Weak solutions to PDE with different ICs

[TaPaKaH]

Homework Statement

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.

Homework 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)##.

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.

 

[TaPaKaH]

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##.