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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!

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  • $\begingroup$ 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). $\endgroup$
    – Joako
    Dec 11, 2023 at 18:55
  • $\begingroup$ 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. $\endgroup$
    – Joako
    Dec 11, 2023 at 18:55

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