Articles of sobolev spaces

How to prove $\int_{\Omega}\frac{1}{\kappa}uv\ +\ \int_{\Omega}\kappa\nabla u\cdot\nabla v$ is an inner product in $H^1$

I need help with this problem from my homework Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there’s constants $M, \beta > 0$, such that $\beta \leq \kappa(x) \leq M$ for each $x\in\Omega$. Now, consider $$\langle u, v\rangle_{\kappa}\ :=\ \int_{\Omega}\frac{1}{\kappa}uv\ +\ \int_{\Omega}\kappa\nabla u\cdot\nabla […]

sobolev spaces – product of two functions

I am working in a exercise, to my solution works I need the following affirmation is true: Let $\varphi : R \rightarrow R$ a convex and smooth function. Let $u \in H^{1}(U)$ a bounded function and $v \in H^{1}_{0}(U)$ a non negative function. Then $\varphi^{‘}(u)v \in H^{1}_{0}(U)$. I am trying to use the definiton , […]

Estimate $L^{2p}$ norm of the gradient by the supremum of the function and $L^p$ norm of the Hessian

Prove $$\|Du\|_{L^{2p}(U)} \le C\|u\|_{{L^\infty}(U)}^{1/2} \|D^2 u\|_{L^p(U)}^{1/2}$$ for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise 10(b). Here is what I did so far: \begin{align} \int_U |Du|^{2p} \, dx &= \sum_{n=1}^\infty \int_U u_{x_i} |Du|^{2p-1} \, dx \\ &= -\sum_{i=1}^n \int_U u(x_i)(2p-1)|Du|^{2p-2} |D^2 u| \, dx […]

Weak derivative in Sobolev spaces

A function $u: \Bbb R \longrightarrow \Bbb R$ is weakly differentiable with weak derivative $v$ if there exists a function $v: \Bbb R \longrightarrow \Bbb R$ such that $$\int_{-\infty}^\infty u \phi’ ~dx = – \int_{-\infty}^\infty v \phi ~dx$$ for all smooth functions $\phi: \Bbb R \longrightarrow \Bbb R$ that vanish outside some bounded set. Functions […]

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u’v_0’+uv_0), \forall u \in H^1(0,1)$. Further Show that $v_0$ is the solution of some differential equation with appropriate boundary conditions. Compute $v_0$ explicitly. Let $f: H^1(0,1) \to \mathbb{R}$ be defined by $f(u)=u(0)$. Then I showed that $f$ is linear and continuous. Hence there exists […]

Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?

I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution $$v=\Gamma\star f$$ They claim that $v\in W^{1.p}(\mathbb{R}^n)$. Does anyone knows why this is true with $f$ being only in $W^{-1,p}(\mathbb{R}^n)$? Update: Maybe this can thrown some light upon the matter. […]

Composition of a weakly convergent sequence with a nonlinear function

Let $\Omega\subset\mathbb{R}^n $ be bounded smooth domain. Given a sequence $u_m$ in Sobolev space $H=\left \{v\in H^2(\Omega ):\frac{\partial v}{\partial n}=0 \text{ on } \partial \Omega \right \}$ such that $u_m$ is uniformly bounded i.e. $\|u_m\|_{H^2}\leq M$ and given the function $f(u)=u^3-u$. If I know that $u_m\rightharpoonup u(u\in H)$ in $L^2$ sense i.e. $\int_{\Omega}u_m v\to \int_{\Omega}u […]

A bounded sequence

I have a question please : Let $f:[0,2\pi]\times \mathbb{R} \rightarrow \mathbb{R}$ a differential function satisfying : $\displaystyle k^2\leq \liminf_{|x|\rightarrow \infty} \frac{f(t,x)}{x}\leq \limsup_{|x|\rightarrow \infty}\frac{f(t,x)}{x} \leq (k+1)^2$ Let $(x_n)\subset H^1([0,2\pi],\mathbb{R})=\lbrace x\in L^2([0,2\pi],\mathbb{R}),x’\in L^2([0,2\pi],\mathbb{R}),x(0)=x(2\pi)\rbrace$ sucht that $\|x_n\|\rightarrow \infty$ when $n \rightarrow \infty$ Why : the sequence $\left(\displaystyle\frac{f(t,x_n)-k^2 x_n}{\|x_n\|}\right)$ is bounded ? Is this answer given by :@TZakrevskiy true […]

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem

Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} \subset U$ with $\overline{V}$ compact). My problem is to show that $$\|f\|_{W^{m,p}(U)}\leq\delta\tag{2}.$$ Evans book (p. 252) says that $(2)$ is obtained by taking the supremum in $(1)$ over sets $V\subset\subset […]

weak derivative of a nondifferentiable function

I am reading a book on Sobolev and having trouble understanding a notion of weak derivative. I consider a function $(x-1)^+=\max(x-1,0),x\in[0,2]$, I have a problem at $x=1$, so it is continuous and “almost everywhere differentiable” now, does it have a weak derivative everywhere? Can we “quantify” that? I know that the derivative in classical sense […]