0
votes
1answer
17 views
Approximation for $L_{\text{loc}}^{\infty}(U)$ is this proof correct?
Let $U\subset\mathbb{R}^n$ be open and bounded. I am trying to extend Evans' proof (in his PDE book) for approximating functions in $L_{\text{loc}}^{p}(U)$ for the case that $p=\infty$ using ...
1
vote
2answers
35 views
Improvment of $W^{1,p}$ regularity of a elliptic equation solution.
I'm looking for some reference for results like
$$ \mbox{div}(A(x) \nabla u) = 0, \ \ u \in H^1=W^{1,2} \Rightarrow u \in W^{1,p}, p>2 $$
where $A(x)$ is elliptic, this is, $Id\lambda \le A(x) \le ...
1
vote
0answers
43 views
Sobolev spaces - about weak derivative [duplicate]
Let $U$ a bounded and open subset of $R^n$. Let $u \in H^{1}(U)$ a bounded function , $v \in H^{1}_{0}(U)$ a non negative function. Consider $\varphi : R \rightarrow R$ a convex and smooth ...
0
votes
1answer
42 views
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 ...
2
votes
2answers
35 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 ...
3
votes
0answers
75 views
Weak solution to PDE boundary value problems
Take $\lambda>0$ and let $\Delta$ be the Laplacian operator in dimension $n$ and let $\Delta^2:=\Delta\circ\Delta$. Consider the boundary value problem $\Delta^2 u+\lambda u=f$ on some open subset ...
2
votes
0answers
21 views
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
0
votes
2answers
44 views
Relationship between sobolev spaces
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. I would like to understand something about the relationships between these two spaces: $$H^1_0(\Omega) \cap H^2(\Omega)\quad \text{and}\quad ...
7
votes
0answers
223 views
+150
Hille Yosida theorem application
Disclaimer: pretty long and specific (contraction semi groups involved).
I have fourth order parabolic equation
$$
u_t + \Delta^2 u = 0
$$
on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
6
votes
2answers
58 views
Decay of $H^1(\mathbb{R}^n)$ functions
Is it true (is there a commonly known theorem) that says:
$f \in H^1(\mathbb R ^n)$ $\Rightarrow$ $\displaystyle \lim_{|x| \to \infty} f(x) = 0$ pointwise (where $H^1$ denotes the Sobolev space ...
1
vote
1answer
27 views
Difference Quotient Proof
Theorem: Let $u \in W^{1,p}(U)$ and let $V \subset \subset U$ (I.e. there is a compact set containing $V$ that is in $U$). Then for $1\le p <\infty$ there exists a constant $C$ such that for $0 ...
0
votes
0answers
26 views
Simple heat equation, solution regularity
I have a small problem with a regularity result for a simple parabolic heat equation:
Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
1
vote
2answers
44 views
What happens when you change space of test functions associated with weak derivatives?
Recall that $u \in L^2(0,T;H^1)$ has weak derivative $u' \in L^2(0,T;H^{-1})$ iff
$$\int_0^T uv' = -\int_0^T u'v$$
holds for all $v \in C_0^\infty(0,T).$
What happens if we only require that this ...
1
vote
1answer
31 views
Sobolev spaces - embedding - exercise
I have to show that $H^{2}(\Omega) \subset \subset H^1(\Omega)$. I think the Arzela - Ascoli theorem can help.. .I dont know how to start this exercise . I am beginner in Sobolev spaces.. someone can ...
5
votes
1answer
72 views
Why are weak solutions to PDEs good enough?
Obviously solutions in $C^k$ are nicer but it seems people are happy with obtaining weak solutions in some Sobolev space that only satisfy the weak formulation. Why should this, in the real world, be ...
0
votes
0answers
40 views
about a proof in chap 5 in PDE evans book
in the proof of the theorem 5 page 281 of Evans ( PDE - Evans ) , he write the Morrey estimate. he writes in the estimate : $y \in B(x,r)$
After in the proof we have
$$ |v(y) - v(x)| = |u(y) - ...
3
votes
2answers
46 views
Sobolev space boundary value in PDE
I often read this:
Let $\Omega$ be a open bounded set. There is a unique $u \in H^1(\Omega)$ such that
$$-\Delta u = f \text{ on $\Omega$}$$
$$u|_{\partial \Omega} = g$$
But how can we write ...
-1
votes
2answers
47 views
When does $v \in H_0^1$ and $|v|_{L^2}=0$ imply that $\|v\|_{H_0^1}=0$?
Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth.
Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in ...
1
vote
2answers
88 views
sobolev space-equivalence of scalar product
Let $f \in L^2(\mathbb{R}^n).$ Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique solution $u \in H^1(\mathbb{R}^n)$?
2- Prove that there exist a constant $C \geq 0$ ...
2
votes
1answer
93 views
question 9 - chap 5 evans PDE
The question is :
Integrate by parts to prove :
$$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$
for $ 2 \leq p < \infty$ ...
1
vote
1answer
85 views
Examples of truly abstract evolution PDEs?
Let $V \subset H \subset V^*$. Consider the parabolic PDE
$$y' = A(t)y + f$$
which is found in many books. Usually under some assumptions on $A(t)$ and $f$, there is a solution $y \in L^2(0,T;V)$ with ...
1
vote
0answers
54 views
Sobolev trace theorem for manifolds with boundary
Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality
$$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$
will hold.
...
0
votes
1answer
40 views
Does elliptic regularity result depend on boundary conditions?
Let $\Omega$ be a domain with boundary $\partial\Omega$. Suppose I am given a weak formulation:
$$b(u,v) = (f,v) \quad\forall v \in H^1(\Omega)$$
Assuming $b$ is nice enough, does the elliptic ...
2
votes
1answer
73 views
Using Galerkin method for PDE with Neumann boundary condition?
I am wanting to show existence of solutions to
$$u_t +L(u) = f \;\;\text{on}\;\; \Omega$$
with initial condition $u|_{t=0} = u_0$ and Neumann boundary condition $\nabla u\cdot \nu = 0$ on ...
0
votes
1answer
36 views
Poincare inequality on $H^1_0(M)$
Is it possible to deduce the Poincare inequality for functions in $H^1_0(M)$ from the Poincare inequality for functions in $H^1(M)$ with mean value 0?
$M$ is a hypersurface with non-empty boundary.
2
votes
1answer
55 views
Proving that a weak solution of a BVP satisfies the boundary condition
I am given the smooth function $u$ which satisfies $\int_U
(\nabla u \cdot \nabla v +uv)\,dx = \int_U
fv\,dx$ for all functions $v$ in the Sobolev space $H^1(U)$, where $f\in ...
2
votes
0answers
48 views
Sobolev Spaces: The difference between $W^{k,p}$ and $W^{k,p}_0$
Let $U$ be an open set in $R^d$. I am confused about the differences between
$$W^{k,p}(U):=\{u\in L^p(U): D^{\alpha}u\in L^p(U) \text{ for all } |\alpha|\le k\}$$ and ...
6
votes
2answers
167 views
Trace regularity result $\lVert n \times u\rVert_{H^{-1/2}}$
There is a result in a paper I am reading :
Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that
$$\lVert n \times u\rVert_{H^{-1/2}(\partial ...
0
votes
1answer
39 views
theorem in capacity theory
I am trying to understand the proof of a theorem of capacity theory
the theorem can be found in the book "nonlinear potential theory of degenerate elliptic equations" the authors are Juha Heinonen, ...
0
votes
1answer
35 views
$L^2(U)$ compact embedded in $H^{-1}(U)$?
Let $U$ be an open subset of $R^d$. We already knew that $L^2(U)$ is a subset of $H^{-1}(U)$. Question: is this a compact embedding?
3
votes
2answers
56 views
Decomposition of functionals on sobolev spaces
It is a well known theorem, that every signed measure can be split into its positive and negative parts (Hahn-Jordan-Decomposition). My question is, if something similar is possible for functionals on ...
0
votes
1answer
42 views
Paradoxical argument when applying Lax-Milgram theorem to a piecewisely defined elliptic problem
$\newcommand{\v}{\boldsymbol}$
To make the problem easier, simply consider a smooth simply-connected domain $\Omega\subset \mathbb{R}^2$. $\overline{\Omega} = ...
1
vote
1answer
54 views
The characterization of Sobolev space
If $\Omega$ is a bounded open set in $\mathbb R^n$ and $u$ is a distribution with supp$u \subset \subset \Omega$. For any $s \in \mathbb R$, if ${(I - \Delta )^{\frac{s}{2}}}u \in L_{loc}^2(\Omega ...
1
vote
0answers
74 views
Hölder norm estimates
How do you prove the following estimate for composition of functions:
If $k\geq 1$, then there exists a constant $c=c(k,\alpha)$ such that
$$
\|f_1\circ g_1-f_2\circ g_2\|_{k,\alpha}\leq ...
1
vote
1answer
57 views
Behavior of the pointwise norm of the gradient w.r.t. to boundary conditions in elliptic PDEs
Let $B\subset \mathbb{R}^2$ be some open ball in the interior of a (nice) domain $\Omega$ and $y_i\in H_0^1(\Omega)\cap H^2(\Omega)$ for $i=1,2$.
If I know that
\begin{align}
&\bullet\quad ...
1
vote
1answer
50 views
How to establish the estimate?
I consider the inequality as follows: let $a>0,b>0$, and satisfy $$2a^2-b^2\leq C(1+a).\tag{1}$$ If we assume that $b\leq a$, then there exist a constant $C_1>0$ such that $$a\leq ...
2
votes
1answer
66 views
Holder estimates for the gradient of the solutions to the linear divergence form elliptic equation?
Now I'm considering the Drichlet problem
\begin{aligned}
(a_{ij}(x)u_{x_i})_{x_j}+b_i(x)u_{x_i}+c(x)u &= f(x),\quad x\text{ in }\Omega \\
u(x) &= g(x),\quad x\text{ on }\partial\Omega.\tag{1}
...
7
votes
2answers
221 views
Sobolev space is an algebra
How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
3
votes
1answer
55 views
The inclusion of Sobolev spaces
Let $\Omega$ denote a bounded open set in $\mathbb R^n$ with sufficiently smooth boundary. If $u\in H_m^0(\Omega ) \cap H_k^{loc}(\Omega )$ with $m,k$ two positive integers, then can we conclude that ...
5
votes
1answer
120 views
Elliptic Regularity Theorem
I want to collect some results on elliptic regularity. The problem I consider is
\begin{aligned}
Lu&=f,&in\,\,\,U,\\
u&=g,&on\,\,\, \partial U.\tag{1}
...
4
votes
0answers
55 views
Splitting the action of functionals in duals of Sobolev spaces
Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
5
votes
0answers
93 views
Help me understand this proof (showing that something is a norm).
I am reading the following paper: Takáč, Peter On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51 (2002), no. 1, 187–237.
I need help to understand the ...
5
votes
0answers
74 views
Can we do some scaling argument in the presence of inhomogeneous norms?
Notation:
$B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$.
$\hat{f}$ stands for the Fourier transform of $f$.
Question. The following inequality holds true for all $f\in ...
3
votes
0answers
68 views
invertible operator Sobolev space
Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
2
votes
0answers
89 views
Why is this function space dense in this Bochner-type space? (Rogers and Renardy)
Let $\phi(t) = \sum_{i=1}^N\beta_i(t)\phi_i(x)$, where $\phi_i$ is basis for space $V$, and $\beta_i \in C_c^\infty(0,T).$
Renardy and Rogers says that
1) Functions of the above form are dense in ...
2
votes
1answer
141 views
Compact Embedding of $W^{1,2}(0,T;\mathbb{R}^d)$ in $C(0,T;\mathbb{R}^d)$
I need to prove (if true) that the space $W^{1,2}(0,T;\mathbb{R}^d)$ is compactly embedded in $C(0,T;\mathbb{R}^d)$. The proof for the continuous embedding part is straightforward and is given in PDE ...
2
votes
1answer
88 views
Is the Sobolev Space $H^k(0,1)$ a banach algebra?
In Adams'book:Sobolev Spaces, I know that if $kp>n,\Omega\subset R^n$ is boundary domain and has cone property, then $W^{k,p}(\Omega)$ could see as a banach algebra. My question is that does it ...
4
votes
1answer
145 views
Understanding the dual
There is an argument that disturbs me somewhat in Rabinowitz : Minimax methods in critical point theory. p.94
We are trying to prove that for certain functions, the Palais - Smale condition can be ...
4
votes
2answers
87 views
Compactness result PDEs
There is an argument that I see is used often in Evans PDE book, that I don't really get. We take a bounded sequence, say $(u_m) \in W^{1,q}(\Omega)$. By some functional analysis results, we know ...
2
votes
1answer
63 views
Existence and uniqueness of PDE with solutions in $W^{k,p}$ with $p \neq 2$?
I just realised that i have never seen the space $W^{k,p}$, $p\neq 2$, used in showing existence/uniqueness to some PDE. Usually books/lectures build up theory about $W^{k,p}$ (like certain compact ...
