Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

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2
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119 views

Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
0
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0answers
44 views

Why is it impossible to reduce a linear PDE of the second order in more than two independent variables to canonical form globally

It is known that in the case of more than two independent variables, it is usually not possible (especially in the case of PDE with the variable coefficients) to reduce a linear partial differential ...
2
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0answers
91 views

Algebraic methods in pde [duplicate]

I'm finding myself with a linear, very symetric system of first orders pde with polynomial coefficients. Wandering on the web, i learnt there is some nice alebraic way to deal with it involving ...
2
votes
1answer
101 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
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0answers
45 views

Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$. Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
0
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0answers
86 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
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1answer
190 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
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1answer
141 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
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0answers
71 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
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0answers
107 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
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0answers
68 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
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1answer
97 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
170 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
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1answer
195 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
266 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
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2answers
95 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
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2answers
112 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
57 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
96 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
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1answer
124 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
240 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
215 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
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1answer
154 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
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1answer
263 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
89 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 $H^1(U)$?
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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
123 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
280 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
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0answers
203 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
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0answers
163 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
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0answers
318 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
395 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
152 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
257 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
246 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
805 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 ...
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0answers
362 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
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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
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1answer
269 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- $$u_\eta- v_\zeta= 0$$ ...
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0answers
133 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 $M _ 2$. ...
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0answers
166 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
411 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
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1answer
496 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
293 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
362 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 ...