Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
1
vote
0answers
39 views
Nodal domains on a surface
What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators?
In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...
1
vote
0answers
46 views
Uniqueness of solution of linear PDE of first order
Let $\vec u\colon \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^k$
be at least $C^1$- smooth vector valued function. Assume they satisfy a first order linear equation
$$\partial_t \vec u(x,t)=\sum_{j=1}^...
1
vote
1answer
97 views
Existence of unique critical points to second order elliptic PDEs
Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1)
$$
\begin{cases}
Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...
1
vote
0answers
15 views
Rankine Hugoniot Condition for non piecewise smooth solution
I studied the following theorem: (Rankine-Hugoniot condition)
Let $u:\mathbb{R} \times [0,+\infty) \to \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution of the conservation law ...
4
votes
1answer
122 views
Method of characteristics beyond the Lipschitz setting
I have come across the following easy-looking problem that is driving me mad.
I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...
0
votes
0answers
26 views
Change of variable for the Stokes equations
I asked this question to the Mathematics community but had no response (https://math.stackexchange.com/q/2885217/521741).
Let $\Omega$ be domain of $\mathbb{R}^n$ and $\Phi : \Omega \to \Phi(\Omega)$ ...
3
votes
0answers
46 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 ...
1
vote
0answers
31 views
Random solute transport equation
After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
0
votes
1answer
47 views
Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)
I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting.
I want to verify and compare different Discretizations of the ...
0
votes
0answers
31 views
What is the minimum setting in which regularity results are available for the solutions of Poisson's equation?
Let the generator $L$ of a diffusion process be given in Hörmander form, i.e.
$$L=\frac{1}{2}\sum_{i=1}^k X_i^2+X_0,$$
where $k\leq n$ and $X_i$, $i=0,1,...,k$, are vector fields on $\mathbb{R}^n$. ...
0
votes
2answers
118 views
Solving the Poisson equation using a random walk on $\mathbb Z ^d$
How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
2
votes
0answers
69 views
Observability inequality for the heat equation
I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions:
\begin{array}{c}
\varphi _{t}+\...
1
vote
0answers
116 views
Underdetermined PDE
Let $\Omega\subset\mathbb{R}^d$, $d>1$ be an open and bounded Lipschitz domain. It is well-known, that for a given function $f\in L^q(\Omega)$ with $1<q<\infty$ and and such that $\int_\Omega ...
0
votes
1answer
112 views
Unique continuation for the wave equation
Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation
$$u_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$
where $f$ and $1-c$ ...
0
votes
1answer
72 views
System of first order linear coupled PDEs [closed]
I'm in trouble finding the solution of this system of 2 PDEs:
\begin{equation}
\frac{\partial u_1}{\partial t} + a_1 \frac{\partial u_1}{\partial x} = b (u_1-u_2)\\
\frac{\partial u_2}{\partial t} + ...
2
votes
2answers
263 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 ...
1
vote
1answer
90 views
schauder regularity heat equation
Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.
It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\...
5
votes
1answer
121 views
The division problem for tempered functions
It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...
1
vote
0answers
40 views
Convex solutions of linear hyperbolic PDEs in a planar domain
Consider a linear homogeneous 2nd-order PDE in a convex planar domain $\Omega$ :
$$a(x,y)\frac{\partial^2u}{\partial x^2}+2b(x,y)\frac{\partial^2u}{\partial x\partial y}+c(x,y)\frac{\partial^2u}{\...
2
votes
1answer
51 views
Positive form for a homogeneous elliptic pde
I have a pde of the following form:
\begin{align}
&P(x,D)u = f \text{ on } \Omega, \\
&P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha},
\end{align}
where one can assume that $f$ ...
1
vote
0answers
134 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.
...
8
votes
3answers
313 views
What does the flow of the principal symbol of the differential operator tell us about the PDE?
Disclaimer: Let me apologize in advance for asking this slightly vague question
Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...
0
votes
1answer
121 views
Is fractional Laplacian invariant under rotation?
If $\Delta u=0$, then $\Delta u(Ox)=0$, where $O$ is an orthogonal matrix. From here, do we know whether fractional Laplacian is invariant under rotation? We use the usual definition of fractional ...
4
votes
0answers
74 views
Linear PDE with non constant coefficients and properties of Green's Function
Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case
\begin{...
1
vote
0answers
31 views
Regularity of Poisson problem with rough coefficients and mixed boundary conditions
Let $d \in \mathbb N$ and $\Omega \subseteq \mathbb{R}^d$ be open, bounded and connected. Let the boundary $\partial \Omega$ be piecewise Lipschitz and partitioned into a Neumann part $\partial \...
1
vote
1answer
105 views
Question on expansion into Neumann eigenfunctions
Let $\Omega$ be an open bounded domain with a boundary $\partial\Omega$. Consider the following Neumann eigenvalue problem for Laplacian: find $(\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$
\...
1
vote
0answers
56 views
Why does a homogeneous first-order linear PDE have $n-1$ functionally independent solutions? [closed]
Why does a homogeneous first-order linear PDE$$\sum_{i=0}^n\xi_i(\mathbf{x})\frac{\partial u}{\partial x_i}=0,$$where $\mathbf{x} = (x_1, x_2, …, x_n)$, have $n-1$ functionally independent solutions ...
2
votes
0answers
89 views
What type of boundary (if any) problem for this family of elliptic PDEs? “half boundary”?
Classic literature for a general elliptic PDE with Dirichlet boundary condition is typically studied with the following set up: Let $\Omega \subset R^n$ be some open bounded domain and $\partial \...
0
votes
1answer
77 views
How do I show continuity of the mixed and weak solution to Zaremba problem?
I am interested in showing continuity/boundedness of the weak solution to the following problem pde:
\begin{align*}
0 &= \mathbf{q} + \mathbf{\nabla}u && \quad x\in \Omega,\\
0 &= \...
2
votes
0answers
81 views
Singularity of the solution of a PDE whose coefficients have zeros
The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):
$$\mathcal{A}p=0, \quad p\in C^2(\...
0
votes
1answer
329 views
Is this function positive?
Could someone tell me if my argument is correct?
Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$, I have a system of two coupled PDE's and I proved that its solution $(u_0(t, r), u_1(t, ...
5
votes
1answer
221 views
Existence of second order potential for PDE
There is a statement in the literature (see the paragraph between equations (18) and (19) in http://aip.scitation.org/doi/10.1063/1.523863), which I would like to generalise, but I don't have a nice ...
1
vote
0answers
96 views
Maximum principle on noncompact manifold with boundary
On a complete noncompact manifold with ricci curvature bounded below, we have Yau's generalized maximum principle. What if we have we have noncompact manifold with compact boundary and one complete ...
1
vote
3answers
188 views
BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?
I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...
1
vote
0answers
37 views
Nonautonomous wave equation of memory type
I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation
$$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$
This problem can be written ...
3
votes
2answers
166 views
General formula for integrating factor of an homogeneous differential 1 form
This question is probably very elementary but I don't know how to tackle the conversely part of the following result. Let $M(x,y)$ and $N(x,y)$ be two differentiable and homogeneous functions of the ...
0
votes
1answer
91 views
Poisson Equation across a Hypersurface [closed]
Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem
$ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
2
votes
0answers
198 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 = ...
1
vote
0answers
133 views
Uniqueness in a linear elliptic PDE
Let $a \in L^{\infty}(\Omega)$ and suppose $a$ is continuous near the boundary $\partial \Omega$. Assume $w\in H^1(\Omega)$ solves
$\Delta w=a w$ in $\Omega$ with $w=\frac{\partial w}{\partial \nu}=...
1
vote
1answer
80 views
Decay time to constant function of heat kernel on 2-sphere
Let us consider solving the heat equation on the sphere given a delta function as initial data.
$$(\partial_t - \Delta)K(x,y;t) = 0 $$
$$K(x,y;0) = \delta(x,y)$$
One would expect that for large ...
2
votes
1answer
124 views
Growth at infinity of a solution to a parabolic PDE
Let us consider the equation:
\begin{align*}
(\partial_t - \Delta - b(t,x) \partial_x) u(t,x)& = f(t,x) \\
u(0,x) & = u_0
\end{align*}
defined on the whole real line (so in one dimension - but ...
3
votes
2answers
443 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^{...
1
vote
1answer
79 views
Examples of the time-dependent linear wave equation
I am looking for examples of the non-autonomous linear wave equation that have some relevant applications.
What articles and sources (maybe, some catalogue like Handbook of Nonlinear Partial ...
0
votes
1answer
50 views
Improved maximum principle estimates (deleting first mode)
Recall given any function $v(x)$ defined on $B$ (the unit ball centred at the origin in $ R^N$) we can write
$$v(x) = \sum_{k=0}^\infty a_k(r) \psi_k(\theta)$$
where $ r=|x|$ and $ \theta = \frac{...
1
vote
0answers
47 views
An existence result for solutions of elliptic equations with a mixed boundary problem
Assume that $\Omega$ is a bounded domain such that
$\partial\Omega=\Gamma_1\cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint and closed. Let us consider the following elliptic equations.
...
4
votes
1answer
541 views
Geometry of curves on the sphere
Let P be a finite set of points on the unit sphere $S^2$ such that
for every $p\in P$, there exists a closed curve $\gamma_p \subset S^2$ which has a self intersection at $p$ and passes through $-p$....
1
vote
1answer
219 views
Functions orthogonal to harmonic functions
Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose
$\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\...
5
votes
0answers
95 views
Continuous right inverse to the Laplacian operator on $C^\infty$
For each $f\in C^\infty(\mathbb R^n)$, there exists $u\in C^\infty(\mathbb R^n)$ such that $\Delta u=f$. This, I guess, has been known well before the more general Malgrange-Ehrenpreis theorem that ...
3
votes
0answers
69 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 ...
3
votes
1answer
96 views
For an arbitrary $G(x,t)$, does $f_t=2G_xf+Gf_x$, $f(x,0)=0$ have a unique solution for $f$?
I asked the following question at Math Stackexchange a while ago here but did not get a correct answer.
Let $f(x,t)$ and $G(x,t)$ be smooth functions from
$\mathbb R^2\to\mathbb R$.
The ...
