I am considering a question in the calculus of variations. I can understand the concept of the variational derivative, but I am not sure what the $\delta u$ in this question means:
"6. Let $Ω⊆R^n$ be a connected and relatively compact open set with smooth boundary. In this problem we study the gradient flow of the functional $$J[u]=∫_{Ω} {\frac{1}{2}|∇u|^2+F(u)}dVol$$ where $F(u)= \frac{1}{4} (u^2-1)^2$, subject to the mean-zero constraint $$∫_{Ω}u(x,t)dVol(x) =0 \space ∀t$$ We will show, following [2] and [3], that accomodating the mean-zero constraint requires modifying the duality pairing you used to define the variational derivative in class. Compute the variational derivative of J[u] using the expression from the lecture notes: $$⟨δJ/δu,δu⟩_{L^2}=DJ(u)∙δu \space ∀δu∈C_{c}^∞ (Ω)$$ where DJ(u) denotes the Fréchet derivative of J[u] at u. Explain why the gradient flow $$u_t=-δJ/δu$$ does not in general satisfy the mean-zero constraint, even if u satisfies a homogeneous Neumann condition. Do you recognize the gradient flow PDE?"