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I am working on an energy functional arising from an inhomogeneous Neumann Problem. The boundary term makes it difficult to compare the energies while truncating. Concretely,

Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c \in \mathbb{R}$. Suppose for any $\epsilon >0$ there exists a set $A_{\epsilon} \subset U$ with $f> c+\epsilon $ a.e. on $A_{\epsilon}$ and $m(U-A_{\epsilon})<\epsilon. $ Define $$g(x)= \max(f(x),c+\epsilon) \quad \text{for} \,\,x \,\, \text{in} \,\, U.$$

As $f$ and $g$ are in $H^1(U)$, their traces $T(f)$ and $T(g)$ are in $H^{\frac{1}{2}} (\partial U)$. We know $g \geq f$ in $U$. Thus, $T(g)\geq T(f) $.

But, can we characterize when does for any $h \in H^1(U)$, we have $$\int_{\partial U} T(g) T(h) \leq \int_{\partial U} T(f) T(h)? \tag{1}\label{1}$$

This boundary term in a Neumann Energy Functional typically competes with terms like the $H^1$ norm. So, while $||g||_{H^1} \leq ||f||_{H^1}$ for $c+\epsilon<0$, is there a way to conclude $\eqref{1}$, or to show that the overall energy decreases for $g$?

I am working on an energy functional arising from an inhomogeneous Neumann Problem. The boundary term makes it difficult to compare the energies while truncating. Concretely,

Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c<0$. Suppose for any $\epsilon >0$ there exists a set $A_{\epsilon} \subset U$ with $f> c+\epsilon $ a.e. on $A_{\epsilon}$ and $m(U-A_{\epsilon})<\epsilon. $ Define $$g(x)= \max(f(x),c+\epsilon) \quad \text{for} \,\,x \,\, \text{in} \,\, U.$$

As $f$ and $g$ are in $H^1(U)$, their traces $T(f)$ and $T(g)$ are in $H^{\frac{1}{2}} (\partial U)$. We know $g \geq f$ in $U$. Thus, $T(g)\geq T(f) $.

But, can we characterize when does for any $h \in H^1(U)$, we have $$\int_{\partial U} T(g) T(h) \leq \int_{\partial U} T(f) T(h)? \tag{1}\label{1}$$

This boundary term in a Neumann Energy Functional typically competes with terms like the $H^1$ norm. So, while $||g||_{H^1} \leq ||f||_{H^1}$ for $c+\epsilon<0$, is there a way to conclude $\eqref{1}$, or to show that the overall energy decreases for $g$?

I am working on an energy functional arising from an inhomogeneous Neumann Problem. The boundary term makes it difficult to compare the energies while truncating. Concretely,

Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c \in \mathbb{R}$. Suppose for any $\epsilon >0$ there exists a set $A_{\epsilon} \subset U$ with $f> c+\epsilon $ a.e. on $A_{\epsilon}$ and $m(U-A_{\epsilon})<\epsilon. $ Define $$g(x)= \max(f(x),c+\epsilon) \quad \text{for} \,\,x \,\, \text{in} \,\, U.$$

As $f$ and $g$ are in $H^1(U)$, their traces $T(f)$ and $T(g)$ are in $H^{\frac{1}{2}} (\partial U)$. We know $g \geq f$ in $U$. Can we infer $T(g)\geq T(f) $ immediately?

If not, are there any mild additional regularity assumptions on $f$ that would allow us to compare these traces? Moreover, can we characterize when does for any $h \in H^1(U)$, we have $$\int_{\partial U} T(g) T(h) \geq \int_{\partial U} T(f) T(h)?$$

I am working on an energy functional arising from an inhomogeneous Neumann Problem. The boundary term makes it difficult to compare the energies while truncating. Concretely,

Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c \in \mathbb{R}$. Suppose for any $\epsilon >0$ there exists a set $A_{\epsilon} \subset U$ with $f> c+\epsilon $ a.e. on $A_{\epsilon}$ and $m(U-A_{\epsilon})<\epsilon. $ Define $$g(x)= \max(f(x),c+\epsilon) \quad \text{for} \,\,x \,\, \text{in} \,\, U.$$

As $f$ and $g$ are in $H^1(U)$, their traces $T(f)$ and $T(g)$ are in $H^{\frac{1}{2}} (\partial U)$. We know $g \geq f$ in $U$. Thus, $T(g)\geq T(f) $.

But, can we characterize when does for any $h \in H^1(U)$, we have $$\int_{\partial U} T(g) T(h) \leq \int_{\partial U} T(f) T(h)? \tag{1}\label{1}$$

This boundary term in a Neumann Energy Functional typically competes with terms like the $H^1$ norm. So, while $||g||_{H^1} \leq ||f||_{H^1}$ for $c+\epsilon<0$, is there a way to conclude $\eqref{1}$, or to show that the overall energy decreases for $g$?

I am working on an energy functional arising from an inhomogeneous Neumann Problem. The boundary term makes it difficult to compare the energies while truncating. Concretely,

Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c \in \mathbb{R}$. Suppose for any $\epsilon >0$ there exists a set $A_{\epsilon} \subset U$ with $f> c+\epsilon $ a.e. on $A_{\epsilon}$ and $m(U-A_{\epsilon})<\epsilon. $ Define $$g(x)= \max(f(x),c+\epsilon) \quad \text{for} \,\,x \,\, \text{in} \,\, U.$$

As $f$ and $g$ are in $H^1(U)$, their traces $T(f)$ and $T(g)$ are in $H^{\frac{1}{2}} (\partial U)$. We know $g \geq f$ in $U$. However, the trace operator is not monotone, so we can't compare them immediately.

Are there any mild additional regularity assumptions on $f$ that would allow us to compare these traces? Moreover, can we characterize when does for any $h \in H^1(U)$, we have $$\int_{\partial U} T(g) T(h) \geq \int_{\partial U} T(f) T(h)?$$

I am working on an energy functional arising from an inhomogeneous Neumann Problem. The boundary term makes it difficult to compare the energies while truncating. Concretely,

Suppose $U$ is an open bounded set in $\mathbb{R}^n$ ($n \geq 3)$ with Lipschitz boundary. Let $f \in H^1(U)$ with $f>c$ a.e in $U$ for some $c \in \mathbb{R}$. Suppose for any $\epsilon >0$ there exists a set $A_{\epsilon} \subset U$ with $f> c+\epsilon $ a.e. on $A_{\epsilon}$ and $m(U-A_{\epsilon})<\epsilon. $ Define $$g(x)= \max(f(x),c+\epsilon) \quad \text{for} \,\,x \,\, \text{in} \,\, U.$$

As $f$ and $g$ are in $H^1(U)$, their traces $T(f)$ and $T(g)$ are in $H^{\frac{1}{2}} (\partial U)$. We know $g \geq f$ in $U$. Can we infer $T(g)\geq T(f) $ immediately?

If not, are there any mild additional regularity assumptions on $f$ that would allow us to compare these traces? Moreover, can we characterize when does for any $h \in H^1(U)$, we have $$\int_{\partial U} T(g) T(h) \geq \int_{\partial U} T(f) T(h)?$$