Consider the steady-state heat equation $-a\Delta u = f \in \Omega$, with Robin-type boundary conditions $a\frac{\partial u}{\partial n} + bu = 0 \partial\Omega$.
Suppose that $f = C > 0$ in $D$ and $f = 0$ in $\Omega \setminus D$.
Is the following inequality true? $\int_D u\,dx > \int_{\Omega\setminus D} u\,dx$.
Note that $a>0, b>0, and D⊂Ω$. I tried to use conservation laws, and I suppose that my question can be solved if positive definiteness of the elliptic operator is used. Here, the problem is to avoid use of positive definiteness, or at least we can use that $(A^{−1}f,f)>0$ for $f≥0$.
Maybe I formulated the problem improperly because the problem is to show diagonal dominance of Gram matrix of the reaction-diffusion operator $A$ that is diagonal dominance of the matrix $(Af_i,f_j)$ for two given positive functions $f_1,f_2$. The question arises if we assume $f_1=C_1$ in $Ω_1$ and $f_2=C_2$ in $Ω_2$. Maybe it is necessary to reformulate the problem if some averaging over the domain is used.