_jx_j}=\int_Ufv$$or (depending on the order of partial integration)$$\sum_{i,j=1}^n\int_Uu_{x_ix_j}v_{x_ix_j}=\int_Ufv.$$Here I make a similar statement that both of the equalities above can be made to hold for any ##v\in W_0^{2,2}(U)## hence if ##u\in W_0^{2,2}(U)## satisfies those for any ##v## then it is a weak solution.
    From here on the existence and uniqueness of ##u## follows from Lax-Milgram Theorem after proving the coercivity and continuity of left-hand side (denoted as bilinear form ##B[u,v]##) and continuity of right-hand side (denoted as ##L^2(U)## inner product ##(f,v)##).

    My problem is that my assumption on ##u\in W_0^{2,2}(U)## might be too general since it makes no difference between cases a) and b) but I can't think of any "more specific" Sobolev space to let ##u## be in.

    Any comments welcome.
     
  2. jcsd
  3. Apr 6, 2013 #2
    I tried to solve a) for ##u,v\in W^{2,2}(U)\cap W_0^{1,2}(U)## but hit a technical problem.
    I defined$$(u,v)=\sum_{i,j=1}^n\int_Uu_{x_ix_j}v_{x_ix_j}$$and tried to show that it's an inner product on ##u,v\in W^{2,2}(U)\cap W_0^{1,2}(U)##, but I'm having difficulties with showing that $$0=(u,u)=\sum_{i,j=1}^n\|u_{x_ix_j}\|_{L^2(U)}$$ implies that ##u\equiv0## almost everywhere on ##U##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Weak solutions to PDE with different ICs
  1. General solution to PDE (Replies: 10)

Loading...