2
votes
1answer
28 views

bilinear continuous, coercive form

Let $k\in \mathbb{R}, k\neq 1$, consider the space $$ V = \{u\in H^1(0,1): u(0) = ku(1)\}$$ Let $$a(u,v) = \int_0^1 (u'v'+ uv)\; dx - \left(\int_0^1 u\; dx\right) \left(\int_0^1 v\; ...
0
votes
1answer
76 views

Fundamental lemma of calculus of variations, gradients

Let $D \subset \mathbb{R}^d$ be a smooth bounded domain. Let $C_c^\infty(D)$ denote smooth and compactly supported functions on $D$. Let $f \in [C_c^\infty(D)]^d$ be a smooth, compactly supported ...
1
vote
1answer
48 views

Convergence of integral means of the gradient of a Sobolev function

Let $B_R(x_0)\subset\mathbb{R}^n$ with $R<1$ for $n\geq3$ and suppose $u\in H^1\big(B_R(x_0);\mathbb{R}^N\big)\cap L^{\infty}\big(B_R(x_0)\big)$. Define, \begin{equation} \phi(R)\equiv ...
2
votes
2answers
105 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...
4
votes
1answer
50 views

Existence of variation

Let $I[w] =\int_U L(Dw,w,x) dx$. Let $1<q<\infty$, and there exist constants $\alpha>0$,$\beta\ge0$ such that $$L(p,z,x)\ge \alpha |p|^q - \beta$$ This implies that if $I[w]$ exists, $$I[w] ...
1
vote
1answer
43 views

Some ideas about $H=W$

Meyers and Serrins theorem says that $H=W$. ie $H^j_p(\Omega) = W_p^j(\Omega)$ . Here the norm of $$\|u\|_{H^j_p(\Omega)} = (\int_\Omega\sum_{|\alpha|\le j}|D^\alpha u|^pdx )^{1/p}$$ where ...
1
vote
0answers
60 views

How can I integrate this?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $\phi_1,v,\phi\in W_0^{1,p}(\Omega)$ with $p\in (1,\infty)$. How can I evaluate the integral: $$\int_0^1F(s)ds$$ where ...
1
vote
0answers
32 views

Minimum is attained in a subset of a Sobolev space

Let $\Omega \subset \mathbb R^n$. I have a functional of the form, $$\int_{\Omega}f(x,u,\nabla u)dx$$ where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...
5
votes
1answer
136 views

Smoothness of a non-local functional

While studying a nonlinear PDE arising from quantum mechanics, I met a statement that I cannot prove easily. Let us write $E=W^{1,2}(\mathbb{R}^3)$ for the usual Sobolev space, and define the ...
2
votes
0answers
79 views

Extension of Uncertainty Relations to a specific potential in Schrödinger Equation

Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle ...
1
vote
2answers
205 views

Weak lower semicontinuity of a functional on Hilbert space?

Let $H:=\left\{u\in L^2(R^N):\nabla u \in L^2(R^N)\right\}$ and a functional $$f(u)=\int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2.$$ If $\{u_n\}\subset H$ is a sequence such that ...