0
votes
0answers
9 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
4
votes
1answer
36 views

First eigenvalue of Laplacian and Poincaré inequality

Any idea on how to solve: $\int_{\Omega} |\nabla u|^2 d^n x=\lambda_1\int_{\Omega}u^2 d^n x$, with $u\in H^1_0(\Omega)$ and $\Omega\subset\mathbb{R}^n$, and $\lambda_1$ the first eigenvalue of the ...
1
vote
1answer
25 views

Extension of a function from the edge.

How can I extend the function $f\in W^{1-\frac{1}{p},p}(\mathbb R\times\lbrace0\rbrace)$ up to $g\in W^{1,p}(\mathbb R\times(0,\infty))$?
1
vote
1answer
32 views

Does $C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})=C^1([0,T];H^{s+1})$?

I have asked this question with other problems, but about this part nobody answers. So I want to ask again, and put some details in it. My question is whether the following equality is correct. If it ...
0
votes
1answer
29 views

Canonical Separation of variables

Do the functions of the form $\psi(x)\phi(y)$ span $L^2(\mathbf{R}^6)$? Insert proper grammar here.
1
vote
0answers
37 views

Estimate difference of solutions of an equation with a bilinear symmetric continuous form

My question refers to differential equations in Sobolev spaces. It is as follows: Let $\Omega \subset \mathbb{R}^n$ be a bounded open set. Let $a: H_0^1(\Omega) \times H_0^1(\Omega) \rightarrow ...
1
vote
1answer
68 views

Establishing a relationship between weak solution in $L_2(\Omega)$ and weak solution in $W^{1,\ 2}(\Omega)$ with classical solution

I have a problem: For $\Omega$ be a bounded domain in $\Bbb R^n$. Denote $L_{2,\ 0}(\Omega)=\overline{C_{0}^{\infty}(\Omega)}$, with norm in $L_2(\Omega)$. We consider: ...
1
vote
1answer
153 views

Definition of weak solution in $W^{1,2}(\Omega)$.

I have a problem: For $\Omega$ be a bounded domain in $\Bbb R^n$. We consider $$\left\{\begin{matrix} \Delta u-\lambda u =f \ \rm in \ \Omega & \\ u\mid_{\partial {\Omega}} =0 ...
2
votes
0answers
72 views

(Sobolev space) A star domain in $\Bbb R^n$

We know the theorem: If $\Omega$ is a star domain in $\Bbb R^n$, then $C^{\infty}(\overline{\Omega})$ is dense in $W^{k,p}(\Omega)$. It means that ...
1
vote
1answer
293 views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
1
vote
2answers
69 views

Book searching in Elliptic Equation

I am learning a course with the subject of Elliptic Equations. If you know about it, please recommend me a book on Elliptic Equations. And if that's possible, someone post these books/author/...that ...
0
votes
1answer
61 views

$(u_n)$ is bounded in $H^1(\mathbb{R}^N)$, some results about the convergence.

If $(u_n)$ is bounded in the Hilbert space $H^1(\mathbb{R}^N)$, we have that, up to a subsequence, \begin{eqnarray} &&u_n \rightharpoonup u\ \mbox{ weakly in }H^1(\mathbb{R}^N),\\ ...
3
votes
1answer
160 views

Equivalent norm in sobolev space H^2

I consider space $H^{2}(0,a)=\{ f\in L^{2}(0,a): f',f''\in L^{2}(0,a) \}$ I define norm $\Vert w \Vert_{H^{2}}:=b\Vert w''\Vert_{L^{2}}$, where b is positive constant. I couldn't proof that it is ...
3
votes
1answer
92 views

EDIT: Estimation of solution of variational problem

We consider the problem $$-u''+a(x)u=f, x\in (0,1)=\Omega , u(0)=\alpha,u(1)=\beta$$ with $f \in L^2(\Omega) , a(x) \geq a_0 > 0, a \in L^{\infty}(\Omega)$ Question 1: prove that the variational ...
3
votes
2answers
102 views

Finding a strong enough solution to a specific PDE problem.

Let $U\subset \mathbb{R}^n$ with smooth boundary $\partial U$. And consider the expression $$\Delta u = f.$$ $$\text{+"convenient boundary conditions"}$$ In my specific case $f\in H^2_0$. Under ...
2
votes
1answer
185 views

Estimating Poincare constant for unit interval

I want to show that the Poincare constant for the $W^{1,2}_0(0,1)$ is smaller than $1$. More specifically, I want to show that there is a constant $C<1$ such that for any $f\in C^\infty_c(0,1)$ ...
1
vote
0answers
79 views

Solving a Sturm-Liouville differential equation variationally

This is a problem from Haim Brezis' functional analysis book (Exercise 8.41). I solved parts of it, but am stuck on some parts/want confirmation on the method. The problem is as follows: Let $q ...
1
vote
1answer
81 views

Critical exponent in O.D.E and Gâteaux derivative

I have this example and I don't understand its resolution: Let $\Omega \subset \mathbb{R}^n , n\geq3$ be a bounded open set (with smooth boundary), let $f\colon\mathbb{R}\rightarrow \mathbb{R}$ be ...
0
votes
1answer
84 views

second derivative of solutions to ODE with Lipschitz coefficients

Just the simple ODE with Lipschitz coefficient $a$ \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} We know that the existence and uniqueness holds and the solution is in $C^1$. ...
4
votes
2answers
116 views

How to define difference quotients for Sobolev functions

There is a definition for difference quotients of Sobolev functions I do not understand. Let $U\subset\Omega\subset\mathbb{R}^n$ be open sets such that the closure of $U$ is compact in $\Omega$ and ...
2
votes
0answers
74 views

ODE with irregular coefficient

Here's a simple ODE \begin{align} &\frac{d}{dx}h(x)=a(h(x))\\ &h(0)=x_0 \end{align} I want the solution $h(x)$ to be (at least) continuous with its first and second order derivative exist only ...
0
votes
1answer
136 views

Norm and invertibility of operator $\left(-\Delta+\lambda I\right)$ with $\lambda>0$.

Let $\lambda>0$ and $n\geq 1$. Prove that the operator $$-\Delta+\lambda I:H^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$$ is invertible and find the norm $$\left|\left|\left(-\Delta+\lambda ...
1
vote
0answers
157 views

Application of method of continuity in partial differential equations

Consider a differential operator $$L_t:= (1-t)(\Delta-\lambda) + t L,\qquad t\in[0,1].$$ For any $u\in C^2_0(\mathbb{R}^2)$, we have $$\lambda^2 \|u\|_2^2 + 2\lambda\sum_{i}\|u_i\|_2^2 + ...
1
vote
2answers
111 views

What is 'imbedding' with Sobolev space and $ L^2 $ space?

I want to know that the meaning of the following. $$ W^{n,1}\textrm{ is continuously imbedded into }L^2$$ Here, $W^{n,1}$ is a Sobolev space.
4
votes
1answer
326 views

The Sobolev Spaces $H_0^s(\Omega)$

Let $\bar{\Omega}$ be a smooth, compact manifold with boundary; we denote the interior by $\Omega$. We can suppose $\bar{\Omega}$ is contained in a compact, smooth manifold $M$, with $\partial\Omega$ ...
1
vote
2answers
206 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 ...