# Let $f$ have compact support and $-u''+u=f$ a.e. on $(0, 1)$. Is the support of $u$ a compact subset of $(0, 1)$?

I'm reading section 8.4 Some Examples of Boundary Value Problems in Brezis' Functional Analysis.

Consider the problem $$(14) \quad \left\{\begin{array}{l} -u^{\prime \prime}+u=f \quad \text { on } I=(0,1), \\ u(0)=u(1)=0, \end{array}\right.$$ where $$f$$ is a given function (for example in $$C(\bar{I})$$ or more generally in $$L^2(I)$$ ). The boundary condition $$u(0)=u(1)=0$$ is called the (homogeneous) Dirichlet boundary condition.

Definition. A classical solution of (14) is a function $$u \in C^2(\bar{I})$$ satisfying (14) in the usual sense. A weak solution of (14) is a function $$u \in H_0^1(I)$$ satisfying $$(15) \quad \int_I u^{\prime} v^{\prime}+\int_I u v=\int_I f v \quad \forall v \in H_0^1(I) .$$

Proposition 8.15. Given any $$f \in L^2(I)$$ there exists a unique solution $$u \in H_0^1$$ to (15). Furthermore, $$u$$ is obtained by $$\min _{v \in H_0^1}\left\{\frac{1}{2} \int_I\left[ (v^{\prime})^2+v^2\right] - \int_I f v\right\};$$ this is Dirichlet's principle.

Proof. We apply Lax-Milgram's theorem (Corollary 5.8) in the Hilbert space $$H=$$ $$H_0^1(I)$$ with the bilinear form $$a(u, v)=\int_I u^{\prime} v^{\prime}+\int_I u v=(u, v)_{H^1}$$ and with the linear functional $$\varphi: v \mapsto \int_I f v$$

We recall the definition of the support of a function

Proposition 4.17 (and definition of the support). Let $$f: \mathbb{R}^d \rightarrow \mathbb{R}$$ be any function. Consider the family $$(\omega_i)_{i \in I}$$ of all open sets on $$\mathbb{R}^d$$ such that for each $$i \in I, f=0$$ a.e. on $$\omega_i$$. Set $$\omega := \bigcup_{i \in I} \omega_i$$. Then $$f=0$$ a.e. on $$\omega$$.

Now we assume that $$f \in L^2 (I)$$ and that $$\operatorname{supp} f$$ is a compact subset of $$I$$.

Is it true that support of the weak solution $$u$$ of $$(14)$$ is a compact subset of $$I$$. If the answer is negative, can the situation be improved if $$f \in C_c (I)$$?

Thank you so much for your elaboration!

• For the unrestricted homogeneous solution of $-u'' +u=0$ at least I don't think could stand compact-supported solutions, since is a 2nd order linear ODE which solutions are power series which cannot become constant in a non-zero measure interval or it will violate the Identity Theorem (check this). Dec 11, 2023 at 18:55
• Now in a bounded domain, if you consider to be zero in the edges equivalent to being of compact-support (so you don't care about what happens outside), it will be a different story. I am not informed enough to make any conclusion, but I hope this comment could help you as a starting point. Dec 11, 2023 at 18:55