0
votes
1answer
21 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
0
votes
1answer
24 views

Application of Sobolev inequality

I saw this inequality without knowing how to justify it. This is probably an application of Sobolev inequality (or not). Here it is: $u,v\in H^1(\mathbb{R}^d)$, then ...
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}, ...
2
votes
1answer
64 views

Proving $u\mapsto |u|^2u$ is Lipschitz on bounded subsets of $H^2(\Omega)\cap H_0^1(\Omega).$

I'm reading a paper and am stumped verifying two details. Let $\Omega$ be a bounded region in $\mathbb{R}^2$ with smooth boundary. I'd like to show that the map $u\mapsto |u|^2u$ is a map from ...
2
votes
2answers
66 views

Variational methods : Why i can't apply this theorem?

Consider the following problem: Find a weak solution for $$u'' + u = \sin t ,\,\, 0 < t < \pi$$ $$u(0) = u(\pi)=0 $$ the corresponding functional for the problem is $\varphi(u) = ...
1
vote
0answers
46 views

Extension of function from $W^{1,p}(\Omega - \{x\})$ to $W^{1,p}(\Omega)$

Suppose $\Omega\subset \mathbb{R}^n$ is open, $p\geq 1$, $n\geq 2$ and $u \in W^{1,p}(\Omega-\{x\})$. Show that $u$ extends to a function in $W^{1,p}(\Omega)$. So far I have; clearly $u$ and each ...
2
votes
0answers
47 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
2
votes
0answers
21 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
2
votes
0answers
42 views
+50

How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
0
votes
1answer
25 views

Leibniz's rule in $W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$

Is it true? If $\Omega\subset\mathbb{R}^n$ is bounded and $u,v\in W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$, then $uv\in W^{1,\,p}(\Omega)\cap L^\infty(\Omega)$ and $$\nabla(uv)=u\nabla v+v\nabla u.$$ ...
2
votes
1answer
33 views

Show that this simple functional is not bounded below

Define $\varphi(u) = \displaystyle\int_{0}^{1} \displaystyle\frac{{|u'| }^2}{2} - \displaystyle\frac{{u }^2}{4} - hu \ dt$, $u \in H^{1}_{0}(0,1)$ where $h: [0,1] \rightarrow R$ is a continuous ...
1
vote
1answer
64 views

Imbedding theorem for Sobolev space $D^{k,p}$

I can't find any reference on the imbedding theorem for the Sobolev space $D^{k,p}$, which is defined by $$D^{k,p}(\Omega)=\{u\in L_{loc}^{1}(\Omega)|D^{k} u\in L^{p}(\Omega)\}.$$ Is it the same as ...
2
votes
2answers
60 views

dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
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
0answers
47 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
0
votes
0answers
14 views

Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
0
votes
0answers
16 views

A parabolic maximum principle (if initial value is bounded, so is solution)?

Let $u \in L^2(0,T;H^1)$ with $u_t \in L^2(0,T;L^2)$ solve the PDE $$u_t - \Delta u = f$$ $$u(0)= u_0$$ on $\Omega \times (0,T)$ where $\Omega$ is a bounded domain. We do NOT have the Poincare ...
2
votes
1answer
71 views

Why is $-\Delta+1$ an isomorphism between Sobolev space $W^{2,p}$ and $L^p\,$?

Consider the linear elliptic operator $L: W^{2,p}(\mathbb{R}^N)\to L^p(\mathbb{R}^N)$ defined by $Lu=-\Delta u + u$. Can we prove that $L$ is an isomorphism for all $p\geq 1$? It can be proved that it ...
0
votes
0answers
25 views

Boundary value problem for two functions

The question is: Let $\mathcal{H}=H_0^1(\Omega)\times H^1(\Omega)$ and consider the solution $(u,v)\in\mathcal{H}$ to the differential problem \begin{equation} -\Delta u=f+a(v-u)\quad\text{in }\Omega ...
1
vote
1answer
42 views

Definition of fractional Laplacian on a compact manifold?

How does one define the fractional Laplacian operator $(-\Delta)^s$ on a compact Riemannian manifold? In $\mathbb{R}^n$, it is defined $$ (-\Delta)^s f(x) = c_{n,s} \int_{\mathbb{R}^n} \frac{f(x) - ...
2
votes
1answer
19 views

how to prove that this weak solution is subharmonic?

My question is about this article http://hal.inria.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf. My question is : Consider a smooth, bounded and convex domain $K$ in $R^n$ such that $K\subset \{ x_1 = ...
0
votes
0answers
23 views

Why do the following regularisations of a function in Sobolev space exist?

Suppose $v_1, v_2$ satisfy $\mu \leq v_1(x,t), v_2(x,t) \leq M$ a.e. in $Q:=\Omega\times(0,T)$ and $$(v_1, \eta_1) \quad\text{and} \quad (v_2, \eta_2) \in L^2(0,T;H^1(\Omega)) \cap L^2(Q).$$ Define ...
0
votes
1answer
33 views

Isomorphism between Sobolev space and L^p

Let $L_1$ be an elliptic PDE operator $L_1:W^{2,p}\rightarrow L^p$ and $L_2=e^fL_1$ where f is a bounded function. I proved $L_1$ is an isomorphism, can I conclude $L_2$ is an isomorphism?
1
vote
1answer
34 views

minimizes the functional to solve a pde

I am trying to do this exercise: Let $\Omega$ a open bounded domain in $R^n$. Consider the Dirichlet problem $$ \left\{ \begin{array}{ccccccc} -\Delta u = \lambda \sin (u) + f , \ \text{in ...
2
votes
0answers
28 views

$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
-1
votes
1answer
27 views

Show that this functional is coercive - variational methods

For $u \in H^{1}_0({\Omega})$ ($\Omega$ is a domain open and bounded in $R^n$). Let $0 < \lambda < \lambda_1$ ($\lambda_1$ is the first eigen value of the laplacean) and a fixed $f \in ...
1
vote
1answer
27 views

Does $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ give something useful?

If $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ is there any way to extract a strongly convergent subsequence of $u_n$ in some useful space for PDEs? Or ...
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 ...
2
votes
0answers
31 views

$f_n \rightharpoonup f$ in $L^q(Q)$ $\forall q < \infty$ and $f_n' \rightharpoonup f'$ in $L^2(0,T;H^{-1})$ implies $f_n \to f$

(... in $C^0([0,T]; H^{-1})$. ) Let $f_n$ be a sequence of functions defined on $Q:=(0,T)\times \Omega$, where $\Omega$ is a bounded domain. I have read this: Since $f_n \rightharpoonup f$ in ...
0
votes
1answer
22 views

Reference needed for: $u \in H^1(0,T;L^2)$ if and only if $\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$

There is a result of the form: a function $u \in H^1(0,T;L^2)$ if and only if $$\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$$ holds for all $h \in [0,T]$. I have only seen one place ...
3
votes
1answer
63 views

Differences between $-\Delta: H_0^1(\Omega)\to H^{-1}(\Omega)$ and $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$

I'll try to explain what I want to know: Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. When we look to $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$, the meaning of $-\Delta$ is ...
1
vote
1answer
58 views

Getting the bound $\frac{1}{h}\int_0^{T-h}\int_t^{t+h}\int_\Omega |\nabla u(\tau)| |\nabla u(t+h) - \nabla u(t)|\;dxd\tau dt \leq C$

Let $u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega)).$ Is it possible to find the following bound: $$\frac{1}{h}\int_0^{T-h}\int_t^{t+h}\int_\Omega |\nabla u(\tau,x)| |\nabla u(t+h,x) - ...
1
vote
2answers
31 views

Is it possible to estimate $| u |^2_{H^1}$ by $|u|_{H^2}$ for bounded functions?

Let $u\in [L^\infty(\Omega)]^m \cap [H^2(\Omega)]^m$ be a vector valued function with bounded $\Omega \subset \mathbb{R}^n$. Moreover, let $\|u\|_{L^\infty} \leq 1$. Is it possible to bound the square ...
0
votes
0answers
57 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
0
votes
1answer
31 views

What is the usual norm of $\mathbb{H_0^1}$?

What is the usual norm of product of sobolev spaces $\mathbb{H_0^1}=H_0^1 \times H_0^1=W^{1,2}\times W^{1,2}$? In my work i need to prove that the norm endowed by the inner product ...
2
votes
1answer
44 views

Closure of partial differential operators on $L^2(\Omega)$

Let $\Omega:=\mathbb{R}^2\setminus\{0\}$. Consider the $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$ $$ H=-\partial_x^2-\partial_y^2+ ...
0
votes
1answer
22 views

About an estimate of theorem 3 in Chapter 12 of Evans' book

This is the proof of theorem 3 in Chapter 12 of Evans' book as the following picture. I really don't understand why $|F(Du,u_t,u)|\le C(|Du|+|u_t|+|u|)$, because he didn't give us any restriction on ...
1
vote
1answer
24 views

On defining appropriate energy. Any principle?

I am reading Evans' book Partial differential equations. but I am really curious about how he define the appropriate energy? Is there any principle or rule to do this things? Because I notice that ...
1
vote
0answers
29 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times ...
1
vote
1answer
49 views

Don't understand proof of a PDE argument, author uses $w(t,\cdot) \in L^6(\Omega)$ when $w(t,\cdot) \in H^1(\Omega)$.

I'm reading this paper. I attach the details below(you don't have to read it all to understand my question). My question is, $w$ is only assumed to be $H^1$ in space, why then in the manipulations ...
1
vote
0answers
43 views

implications of convergence in sobolev spaces

If we are given that $O \subset \Omega$ is open and bounded and $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$. We have a sequence $\{u_{m}\}$ satisfying $$ u_{m} ...
1
vote
1answer
31 views

Is there any Banach space $X$ that $L^2(\Omega)$ is compactly embedded into?

Let $\Omega \subset \mathbb{R}^n$. Is there a good (*) Banach space $X$ that $L^2(\Omega)$ is compactly embedded into: $$L^2(\Omega) \subset\!\subset X$$? If not compactly embedding, I at least would ...
0
votes
1answer
44 views

Bounding $\int_0^T\int_\Omega v|\nabla u|^2$ given that $v \in L^2(0,T;L^2(\Omega))$ and $u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega))$?

If $v \in L^2(0,T;L^2(\Omega))$ and $$u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega)),\tag{1}$$ is possible to get a bound on the integral $$\int_0^T\int_\Omega v|\nabla u|^2$$ of the form ...
3
votes
0answers
34 views

If $u_m \rightharpoonup u$, how to show using monotonicity that $f(u_m) \rightharpoonup f(u)$?

Let $$u_m \rightharpoonup u \quad \text{(weakly) in $L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$}.$$ We are given $f:\mathbb R \to \mathbb R$, a Lipschitz continuous invertible map which is ...
2
votes
1answer
67 views

The dual space of the Sobolev space $H_0^1$

I am slightly confused about the properties of the dual space of the Sobolev space $H_0^1$ as outlined on page 299 in Evans. In particular, following the notation in the book, item 3 says that ...
0
votes
1answer
21 views

Is $[L^2(\Omega), H^2(\Omega)]_{\frac 1 2}=H^1(\Omega)$?

Is the interpolated space of order $\frac 1 2$, $$[L^2(\Omega), H^2(\Omega)]_{\frac 1 2}$$ equal to $H^1(\Omega)$? I can't find any good examples of these interpolation ideas. Assume $\Omega$ is ...
1
vote
1answer
26 views

Transferring a result in PDE from open domain to a manifold

Can anyone recommend me something that in detail, talks about transferring a result in Sobolev space (as opposed to Holder spaces or something like that) that holds for open domains in $\mathbb{R}^n$ ...
2
votes
1answer
78 views

Why are so many inequalities and estimate in PDEs?

I want to study PDEs, but when I read some PDEs books, I notice that there are so many inequalities here and always do estimates. I really want to know why there are so many inequalities here and ...
1
vote
1answer
35 views

Is $\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$ equivalent to $\lVert u \rVert_{H^2(M)}$?

On a bounded Riemannian manifold without boundary, is it true that the norms $$\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$$ is equivalent to the full $H^2$ norm $\lVert u ...
0
votes
1answer
61 views

How to show $\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla v|^2 \varphi \varphi_t - 2\int_Q (\nabla v \cdot \nabla \varphi) (\varphi v_t)$

Let $Q=\bigcup_{t \in (0,T)}\Omega \times \{t\}$. I have seen this identity for all $\varphi \in C_c^\infty(Q)$ such that $0 \leq \varphi \leq 1$: $$\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla ...