Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
2
votes
1answer
70 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 $r$:
$$
\begin{cases}
-\Delta u &= 0, \quad \text {in} \quad B(r), \\
\ \ \ \ \ \, u&= g, \quad \text {in}\quad ...
0
votes
0answers
23 views
Parabolic partial differential equation, initial conditions
Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tg+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
0
votes
0answers
48 views
Equating two systems of PDEs [migrated]
I'm trying to relate two sets of a pair of PDEs, but massively struggling! They should be equivalent up to a linear transformation. Any help would be wonderful!
First set:
$$\frac{\partial ...
0
votes
0answers
81 views
Solvable PDEs and their Green's functions
I have a class of PDEs of the form
$$
-\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0
$$
with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...
1
vote
1answer
183 views
Reference request: Boundary behavior and quantitative lower bound for the principal eigenfunction of an elliptic PDE in a ball $B(r)$
Consider the elliptic eigenvalue problem
$$
\begin{cases}
\int_{B(r)} A(x) \nabla u \cdot \nabla \phi \, dx &= \ \ \frac{\lambda_1}{r^2}\int_{B(r)} u \phi \, dx \\
\qquad \qquad \qquad \quad ...
2
votes
0answers
47 views
The centralizer of Lienard equation
Consider the lienard vector field $\cases{
x'=y -F(x) \\
y'=-x }
$ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
2
votes
0answers
62 views
Linear heat equation with initial condition of generalized function
I am consider a very simple heat equation over the interval $[0, 1]$ with a Neumann BC and a very bad initial condition, written as:
$\partial_tu(t, x) = \partial^2_xu(t, x) + a(t, x)u(t, x)$, for ...
1
vote
0answers
97 views
Comparison principle for partial differential equation with singular coefficients
How (or if) a comparison principle works in the case of equations
singular at some point? For example, I am analyzing a partial
differential equation
$$
...
1
vote
0answers
66 views
Analyticity of one-dimensional PDE solutions with respect to the space variable
Let $n>1$ and $u$ be a solution of a linear PDE with constant coefficients
$$
u_t-\sum_{k=0}^n a_k \partial_x^k u=0,\quad a_k\in \mathbb C,\quad a_n\ne0,
$$
in some neighborhood of a point ...
1
vote
1answer
94 views
Number of linear independent equations
Is there any general rule to find the number of linearly independent equations such that
$$L_i(T_{\mu\nu},\partial_\eta T_{\mu\nu},\partial_\omega\partial_\eta T_{\mu\nu},...)=0$$
where $L_i$ is a ...
1
vote
1answer
162 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
$$\int_\Omega f ...
0
votes
1answer
186 views
Does Frobenius theorem apply to vector-valued function?
We know Frobenius theorem handle pde systems like
$\{Xf=0, Yf=0\}$
requiring Lie bracket $[X,Y]\equiv 0 \mod X, Y$ for completely integrability of the system. However, how to handle systems like ...
4
votes
2answers
236 views
Numerical solution to diffusion-like equation with negative diffusion coefficient region?
I am trying to numerically solve the initial value problem (see later discussion for ICs)
$$ x \frac{\partial f}{\partial t} = \frac{\partial}{\partial x} (1-x^2) \frac{\partial f}{\partial x} - f$$
...
1
vote
2answers
93 views
Bound deg 3 partial differential operator on Laplace eigenfunction?
I am no expert on PDE and analysis but I am looking for certain technique from PDE.
Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...
2
votes
2answers
111 views
Is the left regularizer for elliptic BVP a left inverse for the principal part?
Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
2
votes
1answer
56 views
Solutions of a stochastic reduced wave equation
Given a spherical reference frame $\left(\rho,\phi,\theta\right)$, the reduced wave equation can be written as:
$$\nabla^2U=k^2n^2U$$
in which:
$U=U(\rho,\phi,\theta)$
The solutions of this equation ...
3
votes
1answer
91 views
To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function
I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context ...
3
votes
1answer
119 views
Monotonicity preserving parabolic operators
Let
$$
\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
$$
be uniformly elliptic on ...
2
votes
1answer
239 views
Best approach to solve this PDE
I need to solve this Partial Differential Equation for $\lambda(x,y)$,
$$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...
7
votes
2answers
203 views
Fredholm alternative result for general elliptic system?
Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
4
votes
1answer
149 views
Please recommend some literature on the systematical theory of the elliptic systems!
Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
2
votes
1answer
261 views
Uniqueness of classical solution with degenerate boundary
Consider heat equation on the domain $\Omega = (0,1)\times (0,1)$
in the form of
$$ \partial_{t} u = \frac 1 2 x^{3} (1-x) \partial_{xx} u, \quad
(x,t) \in \Omega$$
with initial data
$u(x,0) = x$ for ...
0
votes
1answer
87 views
The sufficient condition of the Neumann problem
How can we prove that if $\int_U{f}=0$,then the homogeneous Neumann problem $\Delta u=f$on U,and $\frac{\partial u}{\partial n}=0$ on $\partial U$ has a weak solution in $H^1(U)$?
0
votes
1answer
146 views
Concerning Fritz John's article, The Ultrahyperbolic Differential Equation With Four Independent Variables
I am trying to read Fritz John's article, here:
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.dmj/1077490637
And for the proof of Thm 1.1 in page ...
2
votes
1answer
121 views
The maximum in the Poisson problem on the cube with constant source
Question:
Let us consider the Poisson problem on the square with constant source $1$
$$
\begin{cases}
- \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\
u &= 0, \qquad \text{ on } \partial ...
1
vote
1answer
277 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 ...
3
votes
0answers
198 views
well-posedness of the transport equation
I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
5
votes
0answers
161 views
Uniform bound on the eigenfunctions of the Laplacian
Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...
5
votes
0answers
317 views
Lax Pairs for Linear PDEs
I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has ...
1
vote
1answer
162 views
if $\Pi_1$ and $\Pi_2$ be elliptic planes then $\Pi_1 \oplus \Pi_2 $ is still elliptic?
Let $\Omega \in\Lambda^{4}\big(V^{ \star }\big)$ be volume form. Define symplectic bilinear form
$q: \Pi \oplus \Pi \rightarrow R $
$\big( \alpha ,\beta \big) \longrightarrow \alpha ...
1
vote
1answer
146 views
Under which conditions Jacobi PDE system can be represented to symplectic monge Ampere equation?
We know we can reduce Symplectic monge ampere equations to Jacobi PDE system with some compatibility condition. I want to see when the vise versa is correct? and is there any theorem for it.
Here ...
5
votes
1answer
380 views
Regularity of the Maxwell equations
As is well-known, the Maxwell equations can be phrased vectorially as,
\begin{align}
\nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\
\nabla \cdot \mathbf ...
1
vote
1answer
149 views
A Cauchy problem for an iterated Euler-Poisson-Darboux eqaution
Good morning,
I'm interested in solving a Cauchy problem for the iterated singular EPD.
Well, Weinstein (On a class of PDEs of even order, 1955) showed how the decomposition formula leads to the ...
3
votes
1answer
256 views
Upper bounds for the solution of an elliptic PDE depending on a parameter.
Suppose I have the following PDE on $[0,1]^n$
$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$
with periodic boundary conditions and $\int f(x) dx =0$ . ...
0
votes
1answer
238 views
LINEAR Parabolic equations. Smooth dependence from initial data
I am looking for results that show smooth dependence of a solution to a parabolic equation, from the initial data.
More specifically I have the following problem:
CONSIDER spaces $P:=\mathbb{R}^k$ ...
5
votes
1answer
790 views
Short time existence on nonlinear parabolic PDE
I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact?
in which book we have this fact, the number of ...
0
votes
0answers
354 views
What is the Schauder estimate on usual Hölder space for parabolic type equations
What is the Schauder estimate on usual Holder space for parabolic type equations ?
3
votes
0answers
111 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 ...
2
votes
1answer
144 views
Differential finiteness of power series with hypergeometric coefficients
Let $f$ be a formal (or convergent near the origin, as you wish) power series with variables $x_1$, ... , $x_n$ and complex coefficients.
$$ f = \sum_I f_I \mathbf x^I $$
This power series is said to ...
1
vote
1answer
261 views
Linear inhomogeneous PDE
I am trying to understand the behavior of the following linear PDE:
$$\partial_t u(x,t) = \partial_{xx} u(x,t) + f(x) u(x,t)$$
where I set $f(x)=\lambda e^{-x^2} -1$ and with:
IC : $u(x,0)= 1$ on ...
8
votes
3answers
2k views
Physical Interpretation of Robin Boundary Conditions
In a (bounded) domain $\Omega \subset \mathbb{R}^n$, if we're studying the Laplace equation or heat equation or such PDE's we can impose the Dirichlet
$u|_{\partial\Omega} \equiv 0$,
Neumann
$D_{\nu} ...
4
votes
2answers
200 views
hodographic transformation
Let $\phi(x,t)$ be smooth function.
Let $\zeta= x-t$ and $\eta= x+t$. $u=\phi_\zeta$, $v= \phi_\eta$.
Let $u$, $v$ satisfies following equations:
1-
$$u_\eta- v_\zeta= 0$$
...
0
votes
0answers
130 views
coercivity vs boundedness of operator
hi,
the definition of coercivity and boundedness of a linear operator L between two B spaces looks similar:
$\|Lx\|\ge M _ 1\|x\|$ and $||Lx||\le M _ 2\|x\|$ for some constants $M _ 1$ and $M _ 2$. ...
1
vote
0answers
164 views
Is this Stefan-type problem an open problem?
I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary ...
3
votes
1answer
387 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:
...
0
votes
1answer
481 views
Hopf Boundary Point Lemma
The Hopf Boundary Point Lemma
http://en.wikipedia.org/wiki/Hopf_lemma
is a result for the unit normal vector field and the normal derivative.
Is it true if one considers arbitrary directional ...
6
votes
0answers
285 views
Regularity of solutions to a linear degenerate parabolic pde
I've encountered the following problem which is causing me some trouble :
Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function ...
3
votes
0answers
361 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
vote
2answers
467 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 $v$ ...
8
votes
8answers
2k views
Textbooks for PDE between Strauss and Folland
Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...
