Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4,467 questions
0
votes
0
answers
96
views
partial maximum principle for elliptic differential operators
Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $P$ be a self-adjoint, elliptic differential operator defined on $C^\infty(M)$ with smooth coefficients. Suppose as well that the lowest ...
- 1,701
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 ...
- 4,729
6
votes
1
answer
667
views
Possible mistake in De Giorgi's paper on Holder's regularity
$\mu_{n-1}$ is the $n-1$ dimensional measure and $\operatorname{meas}$ is the $n$-dimensional one.
$I(\varrho)$ is the ball of radius $\varrho$ around a fixed point $y$ in the domain $\Omega\subset \...
- 16.8k
4
votes
1
answer
243
views
On-diagonal to off-diagonal heat kernel lower bounds, Davies' argument
Theorem 3.3.4 in Davies' Heat Kernels and Spectral Theory begins with ``on-diagonal'' lower bounds for the heat kernel $K$ of $H$, (i.e. $K = e^{-Ht}$), where $H$ is a uniformly elliptic operator ...
- 7,197
9
votes
1
answer
935
views
Question about an estimate in Hörmander's proof of Cartan's Theorem B
I have been working through the proof of Cartan's Theorem B that Hörmander gives in his book 'Introduction to Complex Analysis in Several Variables'. When I began, I skipped over some of the initial ...
- 1,043
10
votes
1
answer
957
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 $H^1(M)$ and $L^2(M)$.
If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ ...
CommunityBot
- 1
3
votes
1
answer
138
views
Is $R^n$ stochastically complete for the heat kernel of a Schrödinger operator?
Suppose $V:\mathbb{R}^{n} \to \mathbb{R}$ is just a positive polynomial and $K_{t}(x,y)$ is the heat kernel of $H = -\Delta + V$. Then does it follow
$$\int_{\mathbb{R}^{n}} K_{t}(x, \cdot)\,dy = 1?$$...
- 2,715
0
votes
0
answers
100
views
Parabolic PDE; uniform bound on approximations $u'_n$ in $L^2(0,T;V^*)$ without using orthogonal basis?
Let $V \subset H \subset V^*$ be a Gelfand triple, all Hilbert and separable spaces.
I consider the PDE with weak form: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that
$$\langle u'(t), v(...
- 405
2
votes
0
answers
93
views
Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces
Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
- 39.1k
1
vote
2
answers
550
views
$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?
Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?
CommunityBot
- 1
5
votes
1
answer
1k
views
Regularity for transport equation?
In the book of Evans the transport equation,
$$\frac{d}{dt} u + b\cdot \nabla u = 0, \quad u(t=0)=u_0,$$
is solved by the method of charateristics for $b$ and $u_0$ smooth enogh (in terms of $\mathcal{...
CommunityBot
- 1
1
vote
1
answer
198
views
Solution formular for Laplace equation [closed]
I want to slove the Laplace equation on $R^3_+$ with Neumann boundary condition. The equation reads:
$-\Delta u = f$ in $R^3_+$,
$\partial_3 u|_{x_3=0}=g$ on $R^2$.
If $f$, $g$ satisfy compatibility ...
CommunityBot
- 1
3
votes
0
answers
261
views
Idea behind distributional solutions
I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
- 85
1
vote
0
answers
839
views
A variation of Poisson's equation in cylindrical coordinates
Our team of undergraduate physicists are familiar with finding numerical approximations to the following Poisson-like PDE central to our plasma research in a torus:
$\nabla^2 V = \frac{f(V)}{R^2}$
...
CommunityBot
- 1
2
votes
1
answer
380
views
Best approach to solve this PDE
I need to solve this Partial Differential Equation for $\lambda(x,y)$,
$$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...
- 11
1
vote
1
answer
141
views
metric scaling for an inequality
I read a lemma 1.12 in Tobias H.Colding's paper "Ricci curvature and volume convergence"."Suppose that $Ri{c_{{M^n}}} \ge \left( {n - 1} \right)\Lambda {R^{ - 2}}$,p and $q \in M$ with d(p,q)>8R,and ${...
- 6,871
2
votes
0
answers
109
views
Regularity of solution of nonlinear equation
Hi!
Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball
$B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$.
Let $f\in C^{0,\alpha}(B_1)$ ...
- 2,715
1
vote
1
answer
384
views
Reference for: $C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$
If it is true, where may I find a reference/proof for:
$C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$
where $H$ is a Hilbert space.
Thanks
- 25.3k
1
vote
0
answers
122
views
Checking initial condition of PDE is satisfied in Galerkin method
I asked this question here: https://math.stackexchange.com/questions/416885/checking-initial-condition-of-pde-is-satisfied-in-galerkin-method
But I did not receive the solution so I post it here.
...
CommunityBot
- 1
2
votes
2
answers
413
views
Replacing large-dimensional ODE systems with one PDE [closed]
Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?
- 7,197
0
votes
0
answers
113
views
Reference Search for a Functional Minimization Problem
Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is
$$M \equiv\int_{x=-\infty}^\infty u(x) ~ ...
- 151
3
votes
2
answers
236
views
Boundedness of Solutions to $\Delta u = f u$ on $R^2$
Consider the Laplacian $\Delta = d/dx^2 + d/dy^2$ on $\mathbb{R}^2$.
This is true: Let $f$ be a nonnegative function, not identically zero. Then any positive solution of $\Delta u = f u$ is ...
- 11
7
votes
2
answers
905
views
Fredholm alternative result for general elliptic system?
Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
- 3,322
3
votes
0
answers
283
views
Examples of non-uniqueness in reaction-diffusion equations
Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on $\mathbb{R}\...
- 193
4
votes
2
answers
499
views
Analytic dependence on the metric
It is often used implicitly that the maps which associate to metrics curvature quantities (Riemann, Ricci, scalar curvature) and Differential operators like the Laplacian are analytic maps between ...
CommunityBot
- 1
1
vote
0
answers
103
views
Generalized bilinear estimates
Hello. Let $ X^{s,b} $ be the Bourgain space generated by $ \tau - \xi^3 $. It is proved that, for $ s\in (-\frac{1}{2}, 0] $, we have
$$
\|(u^2)_x\| _{X^{s,b'-1}} \leq c \|u\|_{X^{s,b}} \|u\|_{X^{-...
- 25.8k
5
votes
1
answer
471
views
Please recommend some literature on the systematical theory of the elliptic systems!
Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
- 3,322
3
votes
0
answers
252
views
Existence of solutions to a reaction-diffusion problem.
Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on $\mathbb{R}\...
- 193
1
vote
1
answer
543
views
Regularity of the right hand side (the source term) in Evans-Krylov theory
A well-known theorem of Evans and Krylov states that in an equation of the form $F(D^2 u)=g$, provided that the right hand side and $u$ both have Lipschitz gradient, and that $F$ is concave or convex ...
- 4,899
4
votes
2
answers
264
views
Elliptic Harnack inequality for 1D Schrodinger operator?
For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is:
There exist $C_{H} > 0$ and $\delta \...
- 4,899
0
votes
3
answers
502
views
I have this linear PDE...
Hi,
The PDE in question is: $A P_{yy}(y,z) + B P_{zz}(y,z) + ( [ C y -D z] P(y,z) )_y + ( [ D y + C z ] P(y,z) )_z=0,$
where subscript $y,z$ indicates derivatives and $A,B,C,D$ are real. The PDE is ...
- 13k
1
vote
1
answer
172
views
If a function is defined in terms of a solution to an initial value problem, is it also solution to an initial value problem?
Say $f:\mathbb R^{n+1}\to \mathbb R^p$ is a solution to an initial value problem, and $g:\mathbb R^{n+1}\to \mathbb R^q$, so that the components of $g$ can be expressed as polynomials in $f$, $f'$, ...
- 21.5k
0
votes
2
answers
812
views
Estimates for Green's function
Let $n$ - dimension $\geq 3$.
Consider a compact manifold (M,g). Let $\epsilon_0$ denote the injectivity radius of $(M,g)$. Let $B_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < \...
- 4,729
2
votes
2
answers
542
views
The logarithmic fast diffusion equation in one space variable with periodic boundary conditions.
I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$:
$$ u_t = (\ln u)_{xx}$$
which is to run on an interval $ a \leq ...
- 23
0
votes
1
answer
347
views
Dual space of Bochner space: is there an easier proof to show they're isometric?
It is known that $[L^p(0,T;H)]^* = L^q(0,T;H^*)$.
If $p=q=2$ and $H$ is a Hilbert space, is there an easier proof to show that the spaces are isometric? The proof that I know for the general case ...
- 136
9
votes
2
answers
1k
views
PDE with the Jacobian Determinant
Hello,
Could you please help me in answering the following question?
Initially I thought that the following problem can be solved through Monge-Ampere equation, but with Monge-Ampere, I have not ...
- 11
1
vote
0
answers
76
views
h-oscillating function
I need help understanding the following condition:
$u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the flat torus, is ...
- 11
2
votes
1
answer
1k
views
Strong convergence in the Bochner space L^p([0,T],X)
Dear mathoverflowers, I have a question concerning the strong convergence in $L^p([0,T],X)$.
Let $X_1,X$ be two Banach spaces such that $X_1\subset X$ with compact embedding. Let $x_n(t)\in X_1$ be ...
- 5,380
0
votes
0
answers
270
views
Gradient estimates for subsolutions of elliptic equations
Let $M$ be a Riemannian manifold. Assume $u \in C^\infty(M)$ such that $u>0$ and
$\Delta u + \lambda u = 0,$
where $\lambda \geq 0$. There is a poinwise estimate for $|\nabla u|$ in Peter Li's ...
- 56
1
vote
1
answer
339
views
$C_c^{\infty}([0,T];V)$ is dense in $C_c^{1}([0,T];V)$?
Is it true that the space $C_c^{\infty}([0,T];V)$ is dense in $C_c^{1}([0,T];V)$? These are compactly supported functions that are $V$ valued, where $V$ is a Banach or Hilbert space.
- 16.4k
6
votes
1
answer
474
views
Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$
How is the proof that
$$[L^2(0,T;X)]' = L^2(0,T;X')$$
looking like, where $X$ is a Hilbert space? I am asking for the proof that the dual space of $L^2(0,T;X)$ is the space $L^2(0,T;X^*)$.
Is the ...
- 4,729
1
vote
1
answer
294
views
A heat kernel for Schrödinger operator with low-order terms
In "Schrödinger Operator: Heat Kernel and Its Applications", Feng computes the heat kernels associated to Schrödinger operators with at most quadratic potentials.
I am trying to see how these work ...
- 1,687
3
votes
1
answer
2k
views
Regularization by mollifier sequence
A well-known feature used in PDE's is the regularization by convolution with a mollifier sequence $\rho_n$, i.e. $\rho_n(x) := n^d \rho(nx)$ with $x \in \mathbb R^d$, $\rho \in C^\infty_c(\mathbb R^d)$...
- 843
8
votes
1
answer
5k
views
integration by parts for the fractional Laplacian
Is there an integration by parts formula for fractional laplacians in $L^p(\mathbb{R}^N)$, something like
$$
s\in(0,1),\qquad\int\limits_{\mathbb{R}^N}f[(-\Delta)^sg] =\int\limits_{\mathbb{R}^...
- 843
1
vote
2
answers
1k
views
Heat equation with Neumann BC
Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition in a bounded domain $\Omega$.
Is this true to say:
$$\|u(. , t)-v(. , t)\|_p\leq \|u(. , 0)-v(. , 0)\|_p$$ where $u$ and $v$ ...
- 843
1
vote
0
answers
291
views
Laplacian type operator on compact Lie group
Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. ...
- 11
1
vote
1
answer
227
views
Double series solution of wave equation
Let $u(x,y,t)$ be the solution of wave equation $u_{tt}=u_{xx}+u_{yy}, 0\lt x\lt 1, 0\lt y\lt 1, t\ge 0,$ $u(x,y,0)=(x-x^2)(y-y^2), u_t(x,y,0)=0$ and $ u(x,y,t)=0$ on the boundary of the square. Then
...
CommunityBot
- 1
0
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 $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in L^2(...
- 113
3
votes
1
answer
733
views
Trace theorem for manifolds with boundary
Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality
$$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$
will hold.
...
CommunityBot
- 1
1
vote
0
answers
149
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 k^{0.99}}e^{ikx}e^{ily}=\frac{1}{\sqrt{2\...
- 11