2
votes
0answers
50 views
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 ...
1
vote
0answers
38 views
Relation of the kernels of one bounded operator and its extension
Sorry for this long and formal post. The application in PDEs is mentioned just at the end.
Let
$$V \hookrightarrow H \text{ and } Q_H' \hookrightarrow Q',$$
where $V$ and $Q$ are Banach and $H$ ...
1
vote
0answers
19 views
How to perturb a Palais-Smale functional so that the Palais-Smale condition is preserved?
Suppose that $J_0\colon H \to \mathbb{R}$ is a $C^1$ functional on the Hilbert space $H$ satisfying the Palais-Smale condition, that is:
any sequence $u_n\in H$ such that $J_0(u_n)$ is bounded and ...
1
vote
0answers
45 views
Regularization by mollifier sequences
A well-known feature used in PDE's is the regularization by convolution with a mollifier sequence $\rho_n$, i.e. $\rho_n(x) := n^d \rho(nx)$ with $x \in \mathbb R^d$, $\rho \in C^\infty_c(\mathbb ...
2
votes
0answers
52 views
(localized) L^2 norm of quasimode for Laplacian
Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq ...
2
votes
1answer
39 views
Asymptotic behaviour of solutions to elliptic PDE
Let $u$ be a solution (in the distributional sense) of
$$
\Delta u = \delta_r
$$
on $\Omega \subset \mathbb{R}^2$ open, $r \in \Omega$.
Let $w$ be a solution of
$$
Aw = \delta_r
$$
where
$A = ...
5
votes
0answers
120 views
+50
Operator completly continuous
For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP
consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$
and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
4
votes
0answers
68 views
Solution to $\Delta_g u = \delta-1$ on a 2-sphere.
Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
1
vote
1answer
48 views
the first eigenfunction of Dirichlet problem
Let $\Omega$ be a bounded planar domain which has a axis of symmetry and $T:\Bbb{R}^2\longrightarrow\Bbb{R}^2$ symmetry with respect to this axis. Let $u_{1}(x)$ be the first eigenfunction of ...
0
votes
0answers
24 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
41 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
37 views
PDEs: subsequence converges to solution, so whole sequence does too
Suppose we want existence of a function $u$ for the PDE
$$(\frac{d}{dt}u,v) = b(u,v)$$
for all $v$ in a test space.
Sometimes in PDE you have use a Galerkin approximation, so say $u_n$ is a sequence ...
0
votes
2answers
39 views
Continuation of smooth functions on the bounded domain
Given a bounded domain $\Omega\subset \mathbb{R}^n$ and a smooth function $f$ with bounded derivatives on $\Omega$, is it possible to extend $f$ to $\tilde{f} : \mathbb{R}^n \to \mathbb{R}$ such that ...
0
votes
1answer
47 views
Divergence and regularity result on bounded domains
Let $\Omega\subset\mathbb{R}^2$ a bounded set with Lipshitz continuous boundary.
If $$z\in L_0^2(\Omega)=\{v\in L^2(\Omega):\int_\Omega v\,dx=0\},$$
it is true that exists $\phi\in ...
2
votes
1answer
52 views
Concerning the noncompactness of the map, $I: W^{1, 2}(\mathbb{R}^n)\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$
Consider the identity map $I:W^{1,2}(\mathbb{R^n})\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$ where $n\geq 3$. Suppose that this map is not compact that is given some bounded sequence of functions ...
-1
votes
2answers
46 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
1answer
83 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 ...
0
votes
1answer
44 views
relation between Holder continuous and weakly differentiable for the coefficients of a pde
I am reading a book on pdes and the author gives the definition of the weak solution using the adjoint operator. For that expression to make sense, for the case of second order elliptic equation one ...
4
votes
0answers
76 views
What is visualization of gradient flow of a functional?
I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
0
votes
2answers
49 views
a version of taylor theorem
I need to a version Taylor theorem (Taylor expantion) in $H^2(0,1)$.
What is the difference between it and usual Taylor expantion.
Where can I find it?
Thanks.
1
vote
0answers
39 views
Harnack's Inequality and (hypo)elliptic PDE
Background: I am aware of the Harnack's Inequality for linear elliptic equations.
My questions are:
(a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
1
vote
1answer
42 views
Example for the Sobolev embedding theorem
We know that $W^{k,p}\hookrightarrow C^{k-\lfloor\frac{n}{p}\rfloor-1,\gamma}(\bar{\Omega})$ with $kp>n,\gamma=\lfloor\frac{n}{p}\rfloor+1-\frac{n}{p}$, where $n$ is the dimension of $\Omega$, ...
2
votes
1answer
66 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
34 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.
4
votes
1answer
128 views
Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces
I am trying to prove the existence of a weak solution of the problem:
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
on the bounded open set $U\subset\mathbb{R}^n$ ...
0
votes
2answers
45 views
Boundedness of a Solution Operator
Let $v \in L^2(\partial \Omega)$ and define $S(v) := y$ where $y$ satisfies $-\Delta y + y = 0 $ in $\Omega$ and $\frac{\partial y}{\partial \nu} = v$ on $\partial \Omega$. I need to show that $S$ is ...
4
votes
1answer
49 views
Hilbert space and its dual
I have an elliptic equation in the form $$-\Delta u + u =F(u).$$ For any $\phi \in C^{\infty}_{0}$ we rewrite the elliptic equation in weak form $$\int \limits_{\mathbb{R}^n}\left(-\Delta u + ...
2
votes
1answer
141 views
PDE weak solution problem
My professor grades really strictly (details). I would be very happy if you could help me with this problem:
Let $U \subset R^n$ be a bounded set. Consider $ \Delta^2 u = f$ on $U$ and ...
2
votes
1answer
28 views
equivalency weak and strong form
I am trying to show that
if u is smooth enough and $$u\in V, \int_\Omega k \nabla u.\nabla v=\int_\Omega fv ~~~for ~all~v\in V$$
that $$V=\{{v\in H^1(\Omega):v=0 ~on ~\Gamma_1}\}$$
then $u$ is the ...
1
vote
1answer
133 views
Galerkin method for existence for PDE with nonsymmetric bilinear form
Suppose we have a PDE
$$\langle u', v \rangle + a(u,v) = 0$$
where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in ...
0
votes
1answer
45 views
Solving the equation $Pu = f$, given that every $\ell \in \text{Ker}(P')$ has $\ell(f) = 0$
(Stanford Real Analysis Qualifying Exam: Spring 2012) (Ideal time: 24 minutes)
5) Let $X$, $Y$ be separable reflexive Banach spaces. Let $P \colon X \to Y$ be a bounded linear map, and $P'\colon Y^* ...
1
vote
1answer
87 views
Non-homogeneous boundary value problem - weak solution
Consider the following boundary value problem on a bounded subset $U\subset\mathbb{R}^n$:
$$
-\Delta u = f,
$$
for $u: U \rightarrow \mathbb{R}$ where $u = g \in C^2$ on the boundary of $U$.
I have ...
1
vote
1answer
40 views
Generalization of zero-diagonal square matrices to linear operators
Which linear operators in Banach or Hilbert spaces (e.g., partial differential operators or some other operators in functional spaces) are generalizations of square matrices $A=(a_{ij})$ such that ...
0
votes
0answers
37 views
Eigenfunctions of elliptic operator form an orthonormal basis for $L_2$?
Theorem 6.5.1 of Evans PDE is a standard result that says given a symmetric elliptic operator, there exists an orthonormal basis consisting of the Dirichlet eigenfunctions of the operator. But this ...
3
votes
2answers
53 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 ...
2
votes
2answers
62 views
A question concerning measurability of a function
Let $\Omega\subset \mathbb{R}^n$ be bounded and let $X:=H^1(\Omega)$. Let $a:\Omega\times \mathbb{R} \to \mathbb{R}, (x,z)\mapsto a(x,z)$ be a bounded function such that $a(x,.)$ is continuous on ...
0
votes
1answer
46 views
If $v = v_1 + v_2$ and $\lVert v\rVert \leq 1$ then $\lVert v_1\rVert \leq 1$. Why is it true? (Hilbert spaces and orthogonality)
We have $H^1_0 \subset L^2$ where $w_j$ is an orthogonal basis on $H^1_0$ and orthonormal basis on $L^2$.
Let $v= v_1 + v_2$ with $\lVert{v}\rVert_{H^1_0} \leq 1$, where $v_1 \in \text{span}\{w_j\}$ ...
7
votes
1answer
177 views
Do eigenfunctions of elliptic operator form basis of $H^k(M)$?
We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for ...
1
vote
0answers
93 views
Viscous Burgers Equation existence of solution
Can some one prove the local existence of solution for Viscous Burgers equation using fixed point techniques.
The equation is as follows :
$$
\frac{\partial u}{\partial t} - \bigtriangleup u + ...
3
votes
1answer
64 views
Confused about Proof of Thm 4.9 Gilbarg Trudinger
Thm 4.9 in Gilbarg-Trudinger's book states that :
if $B$ is a ball in $\mathbb{R}^n$ centred at $x_0$ and
$f\in C^{\alpha}(B): \sup_{x\in B} (\text{dist}(x, \partial B))^{2-\beta}\vert f(x)\vert ...
1
vote
0answers
103 views
Question about proof of regularity of PDE solution in Evans
We can use a Galerkin method to show that there is a solution to a PDE. So suppose $w_j$ is the basis functions.
I am interested in regularity of solutions. In the book by Evans, he differentiates a ...
0
votes
1answer
24 views
Computing a derivative of map from $V \to V^*$ (PDEs and regularity)
I am reading Rogers and Renardy book on parabolic regularity. There they consider a PDE
$$\dot u = A(t)u + f(t)$$ where $A(t):V \to V^*$ is an operator. In the regularity result, they need $A \in ...
10
votes
1answer
192 views
Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?
Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels.
Feynman-Kac formula is also a pde corresponding to a stochastic process ...
4
votes
1answer
43 views
Weak holomorphicity implies smooth and holomorphic.
This is an extension of a previously asked question:
A function $f\in L^2(D)$ is weakly holomorphic if, for every $\phi\in \mathcal{C}^{\infty}_c(D)$, $$\int_D f\partial_{\bar{z}}\phi = 0.$$ I'm ...
0
votes
1answer
52 views
Weak Holomorphicity: Notation clarification.
A function $f\in L^2(D)$ is weakly holomorphic if, for every $\phi\in \mathcal{C}^{\infty}_c(D)$, $$\int_D f\partial_{\bar{z}}\phi = 0.$$ I'm trying to show that each such $f$ is smooth on the ...
2
votes
1answer
87 views
Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$?
Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$? More precisely, I want to prove that
THEOREM. A sequence $\{f_n\}$ is convergent in $C^k(\bar\Omega)$ (or some more ...
0
votes
1answer
39 views
Which one is the right definition of eigenvalues for a differential operator?
This question might be trivial, but I have problems understanding the definition of eigenvalues for the Laplacian \begin{equation} \Delta : C^2(U) \to C(U). \end{equation} on some open, bounded domain ...
5
votes
1answer
119 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
54 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 ...
3
votes
1answer
100 views
Weighted $L^2$ Estimates for Domains $\Omega\subseteq\mathbb{C}$
EDIT: After mrf's comment below and some discussion with my instructor for the course it was decided that the below was not really an issue. Namely, I went into reading this lecture with the notion ...
