All Questions
Tagged with fa.functional-analysis ap.analysis-of-pdes
546 questions with no upvoted or accepted answers
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61
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$L^p$ estimate for perturbed heat equation
Let us consider the heat equation
$$
\begin{cases}
u_t + f(u)_x - u_{xx} = 0 & x \in (-1,1), \quad t >0\\
u(t,-1) = a(t), \\
u(t,1) = b(t), \\
u(0,x) = u_0(x)
\end{cases}
$$
where $f \in C^\...
- 131
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0
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330
views
Comparison of the spectrum decomposition
In the spectral theorem, we learned that the spectrum of a linear operator $A$ is a disjoint union of: point spectrum (eigenvalues), continuous spectrum (the kernel of $zI-A$ is zero, the closure of $\...
- 1
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37
views
Approximation of "endpoint" initial data for the 3D wave equation
Consider $f\in L^2(\mathbb{R}^3)$ such that, denoting by $A_f$ the solution to the wave equation $\square A=0$ with initial data $A(0)=0$, $(\partial_t A)(0)=f$, we have $A\in L_2(\mathbb{R}^+;L^{\...
- 943
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0
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57
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Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?
Suppose I have an equation in some reflexive separable Banach space $X$:
$$Au=f$$
for given data $f \in X^*$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...
- 107
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88
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Prove that the solution belong to ${L^2}\left( {0,T;{L^2}\left( \Omega \right)} \right)$
Let $k \in {L^\infty }\left( {0,T} \right)$ and we assume that
$$\phi :t \mapsto u\left( t \right) + \int_0^t {k\left( s \right)u\left( s \right)ds} \in {L^\infty }\left( {0,T;{L^2}\left( \Omega \...
- 183
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85
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Could the convex hull of $\operatorname{Lip}_1(\mathbb R)$ be dense in $\operatorname{Lip}_1(\mathbb R^d)$?
$\DeclareMathOperator\Lip{Lip}$My problem is slightly different from the title, but I don't have a more straightforward title. Sorry for that.
For $d\ge 1$, denote $\mathbb S^{d-1}:=\{x\in\mathbb R^...
user128095
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75
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Partial well-posedness results on Schrödinger operators?
Set $ A_i:= -\Delta + V_i :H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3), \ i =1,2 $, where
\begin{equation*}
V_1 = 0, \ \ (\textrm{No interaction}) \\
V_2 = - \frac{\gamma}...
- 269
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answers
83
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$ 0 $ locates in the continuous spectrum of Schrodinger operators?
This is question is motivated by Non-closed range space of Laplace operators?. We aim to determine what kind of potential will make corresponding schrodinger operators possess non-closed range.
For ...
- 269
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55
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Smooth compactly supported function with good scaling with respect to the fractional Laplacian
Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
user139845
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0
answers
109
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Elliptic equation with Neumann boundary condition: RHS in $L^2$ implies solution in $L^\infty$?
Consider the homogeneous Neumann problem $$-\Delta u + ku = f$$ $$\partial_\nu u = 0$$
on a smooth, bounded domain $\Omega$.
If $f \in L^2(\Omega)$, do we obtain the regularity $u \in L^\infty(\...
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326
views
Measurability of the heat semigroup in $L^\infty$
Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...
- 1
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0
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273
views
Local "boundary comparison principle" for harmonic functions
Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...
user123456
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answers
117
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Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 1
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170
views
Strong continuity (weak to strong) of $\langle Au,v\rangle=\int u^3 v dx$
I am currently trying to figure out the following. If I consider the Sobolev space $W^{1,p}_0$ is it possible to show that the operator given by
$$\langle Au,v\rangle=\int u^3 v dx$$
is strongly (weak ...
- 101
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113
views
Conditions for the embedding of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$
Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
user123457
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324
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Adjoint of differential equation
Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation
$$y'(t)=A^*y(t).$$
I ...
- 83
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266
views
Embedding for the Bourgain spaces $X^{s,b}$
Where can I find embedding results for the Bourgain spaces $X^{s,b}$ (for a definition see the bottom of page 2 here).
In particular, I'd like to know if, for $s$ sufficiently large, it is contained ...
user60665
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81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
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308
views
Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
- 617
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0
answers
343
views
A question on weak formulation of the p-laplacian operator
Can it be said that $$\int_{\Omega}\Delta_p u |\phi|^{p-2}\phi dx=\int_{\Omega}\Delta_p \phi |u|^{p-2}u dx\qquad\forall \phi\in C_0^2(\overline{\Omega})$$ is the generalized weak formulation of $$\...
- 157
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119
views
Gauge Fixing Problem on Cylindrical
For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold.
If it is necessary, we could consider the $b_1(Y)=0$ case.
Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
- 1,915
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0
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378
views
compact injection
Put:
$D=\{u\in L^{2}(\mathbb{R}^{n})| x^{\alpha}D^{\beta}_{x}u\in L^{2}(\mathbb{R}^{n}), \forall \alpha,\beta \in \mathbb{N}^{m}:|\alpha|+|\beta|\leq 2 \}$
Why $D \hookrightarrow L^{2}(\mathbb{R}^{n}...
- 135
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470
views
Derivatives of Mollified functions
I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
- 131
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0
answers
371
views
Harmonic function with Dirichlet boundary condition
Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
- 1
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188
views
One parameter family differentiable dependence for linear parabolic pde's
Consider for example, the Black Schole's equation
$$
\partial_tu+0.5\sigma^2s^2\partial_{ss}u+rs\partial_su-ru=0
$$
on $[0,T]\times[0,\infty)$ subject to boundary conditions $u(s,T)=f(s)$.
The ...
- 383
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answers
173
views
Is this has anything to do with Riesz representation?
The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...
- 679
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answers
109
views
solutions of elliptic linear pde depending analytically on a parameter
Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+...
- 1,385
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answers
137
views
Heat asymptotics
Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...
- 11
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0
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358
views
Boundedness of heat semigroup on $L^1(\Omega)$
On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider ...
- 1
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0
answers
151
views
Notion of solution of pde
Let's consider the following Schrodinger equation
$$iu_t+\Delta u+F(u)=0$$
in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in L^\infty(0,...
- 9
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0
answers
184
views
Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380
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answers
200
views
Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$
We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$
I want to show for $r \geq 2$ that
$$\...
- 1
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0
answers
606
views
$b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$
This question stems from the proof of Theorem A.1 on page 425 of this paper.
Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in $C^0(...
- 266
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0
answers
88
views
References for LWP of a NLS Equation
I am studying the LWP of $$i \partial_t \psi + \Delta \psi = \left| \psi \right|^{p-1} \psi + \frac{1}{\left| x \right|^2} \psi$$ in $\mathbb{R}^{1+2}$ given appropriate Cauchy data. It will probably ...
- 516
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0
answers
206
views
About approximate eigenvalue
I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4.
Suppose $X$ is a real Banach Space, $M$ is a ...
- 103
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votes
0
answers
65
views
Interpolation with time continuity
If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".
Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in L^2(0,T;H^{-1}(\...
- 103
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0
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474
views
Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)
Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$.
Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
- 85
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0
answers
156
views
Can a function be constructed from the direction of its gradient?
Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation
$Du=J|Du|$, $u|_{\partial \Omega}=f$
has a solution in a ...
- 23
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0
answers
515
views
If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?
Let $\phi_n$ be a sequence of mollifier converging to the identity
$$
\phi_n(x) \to \delta_{0}(x), \text{pointwise},
$$
with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in C^\infty_{\...
user24451
0
votes
0
answers
113
views
Reference Search for a Functional Minimization Problem
Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is
$$M \equiv\int_{x=-\infty}^\infty u(x) ~ ...
- 151
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votes
0
answers
137
views
$\mathcal{D}(0,T;V)$ is dense in $W(0,T)$
Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in L^2(...
- 113
0
votes
0
answers
166
views
Harnack's Inequality and (hypo)elliptic PDE
Background: I am aware of the Harnack's Inequality for linear elliptic equations.
My questions are:
(a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
- 1
0
votes
0
answers
214
views
Splitting the action of functionals in duals of Sobolev spaces
Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
- 101
0
votes
0
answers
606
views
partial differential equations with mixed boundary conditions
hi,
does anyone know some good references (books, papers) on partial differential equations with mixed boundary conditions ?
actually I am intrested in the following: Let $f(x)=(f_{1}(x),...,f_{n}(...
- 89
-1
votes
2
answers
440
views
Motivation for weak solution of a PDE (initial condition)
The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations"
When looking at a (nonlinear degenerate) PDE like
$$ \...
- 1
-1
votes
1
answer
152
views
Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )
Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2.
More generally, is it true that eigenfunctions ...
