Skip to main content
MathOverflow

Return to Question

Notice removed Draw attention by CommunityBot
Bounty Ended with no winning answer by CommunityBot
Notice added Draw attention by mnmn1993
Bounty Started worth 50 reputation by mnmn1993
deleted 4 characters in body
Source Link
mnmn1993
mnmn1993
  • 54
  • 1
  • 13

My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem.

Let $B=B_1 \subset \mathbb{R}^3$ and $E : \mathbb{R} \to \mathbb{R}$$E : H^1(B) \to \mathbb{R}$ $$E(u)= \int_{B}|\nabla u|^2+(u^2-1)^2 dx - \int_{\partial B}Q(u)d\mathcal{H}^2$$ We assume that $u_0 \in W^{1,2}$ to be the minimizer of the functional $E$ in the configuration space $$K=\{u\in W^{1,2}(B:\mathbb{R})\}.$$ Since $u_0$ is the critical point of the functional, we let $\xi \in K$, we obtain the equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \xi dx - \int_{\partial B }Q'(u)\xi d\mathcal{H}^2 = 0. $$ If we further require that $\xi$ vanishes on the boundary, we have the EL equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \,\xi dx = 0. $$ Suppose we also have that $u \in H^2(B)$, we have $$\int_B -\Delta u \, \xi + 4(u^2-1)u \xi dx + \int_{\partial B } \dfrac{\partial u}{\partial n}\xi- Q'(u)\xi d\mathcal{H}^2 = 0. $$ We finally obtain the equation $$\Delta u = 4(u^2-1)u \,\text{ in } B \,\text{ and }\, \dfrac{\partial u}{\partial n}=Q'(u) \,\text{ on }\, \partial B.$$

My main goal is to prove the minimizer $u_0$ solve the above equation weakly with the desried Neumann boundary condition. However, my question is how to obtain the $H^2$ bound of $u$? I think we can apply standard estimate to obtain $H^2_{loc}$. If we do not have the fact that $u \in H^2(B)$, we may hard to have the existence of $\dfrac{\partial u}{\partial n}$ on the boundary by trace theorem.

My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem.

Let $B=B_1 \subset \mathbb{R}^3$ and $E : \mathbb{R} \to \mathbb{R}$ $$E(u)= \int_{B}|\nabla u|^2+(u^2-1)^2 dx - \int_{\partial B}Q(u)d\mathcal{H}^2$$ We assume that $u_0 \in W^{1,2}$ to be the minimizer of the functional $E$ in the configuration space $$K=\{u\in W^{1,2}(B:\mathbb{R})\}.$$ Since $u_0$ is the critical point of the functional, we let $\xi \in K$, we obtain the equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \xi dx - \int_{\partial B }Q'(u)\xi d\mathcal{H}^2 = 0. $$ If we further require that $\xi$ vanishes on the boundary, we have the EL equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \,\xi dx = 0. $$ Suppose we also have that $u \in H^2(B)$, we have $$\int_B -\Delta u \, \xi + 4(u^2-1)u \xi dx + \int_{\partial B } \dfrac{\partial u}{\partial n}\xi- Q'(u)\xi d\mathcal{H}^2 = 0. $$ We finally obtain the equation $$\Delta u = 4(u^2-1)u \,\text{ in } B \,\text{ and }\, \dfrac{\partial u}{\partial n}=Q'(u) \,\text{ on }\, \partial B.$$

My main goal is to prove the minimizer $u_0$ solve the above equation weakly with the desried Neumann boundary condition. However, my question is how to obtain the $H^2$ bound of $u$? I think we can apply standard estimate to obtain $H^2_{loc}$. If we do not have the fact that $u \in H^2(B)$, we may hard to have the existence of $\dfrac{\partial u}{\partial n}$ on the boundary by trace theorem.

My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem.

Let $B=B_1 \subset \mathbb{R}^3$ and $E : H^1(B) \to \mathbb{R}$ $$E(u)= \int_{B}|\nabla u|^2+(u^2-1)^2 dx - \int_{\partial B}Q(u)d\mathcal{H}^2$$ We assume that $u_0 \in W^{1,2}$ to be the minimizer of the functional $E$ in the configuration space $$K=\{u\in W^{1,2}(B:\mathbb{R})\}.$$ Since $u_0$ is the critical point of the functional, we let $\xi \in K$, we obtain the equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \xi dx - \int_{\partial B }Q'(u)\xi d\mathcal{H}^2 = 0. $$ If we further require that $\xi$ vanishes on the boundary, we have the EL equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \,\xi dx = 0. $$ Suppose we also have that $u \in H^2(B)$, we have $$\int_B -\Delta u \, \xi + 4(u^2-1)u \xi dx + \int_{\partial B } \dfrac{\partial u}{\partial n}\xi- Q'(u)\xi d\mathcal{H}^2 = 0. $$ We finally obtain the equation $$\Delta u = 4(u^2-1)u \,\text{ in } B \,\text{ and }\, \dfrac{\partial u}{\partial n}=Q'(u) \,\text{ on }\, \partial B.$$

My main goal is to prove the minimizer $u_0$ solve the above equation weakly with the desried Neumann boundary condition. However, my question is how to obtain the $H^2$ bound of $u$? I think we can apply standard estimate to obtain $H^2_{loc}$. If we do not have the fact that $u \in H^2(B)$, we may hard to have the existence of $\dfrac{\partial u}{\partial n}$ on the boundary by trace theorem.

Source Link
mnmn1993
mnmn1993
  • 54
  • 1
  • 13

Construction of elliptic equation with Neumann boundary condition from a minimization problem

My question mainly concern about how to construct a elliptic equation with Neumann boundary condition from a minimization problem.

Let $B=B_1 \subset \mathbb{R}^3$ and $E : \mathbb{R} \to \mathbb{R}$ $$E(u)= \int_{B}|\nabla u|^2+(u^2-1)^2 dx - \int_{\partial B}Q(u)d\mathcal{H}^2$$ We assume that $u_0 \in W^{1,2}$ to be the minimizer of the functional $E$ in the configuration space $$K=\{u\in W^{1,2}(B:\mathbb{R})\}.$$ Since $u_0$ is the critical point of the functional, we let $\xi \in K$, we obtain the equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \xi dx - \int_{\partial B }Q'(u)\xi d\mathcal{H}^2 = 0. $$ If we further require that $\xi$ vanishes on the boundary, we have the EL equation $$\int_B \nabla u \cdot \nabla \xi + 4(u^2-1)u \,\xi dx = 0. $$ Suppose we also have that $u \in H^2(B)$, we have $$\int_B -\Delta u \, \xi + 4(u^2-1)u \xi dx + \int_{\partial B } \dfrac{\partial u}{\partial n}\xi- Q'(u)\xi d\mathcal{H}^2 = 0. $$ We finally obtain the equation $$\Delta u = 4(u^2-1)u \,\text{ in } B \,\text{ and }\, \dfrac{\partial u}{\partial n}=Q'(u) \,\text{ on }\, \partial B.$$

My main goal is to prove the minimizer $u_0$ solve the above equation weakly with the desried Neumann boundary condition. However, my question is how to obtain the $H^2$ bound of $u$? I think we can apply standard estimate to obtain $H^2_{loc}$. If we do not have the fact that $u \in H^2(B)$, we may hard to have the existence of $\dfrac{\partial u}{\partial n}$ on the boundary by trace theorem.