All Questions
Tagged with fa.functional-analysis ap.analysis-of-pdes
546 questions with no upvoted or accepted answers
2
votes
0
answers
149
views
Projection semigroup of an isolated eigenvalue
I'm currently working with a paper and I don't get something there. Let $A$ be a closed operator on a Banach space $X$ and $\lambda \in \sigma(A)$ an isolated eigenvalue, i.e. there is a $r > 0$ ...
- 381
2
votes
0
answers
93
views
Reference request: $|\partial_t u - \Delta u|\in L^p(\Omega^T)$ implies Holder estimate
I am currently reading a paper regarding harmonic flow between Riemannian manifolds. Let $\Omega$ be a Riemannian domain of dimension $2$, $\Omega^T=\Omega\times[0,T]$. The equation is
$$
\partial_t ...
- 1,245
2
votes
0
answers
116
views
Density of $C^0(\Bbb R^{n}\times (0,T))$ and $C^{\infty}_c(\Bbb R^{n}\times (0,T))$ in $L_{p,q}(\Bbb R^n_T)$
Let the space $L_{p,q}(\Bbb R^n_T)$ be defined as the set of all measurable $f:\Bbb R^{n}\times(0,T)\to\Bbb R$ such that $||f||_{p,q}<\infty$, where
$$
||f||_{p,q}:=\left(\int_0^T\left( \int_{\Bbb ...
- 1,245
2
votes
0
answers
229
views
Chain rule for Newton-derivative
I'm looking for properties of the Newton-derivative, defined as follows: A function $F \colon X \to Y$ is Newton differentiable at $x\in X$ if there exists $\varepsilon>0$ and a function $G\colon ...
2
votes
0
answers
78
views
Generalization of supersymmetry to dimension 3
in two dimensions there is a simple trick to study the spectrum of operators of the form
$$\textbf{A}:=\left( \begin{matrix}0 && A^* \\ A && 0 \end{matrix}\right)$$
The trick is to ...
- 167
2
votes
0
answers
683
views
Laplace problem with Robin boundary condition on a wedge
I'm trying to understand what the essential differences between Dirichlet/Neumann and Robin boundary conditions are. Therefore, let $\omega \in \left(0, 2\pi\right)$ and let
\begin{equation*}
\Omega = ...
2
votes
0
answers
142
views
Self-adjointness on Banach spaces
Let $A \in L(X,Y)$ be a bounded operator between Banach spaces. Then its dual operator $A' \in L(Y',X')$ has the same spectrum as $A$ by the closed range theorem.
Now, if we have an unbounded ...
- 501
2
votes
0
answers
2k
views
Reference for a proof of the Gagliardo-Nirenberg Interpolation Inequality?
In the book Linear and Quasi-linear Evolution Equations in Hilbert Spaces by Cherrier and Milani, Theorem 1.5.2, we are given the following version of the GN interpolation inequality:
Let $\Omega\...
- 1,247
2
votes
0
answers
90
views
Boundary regularity of solutions to semilinear heat equation
Consider the Cauchy IVP problem
$$u_t - \Delta u + f(t,x,u) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$
with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$
Can you point out a ...
- 303
2
votes
0
answers
218
views
Existence of solutions to time-dependent Schrödinger equations
I would like to know what is known about evolution equations of the form
$$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$
and $y(0)=y_0 \in D(H_0)$
where
$V$ is not a bounded operator, but an unbounded one, $u \...
- 173
2
votes
0
answers
225
views
degree theory argument in elliptic pde; apparent contradiction
i have a question regarding a degree theory argument and an apparent contradiction. Let me point out that I am a complete novice with degree theory and really i am just pushing some symbols with no ...
- 1,385
2
votes
0
answers
115
views
Does this Sobolev-space like construction have a name?
Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
- 305
2
votes
1
answer
605
views
Strong convergence of differential quotient in $L^2(0,T;V^*)$
I have got a problem regarding the weak differentiability of Bochner-integrable functions. Let $(V,H,V^*)$ be a Gelfand-triple and
\begin{align*}
w \in W(0,T) := \{w\in L^2(0,T;V) ~\vert~ \exists w' \...
- 187
2
votes
0
answers
140
views
Question on the differentiability of the solution mapping in the obstacle problem
I'm looking for a reference for the following. Take the obstacle problem:
$$\int_\Omega \nabla u \nabla (v-u) \leq \int_\Omega f(v-u)$$
for a function $u \in K:=\{v \in H^1_0(\Omega) : v \geq \varphi\}...
- 73
2
votes
0
answers
64
views
The continuity of $L^2$ gradient on moving domain
I post this on MSE too. Sorry for this short cross posting. But I stuck in this problem for a while... I don't know what the right tool should be used to handle the moving domain problem...
Let $I:=(...
- 679
2
votes
0
answers
235
views
The Cauchy problem associated with $u_t^\epsilon + H(x,t,u^\epsilon,\nabla u^\epsilon) = \epsilon\Delta u^\epsilon$
Consider the initial value problem $$\begin{cases} u_t^\epsilon + H(x,t,u^\epsilon,\nabla_x u^\epsilon) = \epsilon\Delta_x u^\epsilon & \text{ in } \mathbb{R}^n \times (0,\infty)\\ u^\epsilon = g &...
user81051
2
votes
0
answers
77
views
When do finite dimensional approximations approximate the spectral absicssa of a linear operator?
I apologize if the following is trivial for experts in the field. If so, please feel free to refer me instead to any proper references.
I would like to compute the spectrum of a known non-normal, ...
- 121
2
votes
0
answers
178
views
are these norms equivalent?
If it is known that $\sum_{i,j=1}^{n}a_{ij}\xi_i\xi_j\geq \alpha^2|\xi|^2$, where $\xi=(\xi_1,\xi_2,...,\xi_n)\in\mathbb{R}^n$ then can it be said that $\sum_{i,j=1}^{n}a_{ij}\frac{\partial u}{\...
- 157
2
votes
0
answers
207
views
Smoothing properties of analytic semigroups
Assume $A$ is a second order operator, generator of a positive analytic semigroup on the $L^p$-spaces with $p\in (1,\infty)$ and domain $D(A_p)=W^{2,p}$. Do we have regularity estimates
$\|T_p(t)f\|_{...
- 21
2
votes
0
answers
116
views
The composition of a dissipative operator and a positive operator is dissipative?
Consider the following bilinear system on a open and bounded domain $\Omega$
\begin{equation}
\left\{\begin{array}{r c l}
\displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\
y(0) &...
- 23
2
votes
0
answers
110
views
If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$
Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$.
Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
user76025
2
votes
0
answers
86
views
I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection
I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
user74319
2
votes
0
answers
87
views
1D inhomogeneous linear Schrodinger equation
I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...
2
votes
0
answers
166
views
Getting an a priori energy estimate from PDE weak formulation
On a bounded domain $\Omega$, I have two functions $u$ and $v$ in $L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$ satisfying
$$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int uv$...
2
votes
0
answers
82
views
Properties of a Sobolev bound
I am interested in computing
$$
A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2}
$$
where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound.
...
- 193
2
votes
0
answers
110
views
Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions
I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too.
Consider the following ...
- 271
2
votes
0
answers
150
views
Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
I have a question that it seems simple but I can not solve it.
Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
- 371
2
votes
0
answers
234
views
Concentration compactness on a compact setting
Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\{\varphi_k\}_k \in C^\infty(M)$ such that $\{\varphi_k\}_k$ satisfy the basic concentration ...
- 1,407
2
votes
0
answers
491
views
$L^\infty-L^1$ smoothing effect for the heat equation
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$.
Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation
$$u_t - \Delta u = 0$$
$$u(0) = u_0$$
where $u_0$ is bounded initial data. Here ...
- 21
2
votes
0
answers
244
views
A modification of Minty's trick?
I have the following result:
$$0 \leq \int_0^T (a(t)- |w(t)|)(b(t) - g^{-1}(|w(t)|))\quad\forall w \in L^2(0,T)$$
where $a$ and $b$ are both non-negative.
Does it follow that $b(t) = g^{-1}(a(t))$? ...
- 21
2
votes
0
answers
86
views
Regularity of $u$ in $u_t - \Delta \beta(t,u) = f$, can we get $u_t$ is a function?
I'm looking for reference discussing the regularity of the weak solution $u$ to the equation
$$u_t - \Delta \beta(t, u) = f$$
$$u(0) = u_0$$
where $\beta(t,\cdot)$ is a nonlinear function depending ...
- 51
2
votes
0
answers
223
views
One parameter family of elliptic equations
Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = N(\...
- 21
2
votes
0
answers
467
views
Reference request: The compactness and compact embedding in Besov Space?
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...
- 679
2
votes
0
answers
154
views
Asymptotics of "heat" semigroup
Consider a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Consider a second order elliptic operator $L$ on $L^2(\Omega)$, defined by either the Dirichlet or Neumann boundary ...
- 21
2
votes
0
answers
108
views
Quantitative estimate of heat dispersion - off diagonal estimates
Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = u(...
- 21
2
votes
0
answers
102
views
Sobolev trace of $H^1(\mathcal{M} \times I)$ functions
Let $\mathcal{M}$ be a compact Riemannian manifold and let $I=(0,1)$. I seek a trace theorem saying that functions $u \in H^1(\mathcal{M} \times I)$ have a well-defined trace at $\mathcal{M} \times \{...
- 21
2
votes
0
answers
153
views
Size of the eigenfunction of Laplacian (reference request)
It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then
$$||\phi||_{L^\...
- 1,692
2
votes
0
answers
119
views
Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)
Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve
$$\int_0^\infty\int_\Omega \nabla v\nabla ...
- 49
2
votes
0
answers
553
views
Sobolev space for manifold with boundary
For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
- 1,069
2
votes
0
answers
426
views
Strichartz estimates for the wave equation
Strichartz estimates for the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$ can be stated as
$$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t u\Vert_{C^...
- 21
2
votes
0
answers
144
views
Is logarithmic convexity of the heat kernel with complex time a general fact?
Suppose $H$ is a non-negative self-adjoint operator acting on $L^2(\mathbb{R}^n)$, and generates an analytic semigroup with kernel satisfying a Gaussian upper bound, i.e., if we denote $K(t,x,y)$ the ...
- 879
2
votes
0
answers
382
views
Sobolev space and trace theorems on a non-compact Riemannian manifold with boundary ($M \times (0,\infty)$)
Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$.
I am ...
- 53
2
votes
0
answers
385
views
Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface
Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem
$$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma \...
- 201
2
votes
0
answers
282
views
Reference request: functional analysis results used in Taubes paper (1980)
I am studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I am looking for a reference of three following theorems:
Let $f(x)$ be a convex funtional ...
- 31
2
votes
0
answers
142
views
Uniform bounds for a coupled parabolic system of PDE (linear)
Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon \...
- 43
2
votes
0
answers
142
views
Holder continuity of Poisson equation with divergence free drift
I am interested in the following PDE.
Suppose $u_m$ is a smooth solution of a elliptic equation of the form
$$ -\Delta u_m(x) + a_m(x) \cdot \nabla u_m(x) = f_m(x) \qquad B_1 $$ with $ u_m=0 $ on $\...
- 539
2
votes
0
answers
149
views
Variational inequality on Manifold
Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla \...
- 21
2
votes
0
answers
185
views
Showing $\langle \frac{\partial b(v)}{\partial t}, v \rangle_{H^{-1}(\Omega), H^1(\Omega)} = \frac{d}{dt}\int_{\Omega}\Psi^*(b(v))$
Let $b$ be continuous and increasing with $b(0) = 0$. Define $\Psi(t) = \int_0^t b(s)\;ds$ and $\Psi^*(s) = \sup_{r \in \mathbb{R}} (sr-\Psi(r))$.
(Note $\Psi^*(b(s)) + \Psi(s) = sb(s)$).
Let $v \...
- 173
2
votes
0
answers
146
views
Minimizing some $H^{-1}$ functional over (a subset of) probability densities in $R^d$
Let me consider the following subset of probability measures in $R^d$
$$
\mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\}
...
- 5,380
2
votes
0
answers
155
views
Showing a normal-derivative operator is a (sort of) contraction (related to Crandall-Liggett and PDEs)
Denote by $\mathbb{E}(g)$ the solution of the PDE
$$\Delta v(x,y) = 0 \quad\text{in $\Omega$}$$
$$v(x,0) = g(x) \quad\text{on $\partial\Omega$}.$$
Let $X=L^1(\partial\Omega)$. Let $h(t)$ be a ...
- 155