All Questions
546 questions with no upvoted or accepted answers
2
votes
0
answers
186
views
Changing the test function space in a weak formulation of parabolic PDE
Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T a(u(t),\...
- 155
2
votes
0
answers
231
views
A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?
Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - \...
- 183
2
votes
0
answers
128
views
How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?
I am reading this paper.
Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$
On page 5 of ...
- 23
2
votes
0
answers
127
views
slightly subcritical elliptic pde; the linearized equations
Let $ p_m \nearrow \frac{N+2}{N-2}$ and consider the family of elliptic problems
$$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$ where $B$ is the unit ball ...
- 539
2
votes
0
answers
370
views
Subordination identity and heat operator
I was reading the so-called subordination identity that allows one to derive estimates on the Poisson operator $e^{-t\sqrt{-\Delta}}$ from estimates on the heat diffusion operator $e^{t\Delta}$. I am ...
- 21
2
votes
0
answers
190
views
A contradiction to do with continuity? (involves chain rule)
Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE
$$\frac{d}{dt}D^0_t(\cdot) = V(D^...
- 21
2
votes
0
answers
93
views
Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces
Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
- 21
2
votes
0
answers
157
views
linear operator associated with semilinear elliptic pde
I am reading a paper where at some point they analyse the following linear operator:
$$L_\lambda(\phi)= - \Delta \phi - C_\lambda(x) \phi$$
where $ C_\lambda(x)>0$ (smooth) in $ \Omega$ a bounded ...
- 539
2
votes
0
answers
223
views
optimal regularity for Laplace equation with inhomogeneous L^p Robin boundary condition
Consider the problem
$$-\Delta u = 0 \mbox{ in }\Omega,\qquad \partial_\nu u+\tau u=g\mbox{ on }\partial\Omega,$$
where $\Omega\subset R^n$ is a bounded $C^2$-domain, $\tau>0$ is a constant, and $...
2
votes
0
answers
176
views
A limit involving a regularizing kernel
I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#
...
- 2,222
2
votes
0
answers
242
views
Core of divergence form operator with unbounded coefficient
Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that $a(x)...
- 617
1
vote
0
answers
84
views
Does sets of positive capacity rule out constant functions?
Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by
\begin{align*}
\text{Cap}_{p}(K, U) :=
\inf \left\{
\int_U |\...
- 1,101
1
vote
0
answers
73
views
Convexity and subdifferential monotonicity
Do you know any reference where I can find some results in this sense:
Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
- 1,759
1
vote
0
answers
88
views
Schauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:
Let $u$ be a solution of
$$\Delta u = f \quad \text{in } B_1$$
for $f \in L^\infty$. Then $u \in C^{1,1 - \...
- 452
1
vote
0
answers
56
views
Finding thin plate spline subjected to boundary conditions
I am trying to formulate a problem as a PDE. What I want to know is if my formulation is correct, if it admits solution and what am I missing.
This question is related to : Thin-Plate-Spline ...
- 115
1
vote
0
answers
43
views
If a weighted Laplacian's eigenfunction is zero in an open set, when is it identically zero?
Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be:
$m(x) \cdot \text{div} ( s(x) \nabla f(x))$.
What ...
- 93
1
vote
0
answers
91
views
Parabolic regularity for weak solution with $L^2$ data
I want to study the regularity of weak solutions $u\in C([0,T];L^2(\Omega))\cap H^1((0,T);L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ of the heat equation with Neumann boundary conditions:
$$\begin{cases}\...
- 1,759
1
vote
0
answers
83
views
For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$
For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
- 825
1
vote
0
answers
135
views
Conformal laplacian on asymptotically flat manifolds with boundary
Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...
- 969
1
vote
0
answers
113
views
Computing a limit for the Weierstrass function
Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
- 4,153
1
vote
0
answers
105
views
Applications of finite speed of propagation property
Consider the Laplace operator $\Delta:=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$ on $\mathbb{R}^n$. Let $E_{\lambda}$ be the spectral resolution of $\Delta$, and
$$ H_{t}[f]:=\cos{(t\sqrt{-\Delta})}f=\int_{...
- 287
1
vote
0
answers
109
views
PDE coupled with the pronic numbers (related to triangular numbers)
I am studying the linear PDE:
$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
- 383
1
vote
0
answers
47
views
Existence for a nonlinear evolution equation with a monotone operator that is not maximal
We consider the nonlinear evolution equation
$$
\dot{u}(t) + Bu(t) = 0, \quad u(0)=0
$$
with
$$
A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...
- 171
1
vote
0
answers
109
views
$L^2(0,\infty;L^2(\Omega))$ estimate on solution of heat equation with Neumann boundary condition
Let $u$ be a solution of
$$u' - \Delta u = 0 \quad\text{on $\Omega$}$$
$$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$
$$u(t=0)=u_0\quad\text{on $\Omega$}$$
where $\Omega$ is a bounded ...
- 93
1
vote
0
answers
105
views
Friedrichs extension of the Laplacian from a smooth subspace and density of its eigenbasis in the Frechet topology of the subspace as well?
Let $C^\infty_\text{div}(\mathbb{T}^3)$ be the "real" Frechet space of periodic, divergence-free smooth vector fields on $\mathbb{R}^3$. That is, $\mathbb{T}^3$ is the $3$-dimensional torus.
...
- 3,477
1
vote
0
answers
96
views
Representation formula for the continuity equation on a separable Hilbert space
The following is an informal question for which I'd like to (ideally) find a reference. I'm quite a novice in this area but would be happy to find a reference to a theorem along the following lines (...
- 111
1
vote
0
answers
63
views
Solution to $u_t = A(t)u + f(t)$ on bounded domain
I am dealing with the problem
\begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\
\partial_{\nu} u &= 0 &\text{ on } \partial \Omega \...
- 11
1
vote
0
answers
89
views
Reference for an application of J. Simon "Compact sets in the space $L^p(0,T;B)$"
Theorem 5 p.84 of J. Simons paper "Compact sets in the space $L^p(0,T;B)$" states a generalization of the Aubin-Lions lemma which relaxes the required regularity in time to the existence of ...
- 469
1
vote
0
answers
111
views
Schrödinger equation approximation – continuity of eigenvalues with respect to potential
The question has been crossposted from Stackexchange after receiving no answers.
Setup: the time-independent Schrödinger equation (eigenvalue problem):
$(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$
(On ...
- 131
1
vote
0
answers
153
views
Maximal domain of an unbounded linear operator in a weighted Hilbert-space
Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
- 185
1
vote
0
answers
52
views
A question of interpolation space on homogeneous Carnot group
Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
- 287
1
vote
0
answers
74
views
"N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$
Let us consider the Burgers equation
$$u_t + (u^2)_x = 0$$
In
Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...
- 839
1
vote
0
answers
100
views
N-wave solution of conservation law $u_t + (u - u^2)_x = 0$
How can we compute the "N-wave" source-solution of the conservation law
$$u_t + (u - u^2)_x = 0, $$
that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
- 839
1
vote
0
answers
47
views
Scaling limit of transport equation with double-well potential
Let us consider the transport PDE
$$
u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon)
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...
- 839
1
vote
0
answers
180
views
A potential wrong proof of a Lemma
Consider the following lemma: Let $g \in H^s_{x,y}(S)$ where $S = \mathbb{R}^2$ or $S = \mathbb{T}^2$, and $\eta \in C^\infty(\mathbb{R})$, $\operatorname{Supp}(\eta) \subset [-2,2]$, and $\eta \equiv ...
- 159
1
vote
0
answers
59
views
Identification of a limit point of a sequence of solution of ODE
Let $v^0$ and $v^1$ be the following vector fields over $\big(\mathbb{R}_+^*\big)^3$: for $x\in\big(\mathbb{R}_+^*\big)^3$ and $1\leq i\leq 3$,
\begin{align*}
& v^0_i(x)=x_i(x_{i-1}-x_{i+1}) \\
&...
- 449
1
vote
0
answers
150
views
Analyticity of solutions to Schrödinger's equation
Take Schrödinger's equation on $\mathbb{R}$, $i\partial_t\psi(x,t)=H\psi(x,t)$. Assume that $\psi(x,0)$ has compact support. Using known integral formulas for the propagators, it is fairly ...
- 439
1
vote
0
answers
33
views
Limiting absorption principle for higher powers of resolvents
Let $H$, $A$ be self-adjoint operators on a Hilbert space. Moreover, let $I$ be a bounded open interval contained in the spectrum of $H$. Assume that $H$,$A$ satisfy the following positive commutator ...
- 141
1
vote
0
answers
48
views
Question about higher order mean field equation $\left(-\Delta_{g}\right)^{m} u+\lambda=\lambda \frac{e^{2 m u}}{\int_{M} e^{2 m u} d \mu_{g}}$
I'm reading Dr.Luca Martinazzi's paper
Existence of solutions to a higher dimensional
mean-field equation on manifolds
which proves that for $m \geq 1$, there is an existence result for the equation
$$...
- 809
1
vote
0
answers
152
views
Poisson Kernel and solution formula for fractional elliptic problem
$$
k (-\Delta)^s u + u = 0, \qquad x \in U, \\
u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,
$$
with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
- 839
1
vote
0
answers
597
views
What is $T T^*$ argument?
During my studying of many papers, some authors used what so-called $T T^*$ argument. I have no clue about this concept (or mathematical tool). Could you please enlighten me with some explanations or/...
- 159
1
vote
0
answers
57
views
How to show the solution map of NLS is not smooth?
Let $u(\delta, t)$ satisfy
$$iu_t +\Delta u+ |u|^{2k}u=0, \quad u(0)=\delta v_0$$
Note that the mapping:
$$\delta v_0\mapsto u(\delta, t)= S(t)(\delta v_0)-i\int_0^tS(t-\tau)(|u|^{2k}u)(\tau)d\tau $$
...
- 325
1
vote
0
answers
93
views
$H^s$ norm of dispersive semigroup
The Bourgain space is $X^{s,b} := X^{s,b}(\mathbb R \times \mathbb{T}^3)$ is the completion of $C^\infty (\mathbb R; H^s(\mathbb{T}^3))$ under the norm
$$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle} u(t,x)...
- 159
1
vote
0
answers
74
views
Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$
Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
- 63
1
vote
0
answers
177
views
Eigenvalues and eigenvectors of non-symmetric elliptic operators
We know that the operator $A=\Delta$ with domain $D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\}$ (say $\Omega$ is a bounded nice domain) has eigenvalues $\lambda_1>\lambda_2\ge \...
- 11
1
vote
0
answers
79
views
Reference for smoothness of Nemytskii operator on fractional Sobolev spaces
Let $\varphi:\mathbb{R}\to\mathbb{R}$ be smooth and bounded (together with all of its derivatives). Define the operator
$$
\big(N_\varphi x\big)(t)=\varphi\big(x(t)\big)
$$
for $x\in H^s(T^d)$, the ...
- 93
1
vote
0
answers
96
views
BV estimate for conservation law $u_t +( v(x)f(u))_x=0$
Let $u_0 \in BV(\mathbb R)$ and $f:\mathbb R \to \mathbb R$ be Lipschitz. Consider the Cauchy problem
$$
\begin{align*}
u_t +( v(x)f(u))_x&=0\\
u(0,\cdot) &= u_0
\end{align*}
$$
What is the ...
- 303
1
vote
0
answers
126
views
Estimates for the Benjamin-Ono equation
Consider the Cauchy problem for the Benjamin-Ono equation
$$u_t + \frac{1}{2}(u^2)_x + \alpha \mathcal H(u_{xx}) - \beta u_{xx} = 0, \qquad t>0, \ x \in \mathbb R,$$
where $\mathcal H$ is the ...
- 839
1
vote
0
answers
260
views
Closure of smooth functions in Besov spaces
For real numbers $\alpha > \beta$, we know there is a continuous embedding of Besov spaces $B^\alpha_{\infty,\infty}\subset B^\beta_{\infty,\infty}$. We take the closure of the intersection $C^{\...
- 253
1
vote
0
answers
127
views
Laplacian on the sphere and Moving Plane method
Consider the sphere $\mathbb{S}^n$ as a subset of $\mathbb{R}^{n+1}$, thus $\mathbb{S}^n=\{\omega\in \mathbb{R}^{n+1},\sum_{i=1}^{n+1}\omega_i^2=1\}.$
I am interested in studying positive solutions to ...
- 537