Is the following version of the fundamental lemma of the calculus of variations valid?

Question

Let $U$ be an open subset of $\mathbb{R}^n$ with smooth boundary $\partial U$. Consider a function $f$ in $L^2(u)$. Suppose that for every $h$ in the Sobolev space$ H^2_0(U)$ it holds that $$\int_U f \Delta h=0.$$ Where $\Delta$ is the Laplacian operator.n

Can we conclude that $f=0$ almost everywhere on $U$?

Answer 1

I think so. Assume that $f$ is nonnegative and smooth in some open set $S$. Now test against $\Delta h$ where $\Delta h=1$ in an open set $\Omega\subset S$ with homogeneous Dirichlet boundary conditions. Is this ok? Now, one can use the density of $C_c^\infty$ in $L^2$ and some approximation argument, isn't it?

Answer 2

Let $u: \overline {U}\to \mathbb{R}$ be a harmonic function, i.e. $$\tag{1}\Delta u=0\ in\ U$$

Mutliply $(1)$ by $h\in H_0^2(U)$ in both sides and then integrate: $$\tag{2}\int_U\Delta u\cdot h=0,\ \forall\ h\in H_0^2(U)$$

Use the generalized Green identity to conclude from $(2)$ that $$\tag{3}0=\int_U\Delta u\cdot h=-\int_U\nabla u\nabla h=\int_Uu\Delta h,\ \forall\ h\in H_0^2(U)$$

From $(3)$ we have your claim but $u$ does not need to be zero almost everywhere.