# Weak solutions to PDE with different ICs

1. Apr 1, 2013

### TaPaKaH

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.

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 $U$.