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Say we have a function $F(\lambda) = ||f(\lambda)||_{H^{-1}}$ where $\lambda \in \mathbb{R}$ and $H^{-1}(\mathbb{R}^n)$ is the usual dual of the Sobolev space $H^{1}_{0}(\mathbb{R}^n)$. Suppose we are interested in minimizing the function $F$, then is the following justified: $$\partial_{\lambda} F = ||\partial_{\lambda} f||_{H^{-1}}$$ assuming $f$ smoothly depends on the parameter $\lambda?$

Edit [2]: Based on the comments, it does not make sense to take the norm of the derivative. How else might one proceed to minimize the $H^{-1}$ norm? As a toy example, let
$$f(\lambda) = \Delta u + \lambda u^p$$ for some $p>0, \lambda \in \mathbb{R}$ and $u\in H^{1}_0(\mathbb{R}^n).$ Then we can write, $$F(\lambda) = ||\Delta u + \lambda u^p||_{H^{-1}}.$$ For what value of $\lambda$ does $F(\lambda)$ achieve it's minimum?

Say we have a function $F(\lambda) = \|f(\lambda)\|_{H^{-1}}$ where $\lambda \in \mathbb{R}$ and $H^{-1}(\mathbb{R}^n)$ is the usual dual of the Sobolev space $H^{1}_{0}(\mathbb{R}^n)$. Suppose we are interested in minimizing the function $F$, then is the following justified: $$\partial_{\lambda} F = \|\partial_{\lambda} f\|_{H^{-1}}$$ assuming $f$ smoothly depends on the parameter $\lambda?$

Edit [2]: Based on the comments, it does not make sense to take the norm of the derivative. How else might one proceed to minimize the $H^{-1}$ norm? As a toy example, let
$$f(\lambda) = \Delta u + \lambda u^p$$ for some $p>0, \lambda \in \mathbb{R}$ and $u\in H^{1}_0(\mathbb{R}^n).$ Then we can write, $$F(\lambda) = \|\Delta u + \lambda u^p\|_{H^{-1}}.$$ For what value of $\lambda$ does $F(\lambda)$ achieve it's minimum?

Say we have a function $F(\lambda) = ||f(\lambda)||_{H^{-1}}$ where $\lambda \in \mathbb{R}$ and $H^{-1}(\mathbb{R}^n)$ is the usual dual of the Sobolev space $H^{1}_{0}(\mathbb{R}^n)$. Suppose we are interested in minimizing the function $F$, then is the following justified: $$\partial_{\lambda} F = ||\partial_{\lambda} f||_{H^{-1}}$$ assuming $f$ smoothly depends on the parameter $\lambda?$

Or do we need to use some form of min-max principle since $$\min_{\lambda \in \mathbb{R}}||f(\lambda)||_{H^{-1}} = \min_{\lambda \in \mathbb{R}}\max_{||\nabla g||_{L^2}\leq 1}\int f(\lambda) g = \max_{||\nabla g||_{L^2}\leq 1} \min_{\lambda \in \mathbb{R}}\int f(\lambda) g?$$

Edit: Based on the comment, I guess that taking the norm of the derivative is not justified. Perhaps we can use the definition of the derivative to get, $$F(\lambda + h) - F(\lambda) = ||f(\lambda + h)||_{H^{-1}} - ||f(\lambda)||_{H^{-1}}$$ but this does not lead to a general formula that one can use.

Say we have a function $F(\lambda) = ||f(\lambda)||_{H^{-1}}$ where $\lambda \in \mathbb{R}$ and $H^{-1}(\mathbb{R}^n)$ is the usual dual of the Sobolev space $H^{1}_{0}(\mathbb{R}^n)$. Suppose we are interested in minimizing the function $F$, then is the following justified: $$\partial_{\lambda} F = ||\partial_{\lambda} f||_{H^{-1}}$$ assuming $f$ smoothly depends on the parameter $\lambda?$

Edit [2]: Based on the comments, it does not make sense to take the norm of the derivative. How else might one proceed to minimize the $H^{-1}$ norm? As a toy example, let
$$f(\lambda) = \Delta u + \lambda u^p$$ for some $p>0, \lambda \in \mathbb{R}$ and $u\in H^{1}_0(\mathbb{R}^n).$ Then we can write, $$F(\lambda) = ||\Delta u + \lambda u^p||_{H^{-1}}.$$ For what value of $\lambda$ does $F(\lambda)$ achieve it's minimum?

Say we have a function $F(\lambda) = ||f(\lambda)||_{H^{-1}}$ where $\lambda \in \mathbb{R}$ and $H^{-1}(\mathbb{R}^n)$ is the usual dual of the Sobolev space $H^{1}_{0}(\mathbb{R}^n)$. Suppose we are interested in minimizing the function $F$, then is the following justified: $$\partial_{\lambda} F = ||\partial_{\lambda} f||_{H^{-1}}$$ assuming $f$ smoothly depends on the parameter $\lambda?$

Or do we need to use some form of min-max principle since $$\min_{\lambda \in \mathbb{R}}||f(\lambda)||_{H^{-1}} = \min_{\lambda \in \mathbb{R}}\max_{||\nabla g||_{L^2}\leq 1}\int f(\lambda) g = \max_{||\nabla g||_{L^2}\leq 1} \min_{\lambda \in \mathbb{R}}\int f(\lambda) g?$$

Say we have a function $F(\lambda) = ||f(\lambda)||_{H^{-1}}$ where $\lambda \in \mathbb{R}$ and $H^{-1}(\mathbb{R}^n)$ is the usual dual of the Sobolev space $H^{1}_{0}(\mathbb{R}^n)$. Suppose we are interested in minimizing the function $F$, then is the following justified: $$\partial_{\lambda} F = ||\partial_{\lambda} f||_{H^{-1}}$$ assuming $f$ smoothly depends on the parameter $\lambda?$

Or do we need to use some form of min-max principle since $$\min_{\lambda \in \mathbb{R}}||f(\lambda)||_{H^{-1}} = \min_{\lambda \in \mathbb{R}}\max_{||\nabla g||_{L^2}\leq 1}\int f(\lambda) g = \max_{||\nabla g||_{L^2}\leq 1} \min_{\lambda \in \mathbb{R}}\int f(\lambda) g?$$

Edit: Based on the comment, I guess that taking the norm of the derivative is not justified. Perhaps we can use the definition of the derivative to get, $$F(\lambda + h) - F(\lambda) = ||f(\lambda + h)||_{H^{-1}} - ||f(\lambda)||_{H^{-1}}$$ but this does not lead to a general formula that one can use.