Questions tagged [linear-pde]
Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
370
questions
3
votes
1
answer
331
views
Neumann/Robin Laplacian semigroup well-known estimate
Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
- 1,330
3
votes
1
answer
131
views
Maximum principle for an elliptic like operator
I am trying to prove some monotonicity of a solution of a given pde; after considering a quantity like $ \phi(x) = x \cdot \nabla v(x)$ ($v$ is the solution of a given pde) I arrive at something ...
- 1,363
3
votes
2
answers
1k
views
Reference for De Giorgi-Nash-Moser theory
I am interested in Holder regularity for equations of the form
$$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.
This was proved in the seminal paper of John Nash ...
- 483
3
votes
2
answers
261
views
Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1
In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then $\...
- 31
3
votes
1
answer
1k
views
Stochastic Heat Equation
Given the heat equation:
$$\partial_{t}{\varPhi(x,t)}=k^2\partial_{xx}{\varPhi(x,t)}$$
with the boundary conditions:
$$\Phi(x,0)=\Phi_0$$
and a Neumann boundary condition of the kind:
$${\...
- 399
3
votes
1
answer
158
views
Solvability of general linear PDE with constant coefficients
Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$.
Is it true that the equation $$Du=f$$
is solvable in any ...
- 21.1k
3
votes
1
answer
190
views
Linear transport equation with Lipschitz conditions
Given the equation here, I would like to ask the following relaxed question:
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = ...
- 624
3
votes
1
answer
1k
views
Long time behavior of the heat equation on R
Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...
- 1,619
3
votes
1
answer
202
views
Schauder estimates with boundary conditions
For the elliptic equation with non-divergence form
$$
\sum_{i,j=1}^na_{ij}(x)\partial_{ij}^2u=f\text{ in }B(0,1)\quad\text{and}\quad u=g\text{ on }\partial B(0,1),
$$
where $ \{a_{ij}(x)\} $ is a ...
- 1,133
3
votes
1
answer
87
views
Space of holomorphic functions multiplied by smooth functions taking real values
Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \...
- 33
3
votes
1
answer
198
views
Eigenfunctions of elliptic equations
Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. ...
3
votes
2
answers
264
views
A Global Estimates for Linear Elliptic PDE
Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy
$-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$,
where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following ...
3
votes
1
answer
4k
views
When is separation of variables an acceptable assumption to solve a PDE?
We know that one of the classical methods for solving some PDEs is the method of separation of variables. It works for known types of PDEs and many examples of physical phenomena are successfully ...
- 183
3
votes
1
answer
551
views
Extending a harmonic function in a ball to subharmonic in a larger ball
Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$:
$$
\begin{cases}
-\Delta u &= 0, \quad \text {in} \quad B(r), \\
\ \ \ \ \ \, u&= g, \quad \text {in}\quad \...
- 632
3
votes
1
answer
908
views
Solving Stokes Equations using 3D Fourier transforms
How do you calculate the inverse Fourier transform of $\frac{k_ik_j}{k^4}$. I know it has to be a matrix of the form $=δ_{ij}A(r)+r_ir_jB(r)$, but how do you calculate the functions A(r) and B(r)?
I ...
- 31
3
votes
1
answer
326
views
Reference request: Schauder estimates for parabolic equations
Where can I find Schauder estimates for second order linear parabolic equations (in divergence form with potential)?
Any reference would be highly appreciated.
- 193
3
votes
2
answers
1k
views
Orthogonality to harmonic functions
Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define
$a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$
$b_n=f\Delta^{...
3
votes
1
answer
392
views
Harnack Inequality
In the Han and Lin book there is a Harnak inequality for elliptic operators of the kind:
$$
L u = D_i \big(a^{ij}\, D_ju\big),
$$
and the constant $C$ in the Harnack inequality does not depend on the ...
- 823
3
votes
1
answer
596
views
When does an inverse PDE operator have a kernel (i.e. a fundamental solution?)
Let $L$ be an elliptic linear operator on $\mathbb R^n, n\geq3$. For simplicity, let's stick to the following Schrodinger operator
$$
Lu:=-\Delta u+V(x)u
$$
where $V\geq0$ is the electric potential, ...
- 391
3
votes
1
answer
328
views
elliptic regularity of Neumann problem on Square
I asked a similar question the other day, but I will be more precise now.
Consider $ \Omega:=(0,1 ) \times (0,1)$ and consider
$$ - u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) \mbox{...
- 1,363
3
votes
1
answer
340
views
Pseudoinverse of Neumann-Laplacian
Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that
$$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$
Further assume a solvability condition
$$\int_\Omega f ~\mathrm{d}\...
- 31
3
votes
1
answer
419
views
Ancient Heat equation and Liouville's theorem
I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation:
\begin{equation}
u_{t} = \...
- 139
3
votes
1
answer
506
views
Uniqueness of weak solutions of a heat equation
Let $M$ be a smooth compact closed manifold.
Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of
$$u_t - \Delta u - u = 0$$
$$u(0)=u(T)$$
satisfying $\int_M u(t) = 0$ for all $t$. Is ...
- 33
3
votes
1
answer
288
views
Monotonicity preserving parabolic operators
Let
$$
\mathcal{L}\equiv\sum_{i,j}^{n}a_{i,j}\frac{\partial^{2}}{\partial x_{i}\,\partial x_{j}}+\sum_{i}^{n}b_{i}\frac{\partial}{\partial x_{i}}+c
$$
be uniformly elliptic on $\Omega\equiv\mathbb{R}^{...
- 331
3
votes
1
answer
192
views
The maximum in the Poisson problem on the cube with constant source
Question:
Let us consider the Poisson problem on the square with constant source $1$
$$
\begin{cases}
- \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\
u &= 0, \qquad \text{ on } \partial (...
- 1,081
3
votes
0
answers
57
views
Convex combination of cyclically monotone sets
I want to show the following statement, but I am not sure how.
Proposition(?):
Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions.
Suppose
$$...
- 51
3
votes
0
answers
59
views
Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field
Setting
Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
- 866
3
votes
0
answers
93
views
Comparison principle for Elliptic PDE with exponential nonlinearity
Suppose $\varphi$ is a radial (and radially decreasing) solution of
$$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$
with $ R>0$, and $\psi$ is a decreasing radial function ...
3
votes
0
answers
94
views
On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections
Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
3
votes
0
answers
97
views
How to find a particular solution of a non-homogeneous parabolic partial differential equation
Consider the following non-homogeneous parabolic partial differential equation (PDE)
\begin{align}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \...
- 91
3
votes
1
answer
221
views
Is there a metric on Euclidean space that turns the Helmholtz equation into the Laplace equation?
Is there a Riemannian metric $\tilde g$ on $\mathbb R^d$ such that
$$\tag{1}
\Delta_{\tilde g}=e^f(\Delta +1),$$
for some $f\in C^\infty(\mathbb R^d)$? Here $\Delta=\partial_{x_1}^2+\ldots+\partial_{...
- 682
3
votes
0
answers
85
views
Semiclassical analysis and reflection law
I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of ...
- 445
3
votes
0
answers
41
views
Reference request: existence of strong solutions to a linear parabolic problem with mixed boundary conditions
on a domain $\Omega \subset \mathbb{R}^d$ with smooth boundary $\partial\Omega$ subdivided into two parts $\Gamma_D$ and $\Gamma_N$ I am considering the parabolic problem
$$
\partial_t u = \Delta u + ...
- 51
3
votes
0
answers
66
views
How to solve this linear Cauchy Problem
within my thesis, I am struggeling with the following PDE:
$u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$
$u(T,x,y)=1,$
where $a,b,c,d,e,f$ are polynomials and the ...
- 31
3
votes
0
answers
316
views
Critical spaces and energy estimate in NS equation [closed]
There is a ‘rescaling transformation’ that is particularly significant for
the Navier–Stokes equations when they are posed on the whole space, but is
also important in the local regularity theory.
...
- 139
3
votes
0
answers
120
views
Partial regularity for transmission problem in corner domains
Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
- 133
3
votes
0
answers
554
views
2D laplace equation with Robin boundary condition (Green function)
Let's say that I know a fundamental solution for the Laplace equation in the whole plane:
$$\nabla^2u=\delta\quad \text{in the sense of distributions,}$$
and I need a solution for the laplace equation ...
- 269
3
votes
1
answer
159
views
Solving a system of equations involving smooth functions
I have asked the following question in math.stackexchange, but I could not receive the answer. See here.
Suppose $h_{i\overline{j}}$, where $1\leq i, j\leq n$, are functions defined on $\mathbb{C}^n$ ...
- 501
3
votes
0
answers
76
views
Smoothing inside the null space of a partial differential operator
Let $L$ be a linear partial differential operator with smooth coefficients in $U\subset\Bbb R^n$ and let $u\in W^{k,p}_{loc}(U)$ with $k\in\Bbb N$ and $p\in[1,\infty[$ satisfy $Lu=0$ in the ...
- 31
3
votes
0
answers
198
views
Analytic solution to two component, first order, linear PDE system
I would like to obtain analytic solutions to the following PDE system:
\begin{equation}
\rho_t + D(\lambda)\,\rho_\lambda = A(\lambda) \rho, \tag{1}
\end{equation}
with $\rho = (\rho_0,\rho_1)^T$, $D$ ...
- 268
3
votes
0
answers
332
views
Method of characteristic for a system of first order PDEs
I am working with this system of first order PDEs:
\begin{equation}
\left\{
\begin{aligned}
%Suscettibili
&\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,...
- 31
3
votes
0
answers
106
views
Constant in a trace Sobolev theorem for concave domains
I wonder is the following inequality is true/known:
Let $\Omega\subset \mathbb{R}^n$ be a (locally) Lipschitz domain which is the complement of a convex set, then
$$
\int_{\partial\Omega} |u|^2 ds
\...
- 175
3
votes
0
answers
381
views
Laplace Equation with Tangential Derivative Prescribed on the Boundary [closed]
I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :)
Consider the following Laplace boundary value problem (BVP)
$$\matrix{
{{\nabla ^2}\Phi (x,y)...
- 271
3
votes
0
answers
163
views
How can one do change of variables for solutions to a staochastic partial differential equation?
isHow can one do change of variables for solutions to a staochastic partial differential equation? For example, let us consider the following stochastic transport equation:
$$
dy(t,x) + y_x(t,x) + y(t,...
- 31
3
votes
0
answers
150
views
Quadratic forms over symmetric $3\times3$ matrices
This question is motivated by a technique called compensated compactness (CC), which was elaborated by L. Tartar and F. Murat for the analysis of PDEs. The foreword of CC is a problem about quadratic ...
- 51.6k
3
votes
0
answers
427
views
Boundary regularity of the solution to the Beltrami equation
Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference:
Let us consider the orientation-preserving homeomorphic solutions $f: D \...
- 1,451
2
votes
1
answer
531
views
Does this PDE only have the trivial solution?
Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...
- 366
2
votes
1
answer
252
views
How to trap a particle without using potential field which is infinity at some point? (quantum physics) If impossible, how to prove it?
As we all know, the wave function of the stationary state a quantum particle trapped in a rigid box (with infinite potential outside the box) cannot have a non-zero value outside the box. So can we ...
- 21
2
votes
3
answers
730
views
How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets
$Qn#1 $
: Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak ...
- 1,451
2
votes
1
answer
265
views
Even and odd solutions for the Schrödinger equation
We consider $2a$ - periodic smooth solutions for
\begin{eqnarray*}
-\Delta u+V(x)\,u=0\qquad\hbox{in}\:[-a,a]
\end{eqnarray*}
We assume that $V$ is smooth and even (i.e. $V(-x)=V(x)$). We also assume ...
- 319