All Questions
546 questions with no upvoted or accepted answers
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73
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relative compact on nonlinear term
On the paper: Decay of Solutions to Nonlinear Schrodinger
Equations.
Let $u$ be a solution of the equation
$$Hu+|u|^2u=0,$$
where $H$ is a Schrodinger operator, i.e. $-\Delta+V$ and $V$ is a (...
- 1,915
1
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0
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196
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Compact embedding result
Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
- 395
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922
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A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
- 698
1
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0
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299
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How can I obtain this inequality (from Evan's PDEs)
I am self studying PDEs from Evans' "Partial Differential Equations" textbook. Currently, I am going through Theorem 1 from Section 5.7 (Rellich-Kondrachov compactness theorem) and am having ...
- 185
1
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93
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Relative boundedness of the adjoint
Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...
- 33
1
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211
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Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
- 11
1
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117
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trace inequality for Dirichlet Neumann operator
Does there exists a Sobolev trace inequality of the form
$$ \|U(x, 0)\|_{L^{q}((a, b))} \leq C\sqrt{q}\| \nabla U \|_{L^{2} (\mathcal C)} ; \forall U\in H^{1}_{0, L} (\mathcal C)$$ and for any $q>...
- 402
1
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0
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45
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Shifting Sobolev norms in a hyperbolic estimate
Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate:
$$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((...
- 4,153
1
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0
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84
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Coercivity of $\int (\Delta u + u)^2$ on a subspace of $H^2$?
Let $\Omega = [0,L] \times [0,2\pi]$ and split its boundary into $\Gamma_d = \{0,L\} \times [0,2\pi]$, $\Gamma^1_p = [0,L] \times \{0\}$, $\Gamma^2_p = [0,L] \times\{2\pi\}$. Consider the following ...
- 111
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77
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Zero energy resonances for scaling critical Schrodinger operators
Given a real valued potential $V\in L^1(\mathbb{R}^3)$, we say that the Schrodinger operator $-\Delta + V$ has a zero-energy resonance if there exists $\psi\in L^2_{loc}(\mathbb{R}^3)\setminus L^2(\...
- 943
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304
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Harmonic coordinates on asymptotically flat manifold
I am studying the existence of harmonic coordinates at infinity on an asymptotically flat manifold. My Reference papers are, The Mass of Asymptotically Flat Manifold, by Bartnik [B] and The Yamabe ...
- 914
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60
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Existence of solutions to $\lambda u-\frac{1}{(1+(u')^2)^2} \, \Delta u = f$
I'm looking for existence results for the equation
$$\lambda u-\frac{1}{(1+(u')^2)^2} \, \Delta u = f \quad \text{on the domain $[a,b]$}$$
for $u:[a,b] \to \mathbb{R}$, with either zero Dirichlet or ...
- 11
1
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0
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102
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domain dependence of best constant in inequality
Take $N \ge 3$ and consider the inequality
$$ \| \nabla u\|_{L^N} \le C(\Omega) \| \Delta u \|_{L^\frac{N}{2}} $$ for all $ u \in W^{2,\frac{N}{2}}(\Omega) \cap W^{1,N}_0(\Omega)$ where $ \Omega$ is ...
- 1,385
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198
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Morrey space is Banach space
I'm working with Morrey spaces, which are the spaces
$$L^{p,\lambda}(\Omega):= \left\{ u \in L^1_{loc}(\Omega): \sup_{x \in \Omega, r > 0} r^{-\lambda}\int_{B(x,r)\cap \Omega}|u(y)|^pdy< \infty\...
- 121
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0
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63
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Supnorm problem involving kernel of Cauchy problem
Let $M$ be the $2$-dimensional hyperbolic manifold. Let $K(t,x,y)$ be the kernel appearing in the fundamental solution of the Cauchy problem
$$(\partial^2_t-\Delta_M)u=0,\text{ on }\mathbb{R}^+\times ...
- 1,692
1
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0
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79
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Time dependent Hamiltonians
I'm studying time dependent perturbation theory on Reed-Simon book "Method of modern mathematical physics, II". If one considers an Hamiltonian of the form
$$H(t)=H_0+V(t)$$
the corresponding formal ...
- 11
1
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76
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Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
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183
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Trace theorem for boundary value problem
Consider the inhomogeneous boundary value problem on the infinite strip $(x,y)\in \mathbb{R}\times [0,1]$ defined by
$$\begin{cases}\partial_{x}u + \partial_{y}v=f & {(x,y)\in \mathbb{R}\times (0,...
- 873
1
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0
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180
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Implicit function theorem for operators
Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
- 115
1
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124
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Singular value decomposition in two spaces (reference in Russian paper?)
Let $H$ be a Hilbert space and $X$ be a Banach space such that $H \cap X$ is dense in both.
Now, let $T$ be an operator such that $T: H \rightarrow H$ and $T:X \rightarrow X$ exists in the sense that ...
1
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0
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50
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Verifying general assumption for parabolic PDE
I've got some problems verifying an assumption for a parabolic PDE. Namely, let $(V,H,V^*)$ be a Gelfand-Triple, $u_0 \in V$, $\psi\colon V \to \mathbb{R}$ convex and lower-semicontinuous and $a\colon ...
- 187
1
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0
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105
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compactness of sequence of harmonic functions
Let $ \Omega$ denote a smooth bounded domain in $ R^N$ and let $u_m \in C^\infty( \overline{\Omega})$ harmonic functions. We also suppose $ u_m$ is bounded in $L^2(\Omega)$ (uniformly in $m$).
...
- 1,385
1
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331
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Verifying a claim regarding $H^1$ weak convergence and $L^2$ strong convergence on a surface
I'm reading a paper whose first section discussed $H^1$ maps defined on star-shaped sets, but I got stuck in verifying a claim for quite a while. I'm now thinking the claim is wrong, but it's hard to ...
- 1,350
1
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0
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128
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determine when $e^{ikx}$ can be boundary value of a holomorphic function
Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.
My question is, for what curves $...
- 31
1
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0
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90
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Monotone operator subgradient
I am trying to solve a PDE of the form $\mathcal{A}u'(t) + \partial\Psi[u(t)] \ni 0$ where $\mathcal{A}$ is a skew-symmetric, linear, monotone operator, $\Psi$ is convex, and $\partial \Psi$ is the ...
- 11
1
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0
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664
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$W^{2,p}$ regularity of elliptic PDEs with Neumann boundary condition
Given an elliptic PDE with Neumann boundary condition
\begin{align}
\left\{
\begin{aligned}
-\sum_{i,j=1}^N\partial_i(a_{ij}\partial_j u)+cu&=f &&\mbox{in}\,\,\,\Omega, \\
\sum_{i,j=1}^Na_{...
- 393
1
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0
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117
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The eigenfunction of modified $1$-laplace equation?
Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
- 679
1
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0
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192
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The decay rate of the spectrum of the Gaussian kernel on compact manifolds
It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
- 617
1
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0
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789
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$C^{1,2}$ regularity of (weak) solutions to the heat equation
Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation
$$u_t - \Delta u = 0$$
$$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$
$$u(0) = u_0$$...
1
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0
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588
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How is the duality pairing of $H^{1/2}$ and $H^{-1/2}$ defined on a subset of the boundary?
Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$.
Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in H^{1/2}(\...
- 142
1
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0
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92
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Perturbation in Besov space
$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
- 515
1
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0
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83
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Boundedness of a function that satisfies a PDE-type inequality
Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$.
Suppose that
$$\sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + \int_{...
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0
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177
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How does the $L^\infty$ norm of the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ depend upon $\alpha$ and $\lambda$?
Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of
$$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$
$$\partial_\nu u = \alpha \quad \text{on $\...
- 11
1
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0
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352
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Heat semigroup ultracontractive?
Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
- 11
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158
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On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $
This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) $...
- 1,396
1
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0
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86
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Asymptotics of a elliptic pde when exponent gets large
I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $...
- 1,385
1
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154
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One-parameter group of unitary operators and Core
Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
- 11
1
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206
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A Question about compactness of an embedding into $L^p$ spaces
Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} \frac{u^...
- 371
1
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95
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Construct a PDE solution from a net of approximations
Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let $...
- 5,407
1
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0
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62
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Reference request - Compact embedding of intermediate space
Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...
- 679
1
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0
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147
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Energy inequalities for Sobolev spaces of negative integer
I asked this question in mathematics stackexchange and couldn't get an answer.
Let $\phi\in H^{s}$ such that the following energy inequality is true:
$$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\...
- 101
1
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0
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70
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Equivalence of two definitions of weak solution (subtlety with null sets)
Consider
$$y_t - \Delta y = f$$
$$y(0) = y_0$$
with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions:
We have $y \in L^...
- 266
1
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0
answers
191
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$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value
Let $u \in H^1((0,T)\times S)$ be the unique solution of
$$u_{tt} + \Delta u =0$$
$$u|_{t=0}= u_0$$
$$u|_{t=T}=0$$
where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
- 21
1
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0
answers
129
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persistence of regularity for nonlinear Klein-Gordon equation
I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" (...
- 1,051
1
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0
answers
116
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Strong solution to parabolic equation without differentiability assumption on coefficient?
Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain
$$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$
$$u|_{\partial\Omega} = 0$$
where $a$ is real-valued and satisfies
$C_1 \leq a(r) \leq C_2$ ...
- 251
1
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0
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213
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Harmonic extension of $L^\infty$ function is in $L^\infty$?
Let $u \in H^{\frac 12}(\Omega)$ with $\int_\Omega u = 0$ and consider the solution $v \in H^1(C)$ where $C=\Omega \times (0,\infty)$ of
$$-\Delta v(x,y) = 0$$
$$\partial_\nu v = 0$$
$$v(x,0) = u(x)$$...
- 21
1
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0
answers
108
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A bilinear estimate in Lp space
Let $\varphi(D)$ be a Fourier multiplier with symbol $\varphi(\xi) = \xi/(1+|\xi|^2)$. It's easy to prove that
\begin{equation}
\|\varphi(D)u^2\|_{H^s(R)}\lesssim \|u\|^2_{H^s(R)} \quad (*)
\end{...
- 425
1
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0
answers
115
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Free Endpoint of Minimization Problem
Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > 0$...
- 516
1
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0
answers
99
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decomposition of tempered distributions by entire analytic functions
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$
Let $j\in \mathbb N$ ...
- 1,051
1
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0
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136
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A linear operator equation (PDE) with non-monotone term
I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...
- 11